Three evolution problems modeling the interaction between acoustic waves and non-locally reacting surfaces

Abstract

The paper deals with three evolution problems arising in the physical modeling of small amplitude acoustic phenomena occurring in a fluid, bounded by a surface of extended reaction. The first one is the widely studied wave equation with acoustic boundary conditions, but its derivation from the physical model is mathematically not fully satisfactory. The other two models studied in the paper, in the Lagrangian and Eulerian settings, are physically transparent. In the paper the first model is derived from the other two in a rigorous way, also for solutions merely belonging to the natural energy spaces.

1 Introduction and main results

1.1 Presentation of the problem and literature overview

We deal with three evolution problems arising in the physical modeling of small amplitude acoustic phenomena occurring in a fluid, bounded by a surface of extended reaction. The problems are posed in a bounded and simply connected domain $\Omega$ of $\mathbb {R}^3$, with boundary $\Gamma =\partial \Omega$ of class $C^{r,1}$ (see [19]), where $r=2,3,\ldots ,\infty$. Moreover, $\Gamma =\Gamma _0\cup \Gamma _1$, $\overline{\Gamma _0}\cap \overline{\Gamma _1}=\emptyset$ and $\Gamma _1$ is nonempty. These properties of $\Omega$, $\Gamma _0$ and $\Gamma _1$ will be assumed throughout the paper without further reference.

The first problem we shall consider, to which in the sequel we shall refer as to the Eulerian model, is the boundary-value problem

$$\begin{aligned} (\mathcal {E})\qquad {\left\{ \begin{array}{ll} p_t+B{{\,\textrm{div}\,}}\textbf{v}=0 \qquad &{}\text {in } \mathbb {R}\times \Omega ,\\ \rho _0\textbf{v}_t=-\nabla p\qquad &{}\text {in } \mathbb {R}\times \Omega ,\\ {{\,\textrm{curl}\,}}\textbf{v}=0\qquad &{}\text {in } \mathbb {R}\times \Omega ,\\ \mu v_{tt}- {{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v)+\delta v_t+\kappa v+p =0\qquad &{}\text {on } \mathbb {R}\times \Gamma _1,\\ \textbf{v}\cdot {\varvec{\nu }} =0 \quad \text {on }\mathbb {R}\times \Gamma _0,\qquad \textbf{v}\cdot {\varvec{\nu }} =-v_t\qquad &{}\text {on } \mathbb {R}\times \Gamma _1, \end{array}\right. } \end{aligned}$$

where $p=p(t,x)$, $\textbf{v}=\textbf{v}(t,x)$, $v=v(t,y)$, $t\in \mathbb {R}$, $x\in \Omega$, $y\in \Gamma _1$, p and v are taking complex¹ values while $\textbf{v}$ takes values in $\mathbb {C}^3$. The reader should remark the difference between the notations $\textbf{v}$ and v. In problem ($\mathcal {E}$) we, respectively, denote by $\nabla$, ${{\,\textrm{div}\,}}$ and ${{\,\textrm{curl}\,}}$ the gradient, divergence and rotor operators, while ${{\,\mathrm{div_{_\Gamma }}\,}}$ and $\nabla _\Gamma$ stand for the Riemannian versions on $\Gamma$ of the first two operators. Moreover, we denote by ${\varvec{\nu }}$ the outward normal to $\Omega$, B and $\rho _0$ are fixed positive constants and $\mu ,\sigma ,\delta$, $\kappa$ are given real functions on $\Gamma _1$, satisfying the assumption

• (A) $\mu ,\sigma ,\delta ,\kappa \in W^{r-1,\infty }(\Gamma _1)$ with $\min _{\Gamma _1}\mu =\mu _0>0$, $\min _{\Gamma _1}\sigma =\sigma _0>0$,

where $W^{\infty ,\infty }(\Gamma _1)$ stands for $C^\infty (\Gamma _1)$ when $r=\infty$. Also assumption (A) will be kept without further reference throughout the paper.

In this paper, we shall also deal with the constrained Eulerian model, in short ($\mathcal {E}^c$). It is constituted by adding to ($\mathcal {E})$ the integral condition

$$\begin{aligned} \int _\Omega p(t,\cdot )=B\int _{\Gamma _1}v(t),\qquad \text {for all } t\in \mathbb {R}. \end{aligned}$$

(1.1)

The complete physical derivation of problems $(\mathcal {E})$ and $(\mathcal {E}^c)$ was given in [32, Chapter 7]. Here we just recall that p and $\textbf{v}$, respectively, stand for the acoustic excess pressure and the velocity field of a fluid contained in $\Omega$, while v stands for the normal displacement inside $\Omega$ of the moving part $\Gamma _1$ of the boundary. In  [32], a mathematical study of $({\mathcal {E}})$ and $({\mathcal {E}^c})$ was not performed. Moreover, as to the author’s knowledge, these problems do not possess a mathematical literature, although being strongly related to the third problem we shall study.

The second problem we shall consider, in the sequel called the Lagrangian model, is the boundary-value problem

$$\begin{aligned} (\mathcal {L})\qquad {\left\{ \begin{array}{ll} \rho _0\textbf{r}_{tt}-B\nabla {{\,\textrm{div}\,}}\textbf{r}=0\qquad &{}\text {in } \mathbb {R}\times \Omega ,\\ {{\,\textrm{curl}\,}}\textbf{r}=0\qquad &{}\text {in } \mathbb {R}\times \Omega ,\\ \mu v_{tt}- {{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v)+\delta v_t+\kappa v-B{{\,\textrm{div}\,}}\textbf{r}=0\qquad &{}\text {on } \mathbb {R}\times \Gamma _1,\\ \textbf{r}\cdot {\varvec{\nu }} =0 \quad \text {on }\mathbb {R}\times \Gamma _0,\qquad \textbf{r}\cdot {\varvec{\nu }} =-v\qquad &{}\text {on } \mathbb {R}\times \Gamma _1, \end{array}\right. } \end{aligned}$$

where we keep the notation and assumptions of the previous problem and $\textbf{r}=\textbf{r}(t,x)$, $t\in \mathbb {R}$, $x\in \Omega$, takes values in $\mathbb {C}^3$. Since ${{\,\textrm{curl}\,}}\textbf{r}=0$, equation $(\mathcal {L})_1$ could be equivalently written, in a distributional sense, as $\rho _0\textbf{r}_{tt}-B\Delta \textbf{r}=0$, where $\Delta \textbf{r}$ stands for the componentwise Laplacian of $\textbf{r}$. Also the complete physical derivation of problem ($\mathcal {L}$) was given in  [32, Chapter 7], although it was written in the alternative form

$$\begin{aligned} (\mathcal {L}')\qquad {\left\{ \begin{array}{ll} p+B{{\,\textrm{div}\,}}\textbf{r}=0 \qquad &{}\text {in } \mathbb {R}\times \Omega ,\\ \rho _0\textbf{r}_{tt}=-\nabla p\qquad &{}\text {in } \mathbb {R}\times \Omega ,\\ {{\,\textrm{curl}\,}}\textbf{r}=0\qquad &{}\text {in } \mathbb {R}\times \Omega ,\\ \mu v_{tt}- {{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v)+\delta v_t+\kappa v+p =0\qquad &{}\text {on } \mathbb {R}\times \Gamma _1,\\ \textbf{r}\cdot {\varvec{\nu }} =0 \quad \text {on }\mathbb {R}\times \Gamma _0,\qquad \textbf{r}\cdot {\varvec{\nu }} =-v\qquad &{}\text {on } \mathbb {R}\times \Gamma _1, \end{array}\right. } \end{aligned}$$

which is more similar to problem $(\mathcal {E})$ than ($\mathcal {L}$). Since in problem ($\mathcal {L}'$) one can use the first equation to eliminate p, the two formulations are completely equivalent. In them p and v have the same physical meaning as in problem $(\mathcal {E})$, even though a Lagrangian framework is considered, while $\textbf{r}$ stands for the displacement of fluid particles from a reference position. In the paper we shall only address problem ($\mathcal {L}$). Also this last problem does not possess, as to the author’s knowledge, a mathematical literature. It is strongly related, as we shall see, with the third problem we shall consider, to which in the sequel we shall refer as to the potential model. It is the boundary-value problem

$$\begin{aligned} (\mathcal {P})\qquad {\left\{ \begin{array}{ll} \rho _0u_{tt}-B\Delta u=0 \qquad &{}\text {in } \mathbb {R}\times \Omega ,\\ \mu v_{tt}- {{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v)+\delta v_t+\kappa v+\rho _0 u_t =0\qquad &{}\text {on } \mathbb {R}\times \Gamma _1,\\ \partial _{\varvec{\nu }} u=0 \quad \text {on }\mathbb {R}\times \Gamma _0,\qquad v_t =\partial _{\varvec{\nu }} u\qquad &{}\text {on } \mathbb {R}\times \Gamma _1, \end{array}\right. } \end{aligned}$$

where $u=u(t,x)$, $t\in \mathbb {R}$, $x\in \Omega$, $\Delta$ denotes the Laplacian operator, and we keep the notation and assumption introduced for $(\mathcal {E})$ and $(\mathcal {L})$. In the paper we shall also deal with the constrained potential model, in short ($\mathcal {P}^c$). It is constituted by adding to ($\mathcal {P})$ the integral condition

$$\begin{aligned} \rho _0\int _\Omega u_t(t)=B\int _{\Gamma _1}v(t),\qquad \text {for all } t\in \mathbb {R}. \end{aligned}$$

(1.2)

Unlike ($\mathcal {E}$) and ($\mathcal {L}$), problem ($\mathcal {P}$) possesses a wide mathematical literature, in which the wave equation ($\mathcal {P}$)$_1$ is equivalently written as $u_{tt}-c^2\Delta u=0$, with $c^2=B/\rho _0$, and it is known as wave equation with acoustic boundary conditions. This type of boundary conditions has been introduced by Beale and Rosencrans, for bounded or external domains, in [5,6,7] when $\sigma \equiv 0$ and $\Gamma _0=\emptyset$. In this case the boundary $\Gamma =\Gamma _1$ is called, using the terminology of [30, pp. 256], locally reacting, since each point of it reacts like a harmonic oscillator.

The same model, in the case $\delta =0$, has been proposed in [42] when $\Omega$ is a strip in $\mathbb {R}^2$ or $\mathbb {R}^3$, and in [33] when $\Omega$ is the half-space in $\mathbb {R}^3$, to describe acoustic wave propagation in an ice-covered ocean. We refer the interested reader to [9] for a historical overview of these and related problems in Mathematical Physics.

After their introduction, acoustic boundary conditions for locally reacting surfaces have been the subject of several papers. See, for example, [2, 10, 11, 14, 17, 18, 20, 22,23,24,25,26,27, 29, 31]. When one dismisses the simplifying assumption that neighboring points do not interact, such surfaces, again using the terminology of [30, p. 266]), are called of extended reaction. We shall call those which react like a membrane non-locally reacting, since other types of reactions can be considered.

The simplest case in which $\sigma \equiv \sigma _0$, so the operator ${{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma )$ reduces, up to $\sigma _0$, to the Laplace–Beltrami operator $\Delta _\Gamma$, was briefly considered in [5, §6], then studied in [15, 16, 36,37,38] and in the recent paper [8]. In all of them the authors assume that $\Gamma _0\not =\emptyset$ and that the homogeneous Neumann boundary condition on $\Gamma _0$ is replaced by the (mathematically more attracting) homogeneous Dirichlet boundary condition. See also [3, 4] for somehow related problems.

Concerning the physical derivation of problem $({\mathcal {P}})$, two rough physical derivations of it were given in [32, Chapter 7], starting from the Eulerian and the Lagrangian models. Although being essentially correct, these derivations were not so clear from the mathematical point of view.

The first aim of the present paper is then to rigorously study the relations among problems $({\mathcal {E}})$, $({\mathcal {E}}^c)$, $({\mathcal {L}})$, $({\mathcal {P}})$ and $({\mathcal {P}}^c)$. To achieve such a result, it is mandatory, from a mathematical point of view, to recall a well-posedness result for $({\mathcal {P}})$ and to prove analogous results for the other problems listed above. This one is the second aim of the paper. As a byproduct of the analysis, we shall also get optimal regularity results for problems $({\mathcal {E}})$, $({\mathcal {E}}^c)$ and $({\mathcal {L}})$. To illustrate these results in the clearest possible way, we start from well-posedness.

In the already quoted paper [32], the authors widely studied, under more general assumptions and with $c^2=B/\rho _0$, the initial-value problem associated to $(\mathcal {P})$, i.e., the initial- and boundary-value problem

$$\begin{aligned} (\mathcal {P}_0)\qquad {\left\{ \begin{array}{ll} \rho _0u_{tt}-B\Delta u=0 \qquad &{}\text {in } \mathbb {R}\times \Omega ,\\ \mu v_{tt}- {{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v)+\delta v_t+\kappa v+\rho _0 u_t =0\qquad &{}\text {on } \mathbb {R}\times \Gamma _1,\\ \partial _{\varvec{\nu }} u=0 \quad \text {on }\mathbb {R}\times \Gamma _0,\qquad v_t =\partial _{\varvec{\nu }} u\qquad &{}\text {on }\mathbb {R}\times \Gamma _1,\\ u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x) &{} \text {in }\Omega ,\\ v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x) &{} \text {on }\Gamma _1. \end{array}\right. } \end{aligned}$$

The starting point of the present paper, see Sect. 3 below, will be to recall several results from [32], with some simplifications essentially due to the higher regularity of $\Gamma$ assumed here. In this section we only need to recall the main well-posedness result in the quoted paper. To state it we introduce, for $n\in \mathbb {N}$, $n\le r$, the Hilbert space

$$\begin{aligned} \mathcal {H}_{\mathcal {P}}^n=H^n(\Omega )\times H^n(\Gamma _1)\times H^{n-1}(\Omega )\times H^{n-1}(\Gamma _1), \end{aligned}$$

(1.3)

using the standard notation $H^0=L^2$ which will be maintained throughout the paper. We also introduce, for $n\in \mathbb {N}\cup \{\infty \}$, $n\le r$, the Fréchet space

$$\begin{aligned} X_{\mathcal {P}}^n=\bigcap _{i=0}^n C^i(\mathbb {R};H^{n-i}(\Omega )\times H^{n-i}(\Gamma _1)). \end{aligned}$$

(1.4)

The phase space for problem $({\mathcal {P}})$ is then $\mathcal {H}_{\mathcal {P}}^1$. We shall consider weak solutions of $({\mathcal {P}})$ and $({\mathcal {P}}^c)$, that is couples $(u,v)\in X_{\mathcal {P}}^1$ which satisfy $({\mathcal {P}})$ in a suitable distributional sense, see Definition 3.1 below. Moreover, we shall mean initial conditions in problem $({\mathcal {P}}_0)$ in the sense of the space $X_{\mathcal {P}}^1$.

Theorem 1.1

(Well-posedness for $({\mathcal {P}}_0)$) For all data $U_{0{\mathcal {P}}}=(u_0,v_0,u_1,v_1)\in \mathcal {H}_{\mathcal {P}}^1$ problem $({\mathcal {P}}_0)$ has a unique weak solution $(u,v)\in X_{\mathcal {P}}^1$, continuously depending on $U_{0{\mathcal {P}}}$ in the topologies of the respective spaces. Moreover, $(u,v)\in X_{\mathcal {P}}^2$ if and only if $U_{0{\mathcal {P}}}\in \mathcal {H}_{\mathcal {P}}^2$, $\partial _{\varvec{\nu }}u_0=0$ on $\Gamma _0$ and $\partial _{\varvec{\nu }}u_0=v_1$ on $\Gamma _1$, such data being dense in $\mathcal {H}_{\mathcal {P}}^1$. In this case u and v satisfy $({\mathcal {P}})_1$ a.e. in $\mathbb {R}\times \Omega$, $({\mathcal {P}})_2$-$({\mathcal {P}})_3$ a.e. on $\mathbb {R}\times \Gamma _1$ and (u, v) continuously depends on data in the topologies of $\mathcal {H}_{\mathcal {P}}^2$ and $X_{\mathcal {P}}^2$. Finally, for all $U_{0{\mathcal {P}}}\in \mathcal {H}_{\mathcal {P}}^1$ and $s,t\in \mathbb {R}$, (u, v) satisfies the energy identity²

$$\begin{aligned} \int _\Omega \!\!\left( \frac{\rho _0}{2}|\nabla u|^2\!+ \!\frac{\rho _0^2}{2B} |u_t|^2\right) +\frac{1}{2} \int _{\Gamma _1}\!\!\!\! \left( \sigma |\nabla _\Gamma v|_\Gamma ^2+\mu |v_t|^2 +\kappa |v|^2\right) \Bigg |_s^t\! =\!-\!\int _s^t\!\!\int _{\Gamma _1}\!\!\delta |v_t|^2. \end{aligned}$$

(1.5)

1.2 Main results I: well-posedness

We start from problem $({\mathcal {P}}^c)$. The initial-value problem associated to it is obtained by adding the integral condition (1.2) to $({\mathcal {P}}_0)$. In the sequel this problem will be denoted by $({\mathcal {P}}^c_0)$.

Setting, for $n\in \mathbb {N}$, $n\le r$, the closed subspaces (respectively) of $\mathcal {H}_{\mathcal {P}}^n$ and $X_{\mathcal {P}}^n$,

$$\begin{aligned} \mathcal {H}_{\mathcal {P}^c}^n&=\left\{ (u,v,w,z)\in \mathcal {H}_{\mathcal {P}}^n: \rho _0\int _\Omega w=B\int _{\Gamma _1}v\right\} ,\nonumber \\ X_{\mathcal {P}^c}^n&=\{(u,v)\in X_{\mathcal {P}}^n: (1.2) \text { holds}\}, \end{aligned}$$

(1.6)

weak solutions of $({\mathcal {P}}^c)$ will be weak solutions of $({\mathcal {P}})$ belonging to $X_{\mathcal {P}^c}^1$, and the phase space associated to $({\mathcal {P}}^c)$ is simply $\mathcal {H}_{\mathcal {P}^c}^1$.

As an easy application of Theorem 1.1, in Sect. 3.4 we shall prove the following result.

Corollary 1.2

(Well-posedness for $({\mathcal {P}}_0^c)$) The statement of Theorem 1.1 continues to hold when one replaces $({\mathcal {P}}_0)$, $\mathcal {H}_{\mathcal {P}}^n$ and $X_{\mathcal {P}}^n$ with $({\mathcal {P}}_0^c)$, $\mathcal {H}_{\mathcal {P}^c}^n$ and $X_{\mathcal {P}^c}^n$.

We now turn to the Lagrangian model. The initial-value problem associated to it is the problem

$$\begin{aligned} (\mathcal {L}_0)\qquad {\left\{ \begin{array}{ll} \rho _0\textbf{r}_{tt}-B\nabla {{\,\textrm{div}\,}}\textbf{r}=0\qquad &{}\text {in }\mathbb {R}\times \Omega ,\\ {{\,\textrm{curl}\,}}\textbf{r}=0\qquad &{}\text {in }\mathbb {R}\times \Omega ,\\ \mu v_{tt}- {{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v)+\delta v_t+\kappa v-B{{\,\textrm{div}\,}}\textbf{r}=0\qquad &{}\text {on }\mathbb {R}\times \Gamma _1,\\ \textbf{r}\cdot {\varvec{\nu }} =0 \quad \text {on }\mathbb {R}\times \Gamma _0,\qquad \textbf{r}\cdot {\varvec{\nu }} =-v\qquad &{}\text {on }\mathbb {R}\times \Gamma _1,\\ \textbf{r}(0,x)=\textbf{r}_0(x),\quad \textbf{r}_t(0,x)=\textbf{r}_1(x) &{} \text {in }\Omega ,\\ v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x) &{} \text {on }\Gamma _1. \end{array}\right. } \end{aligned}$$

Denoting for simplicity $H^n(\Omega )^3=H^n(\Omega ;\mathbb {C}^3)$, we introduce, for $n\in \mathbb {N}_0$, its closed subspace

$$\begin{aligned} H^n_{{{\,\textrm{curl}\,}}0}(\Omega )=\{\textbf{r}\in H^n(\Omega )^3: {{\,\textrm{curl}\,}}\textbf{r}=0\}, \end{aligned}$$

(1.7)

where, at least when $n=0$, ${{\,\textrm{curl}\,}}\textbf{r}$ is taken in the sense of distributions. We also set, for $n\in \mathbb {N}_0$, $n\le r$, the Hilbert space $H^n_{\mathcal {L}}=H^n_{{{\,\textrm{curl}\,}}0}(\Omega )\times H^n(\Gamma _1)$ and, when $n\in \mathbb {N}$, $n\le r$, its closed subspace

$$\begin{aligned} \mathbb {H}^n_{\mathcal {L}}=\{(\textbf{r},v)\in H^n_{\mathcal {L}}: \quad \textbf{r}\cdot \varvec{\nu }=-v \quad \hbox { on}\ \Gamma _1,\quad \textbf{r}\cdot \varvec{\nu }=0\quad \hbox { on}\ \Gamma _0\}, \end{aligned}$$

(1.8)

where $\textbf{r}$ is taken in the trace sense on $\Gamma$. For the same n’s, we also set the Hilbert space

$$\begin{aligned} \mathcal {H}^n_{\mathcal {L}}=\{(\textbf{r},v,\textbf{s},z)\!\in \! H^n_{{{\,\textrm{curl}\,}}0}(\Omega )\!\times \! H^n(\Gamma _1)\!\times \!H^{n-1}_{{{\,\textrm{curl}\,}}0}(\Omega )\!\times \! H^{n-1}(\Gamma _1): (\textbf{r},v)\in \mathbb {H}^n_{\mathcal {L}}\}, \end{aligned}$$

(1.9)

endowed with the norm inherited from the product. Moreover, for $n\in \mathbb {N}\cup \{\infty \}$, $n\le r$, we set the Fréchet space

$$\begin{aligned} X_{\mathcal {L}}^n=\bigcap _{i=0}^{n-1} C^i(\mathbb {R};\mathbb {H}_{\mathcal {L}}^{n-i})\cap C^n(\mathbb {R}; H_{\mathcal {L}}^0). \end{aligned}$$

(1.10)

The phase space for $({\mathcal {L}})$ is $\mathcal {H}_{\mathcal {L}}^1$. In Sect. 4.4 we shall define weak solutions for $({\mathcal {L}})$ as couples $(\textbf{r},v)\in X_{\mathcal {L}}^1$ satisfying $({\mathcal {L}})$ is a suitable distributional sense, see Definition 4.9. We shall mean initial conditions in $({\mathcal {L}}_0)$ in the sense of the space $X_{\mathcal {L}}^1$. The definition of weak solutions, together with their uniqueness, see Theorem 4.16 below, is an essential outcome of the paper. In Sect. 4.5 we shall prove the following result.

Theorem 1.3

(Well-posedness for $({\mathcal {L}}_0)$) For all data $U_{0{\mathcal {L}}}=(\textbf{r}_0,v_0,\textbf{r}_1,v_1)\in \mathcal {H}_{\mathcal {L}}^1$ problem $({\mathcal {L}}_0)$ has a unique weak solution $(\textbf{r},v)\in X_{\mathcal {L}}^1$, continuously depending on $U_{0{\mathcal {L}}}$ in the topologies of the respective spaces.

Moreover, $(\textbf{r},v)\in X_{\mathcal {L}}^2$ if and only if $U_{0{\mathcal {L}}}\in \mathcal {H}_{\mathcal {L}}^2$ and $(\textbf{r}_1,v_1)\in \mathbb {H}_{\mathcal {L}}^1$, such data being dense in $\mathcal {H}_{\mathcal {L}}^1$. In this case $\textbf{r}$ and v satisfy $({\mathcal {L}})_1$–$({\mathcal {L}})_2$ a.e. in $\mathbb {R}\times \Omega$, $({\mathcal {L}})_3$–$({\mathcal {L}})_5$ a.e. on $\mathbb {R}\times \Gamma$ and $(\textbf{r},v)$ continuously depends on data in the topologies of $\mathcal {H}_{\mathcal {L}}^2$ and $X_{\mathcal {L}}^2$.

Finally, for all $U_{0{\mathcal {L}}}\in \mathcal {H}_{\mathcal {L}}^1$ and $s,t\in \mathbb {R}$, $(\textbf{r},v)$ satisfies the energy identity

$$\begin{aligned} \int _\Omega \!\!\!\left( \tfrac{\rho _0}{2}|\textbf{r}_t|^2+ \tfrac{B}{2} |{{\,\textrm{div}\,}}\textbf{r}|^2\right) +\tfrac{1}{2} \int _{\Gamma _1}\!\!\!\left( \sigma |\nabla _\Gamma v|_\Gamma ^2+\mu |v_t|^2 +\kappa |v|^2\right) \Bigg |_s^t\!\! =-\int _s^t\!\!\int _{\Gamma _1}\!\!\delta |v_t|^2. \end{aligned}$$

(1.11)

Our second main result concerns the initial-value problems associated to $({\mathcal {E}})$ and $({\mathcal {E}^c})$. The first one is the problem

$$\begin{aligned} (\mathcal {E}_0)\qquad {\left\{ \begin{array}{ll} p_t+B{{\,\textrm{div}\,}}\textbf{v}=0 \qquad &{}\text {in }\mathbb {R}\times \Omega ,\\ \rho _0\textbf{v}_t=-\nabla p\qquad &{}\text {in }\mathbb {R}\times \Omega ,\\ {{\,\textrm{curl}\,}}\textbf{v}=0\qquad &{}\text {in }\mathbb {R}\times \Omega ,\\ \mu v_{tt}- {{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v)+\delta v_t+\kappa v+p =0\qquad &{}\text {on }\mathbb {R}\times \Gamma _1,\\ \textbf{v}\cdot {\varvec{\nu }} =0 \quad \text {on }\mathbb {R}\times \Gamma _0,\qquad \textbf{v}\cdot {\varvec{\nu }} =-v_t\qquad &{}\text {on }\mathbb {R}\times \Gamma _1,\\ p(0,x)=p_0(x),\quad \textbf{v}(0,x)=\textbf{v}_0(x) &{} \text {in }\Omega ,\\ v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x) &{} \text {on }\Gamma _1. \end{array}\right. } \end{aligned}$$

The second one is simply the problem obtained by adding the integral condition (1.1) to $({\mathcal {E}}_0)$. In the sequel we shall refer to this second problem as to problem $({\mathcal {E}}^c_0)$. To deal with $({\mathcal {E}}_0)$ and $({\mathcal {E}}^c_0)$, we set, for $n\in \mathbb {N}$, $n\le r$, the Hilbert spaces

$$\begin{aligned}&\mathcal {H}_{\mathcal {E}}^n=H^{n-1}(\Omega )\!\times \! H^{n-1}_{{{\,\textrm{curl}\,}}0}(\Omega )\!\times \! H^n(\Gamma _1)\!\times \!H^{n-1}(\Gamma _1),\nonumber \\&\mathcal {H}_{\mathcal {E}^c}^n=\left\{ (p,\textbf{v},v,z)\in \mathcal {H}_{\mathcal {E}}^n: \int _\Omega p=B\int _{\Gamma _1}v\right\} . \end{aligned}$$

(1.12)

We also set, for $n\in \mathbb {N}\cup \{\infty \}$, $n\le r$, the Fréchet spaces

$$\begin{aligned} X_{\mathcal {E}}^n\!=\!\bigcap _{i=0}^{n-1}\! C^i(\mathbb {R};H^{n-1-i}(\Omega ))\!\times \! \bigcap _{i=0}^{n-1} \!C^i(\mathbb {R};H^{n-1-i}_{{{\,\textrm{curl}\,}}0}(\Omega ))\!\times \! \bigcap _{i=0}^n \!C^i(\mathbb {R};H^{n-i}(\Gamma _1)) \end{aligned}$$

and $X_{\mathcal {E}^c}^n=\{(p,\textbf{v},v)\in X_{\mathcal {E}}^n: (1.1) \text { holds}\}.$ The phase spaces for $({\mathcal {E}})$ and $({\mathcal {E}^c})$ are $\mathcal {H}_{\mathcal {E}}^1$ and $\mathcal {H}_{\mathcal {E}^c}^1$. In Sect. 5.4 we shall define weak solutions of $({\mathcal {E}})$ (respectively of $({\mathcal {E}^c})$) as triples $(p,\textbf{v},v)$ in $X_{\mathcal {E}}^1$ (in $X_{\mathcal {E}^c}^1$) satisfying $({\mathcal {E}})$ is a suitable distributional sense, see Definition 5.5. We shall mean initial conditions in $({\mathcal {E}}_0)$ and $({\mathcal {E}}^c_0)$ in the sense of the space $X_{\mathcal {E}}^1$. The definition of weak solutions, together with their uniqueness, see Theorem 5.9 below, is an essential outcome of the paper. In Sect. 5.5 we shall then prove the following result.

Theorem 1.4

(Well-posedness for $({\mathcal {E}}_0)$ and $({\mathcal {E}}^c_0)$) For all data $U_{0{\mathcal {E}}}\in \mathcal {H}_{\mathcal {E}}^1$, $U_{0{\mathcal {E}}}=(p_0,\textbf{v}_0,v_0,v_1)$, problem $({\mathcal {E}}_0)$ has a unique weak solution $(p,\textbf{v},v)\in X_{\mathcal {E}}^1$, continuously depending on $U_{0{\mathcal {E}}}$ in the topologies of the respective spaces. Moreover, $(p,\textbf{v},v)\in X_{\mathcal {E}}^2$ if and only if $U_{0{\mathcal {E}}}\in \mathcal {H}_{\mathcal {E}}^2$ and $(\textbf{v}_0,v_1)\in \mathbb {H}_{\mathcal {L}}^1$, such data being dense in $\mathcal {H}_{\mathcal {E}}^1$. In this case p, $\textbf{v}$ and v satisfy $({\mathcal {E}})_1$–$({\mathcal {E}})_3$ a.e. in $\mathbb {R}\times \Omega$, $({\mathcal {E}})_4$–$({\mathcal {E}})_6$ a.e. on $\mathbb {R}\times \Gamma$ and $(p,\textbf{v},v)$ continuously depends on data in the topologies of $\mathcal {H}_{\mathcal {E}}^2$ and $X_{\mathcal {E}}^2$. For all $U_{0{\mathcal {E}}}\in \mathcal {H}_{\mathcal {E}}^1$ and $s,t\in \mathbb {R}$, $(p,\textbf{v},v)$ satisfies the energy identity

$$\begin{aligned} \int _\Omega \!\!\left( \tfrac{\rho _0}{2}|\textbf{v}|^2+ \tfrac{1}{2B} |p|^2\right) +\tfrac{1}{2} \int _{\Gamma _1}\!\!\left( \sigma |\nabla _\Gamma v|_\Gamma ^2+\mu |v_t|^2 +\kappa |v|^2\right) \Bigg |_s^t =-\int _s^t\int _{\Gamma _1}\delta |v_t|^2. \end{aligned}$$

(1.13)

Finally, all previous assertions continue to hold when replacing $({\mathcal {E}}_0)$, $\mathcal {E}_{\mathcal {E}}^n$ and $X_{\mathcal {E}}^n$, $n=1,2$, with $({\mathcal {E}}^c_0)$, $\mathcal {E}_{\mathcal {E}^c}^n$ and $X_{\mathcal {E}^c}^n$.

1.3 Main results II: relations

Theorems 1.1–1.4 and Corollary 1.2 look like variants of a single result. This fact depends on the relations among all treated problems. To illustrate them, it is useful to disregard initial conditions and introduce, for $n\in \mathbb {N}\cup \{\infty \}$, $n\le r$, the Fréchet spaces

$$\begin{aligned} \mathcal {S}^n_{\mathcal {P}}&=\{(u,v)\in X_{\mathcal {P}}^n: (u,v)\text { is a weak solution of }({\mathcal {P}})\},\nonumber \\ \mathcal {S}^n_{\mathcal {E}}&=\{(p,\textbf{v},v)\in X_{\mathcal {E}}^n: (p,\textbf{v},v)\text { is a weak solution of }({\mathcal {E}})\},\nonumber \\ \mathcal {S}^n_{\mathcal {L}}&=\{(\textbf{r},v)\in X_{\mathcal {L}}^n: (\textbf{r},v)\text { is a weak solution of }({\mathcal {L}})\},\nonumber \\ \mathcal {S}^n_{\mathcal {P}^c}&=\mathcal {S}^n_{\mathcal {P}}\cap X_{\mathcal {P}^c}^1, \qquad \mathcal {S}^n_{\mathcal {E}^c}=\mathcal {S}^n_{\mathcal {E}}\cap X_{\mathcal {E}^c}^1, \end{aligned}$$

(1.14)

endowed with the topologies of the ambient spaces. The last two spaces clearly denote the spaces of weak solutions of $({\mathcal {P}^c})$ and $({\mathcal {E}^c})$. To relate solutions of $({\mathcal {P}})$ and $({\mathcal {P}^c})$ to solutions of other models, we have to consider u up to constant functions in $\mathbb {R}\times \Omega$. Denoting by $\mathbb {C}_{\mathbb {R}\times \Omega }$ the space they constitute, we thus have to consider solutions (u, v) up to $\mathbb {C}_{X_{\mathcal {P}}}=\mathbb {C}_{\mathbb {R}\times \Omega }\times \{0\}$, the elements of which are trivial solutions of $({\mathcal {P}^c})$ and hence of $({\mathcal {P}})$. We then introduce the standard quotient Fréchet spaces (see [34, Chapter I, p.31]) $\dot{\mathcal {S}}^n_{\mathcal {P}}:=\mathcal {S}^n_{\mathcal {P}}/\mathbb {C}_{X_{\mathcal {P}}}$ and $\dot{\mathcal {S}}^n_{\mathcal {P}^c}:=\mathcal {S}^n_{\mathcal {P}^c}/\mathbb {C}_{X_{\mathcal {P}}}$, so trivially $\dot{\mathcal {S}}^n_{\mathcal {P}^c}\subset \dot{\mathcal {S}}^n_{\mathcal {P}}$, which elements are of the form $(u,v)+\mathbb {C}_{X_{\mathcal {P}}}$ for (u, v) in the respective class.

Denoting by $\mathcal {L}(X;Y)$ the space of continuous linear operators between two Fréchet spaces X and Y, and $\mathcal {L}(X)=\mathcal {L}(X;X)$, we can state our third main result.

Theorem 1.5

(The relation between $({\mathcal {P}^c})$ and $({\mathcal {L}})$) For any weak solution (u, v) of $({\mathcal {P}^c})$ and any $t\in \mathbb {R}$, there is unique $\textbf{r}(t)\in H^1(\Omega )^3$ solving the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -B{{\,\textrm{div}\,}}\textbf{r}(t)=\rho _0u_t(t),\quad &{}\text {in }\Omega ,\\ \quad {{\,\textrm{curl}\,}}\textbf{r}(t)=0\quad &{}\text {in }\Omega ,\\ \quad \textbf{r}(t)\cdot \varvec{\nu }=-v(t)\quad &{}\text {on }\Gamma _1,\\ \quad \textbf{r}(t)\cdot \varvec{\nu }=0\quad &{}\text {on }\Gamma _0. \end{array}\right. } \end{aligned}$$

(1.15)

The couple $(\textbf{r},v)\in X_{\mathcal {L}}^1$ is a weak solution of $({\mathcal {L}})$ which also satisfies

$$\begin{aligned} \textbf{r}_t(t)=-\nabla u(t)\qquad \text {for all }t\in \mathbb {R}. \end{aligned}$$

(1.16)

Moreover, the map $(u,v)\mapsto (\textbf{r},v)$ defines a surjective operator $\Psi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}\in \mathcal {L}(\mathcal {S}^1_{\mathcal {P}^c};\mathcal {S}^1_{\mathcal {L}})$ such that $\text {Ker } \Psi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}= \mathbb {C}_{X_{\mathcal {P}}},$ so $\Psi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}$ subordinates a bijective isomorphism ${\dot{\Psi }}_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}\in \mathcal {L}(\dot{\mathcal {S}}^1_{\mathcal {P}^c};\mathcal {S}^1_{\mathcal {L}})$ given by ${\dot{\Psi }}_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}[(u,v)+\mathbb {C}_{X_{\mathcal {P}}}]=\Psi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}(u,v)$ for all $(u,v)\in \mathcal {S}_{\mathcal {P}^c}^1$.

Its inverse is the operator ${\dot{\Psi }}_{_{{\mathcal {L}}{\mathcal {P}^c}}}\in \mathcal {L}(\mathcal {S}^1_{\mathcal {L}};\dot{\mathcal {S}}^1_{\mathcal {P}^c})$, ${\dot{\Psi }}_{_{{\mathcal {L}}{\mathcal {P}^c}}}(\textbf{r},v)=(u,v)+\mathbb {C}_{X_{\mathcal {P}}}$ for all $(\textbf{r},v)\in \mathcal {S}_{\mathcal {L}}^1$, defined by setting u, up to a space-time constant, by

$$\begin{aligned} u(t)=u(0)-\tfrac{B}{\rho _0}\int _0^t {{\,\textrm{div}\,}}\textbf{r}(\tau )\,d\tau ,\quad \text {and}\quad -\nabla u(0)=\textbf{r}_t(0), \end{aligned}$$

(1.17)

so u also satisfies (1.16). Finally, when (u, v) and $(\textbf{r},v)$ are the weak solutions of $({\mathcal {P}}^c_0)$ and $({\mathcal {L}}_0)$ given by Corollary 1.2 and Theorem 1.3, we have

$$\begin{aligned} (\textbf{r},v)=\Psi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}(u,v)\qquad \Longleftrightarrow \qquad -\nabla u_0=\textbf{r}_1\,\,\text {and}\,\, -B{{\,\textrm{div}\,}}\textbf{r}_0=\rho _0u_1. \end{aligned}$$

(1.18)

Remark 1.6

Since, in the equivalent version $({\mathcal {L}}')$ of the Lagrangian model, one has $p=-B{{\,\textrm{div}\,}}\textbf{r}$, equations (1.15)–(1.16) show that the relation established in Theorem 1.5 fulfills the equations $-\nabla u=\textbf{r}_t$ and $p=\rho _0 u_t$. In this way we give a derivation of problem $({\mathcal {P}^c})$ mathematically clearer than the one given in [32]. Moreover, (1.17) makes explicit the construction of the velocity potential u.

We can now give our fourth main result.

Theorem 1.7

(The relations between $({\mathcal {P}})$ and $({\mathcal {E}})$, $({\mathcal {P}^c})$ and $({\mathcal {E}^c})$) For any weak solution (u, v) of $({\mathcal {P}})$ (of $({\mathcal {P}^c})$), the triple

$$\begin{aligned} (p,\textbf{v},v)=(\rho _0 u_t,-\nabla u,v) \end{aligned}$$

(1.19)

is a weak solution of $({\mathcal {E}})$ (respectively of $({\mathcal {E}^c})$). Moreover, the map $(u,v)\mapsto (p,\textbf{v},v)$ defines a surjective operator $\Psi _{_{{\mathcal {P}}{\mathcal {E}}}}\mathcal {L}(\mathcal {S}^1_{\mathcal {P}};\mathcal {S}^1_{\mathcal {E}})$, which restricts to a surjective operator $\Psi _{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}\mathcal {L}(\mathcal {S}^1_{\mathcal {P}^c};\mathcal {S}^1_{\mathcal {E}^c})$, and $\text {Ker } \Psi _{_{{\mathcal {P}}{\mathcal {E}}}}= \text {Ker } \Psi _{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}=\mathbb {C}_{X_{\mathcal {P}}}$. Hence, $\Psi _{_{{\mathcal {P}}{\mathcal {E}}}}$ and $\Psi _{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}$ subordinate bijective isomorphisms

(1.20)

The inverse of ${\dot{\Psi }}_{_{{\mathcal {P}}{\mathcal {E}}}}$ is the operator

$$\begin{aligned} {\dot{\Psi }}_{_{{\mathcal {E}}{\mathcal {P}}}}\in \mathcal {L}(\mathcal {S}^1_{\mathcal {E}};\dot{\mathcal {S}}^1_{\mathcal {P}}),\quad {\dot{\Psi }}_{_{{\mathcal {E}}{\mathcal {P}}}}(p,\textbf{v},v)=(u,v)+\mathbb {C}_{X_{\mathcal {P}}}\,\, \text {for all } (p,\textbf{v},v)\in \mathcal {S}_{\mathcal {E}}^1, \end{aligned}$$

(1.21)

defined by setting u, up to a space-time constant, by

$$\begin{aligned} u(t)=u(0)-\tfrac{1}{\rho _0}\int _0^t p(\tau )\,d\tau ,\quad \text {and}\quad -\nabla u(0)=\textbf{v}(0), \end{aligned}$$

(1.22)

so u also satisfies

$$\begin{aligned} -\nabla u(t)=\textbf{v}(t)\qquad \text {for all }t\in \mathbb {R}. \end{aligned}$$

(1.23)

The inverse of ${\dot{\Psi }}_{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}$ is the operator

$$\begin{aligned} {\dot{\Psi }}_{_{{\mathcal {E}^c}\!{\mathcal {P}^c}}}\in \mathcal {L}(\dot{\mathcal {S}}^1_{\mathcal {E}^c};\mathcal {S}^1_{\mathcal {P}^c}),\quad {\dot{\Psi }}_{_{{\mathcal {E}^c}\!{\mathcal {P}^c}}}=({\dot{\Psi }}_{_{{\mathcal {E}}{\mathcal {P}}}})_{|\dot{\mathcal {S}}_{\mathcal {E}^c}^1}. \end{aligned}$$

(1.24)

Finally, when (u, v) and $(p,\textbf{v},v)$ are the weak solutions of $({\mathcal {P}}_0)$ and $({\mathcal {E}}_0)$, or of $({\mathcal {P}}^c_0)$ and $({\mathcal {E}}^c_0)$, given by Theorem 1.1, Corollary 1.2 and Theorem 1.4, we have

$$\begin{aligned} (p,\textbf{v},v)=\Psi _{_{{\mathcal {P}}{\mathcal {E}}}}(u,v)\qquad \Longleftrightarrow \qquad -\nabla u_0=\textbf{v}_0\,\,\text {and}\,\, p_0=\rho _0u_1. \end{aligned}$$

(1.25)

Remark 1.8

Equations (1.22)–(1.23) show that the relation established in Theorem 1.7 fulfills the equations $-\nabla u=\textbf{v}$ and $\rho _0 u_t=p$. In this way we give a mathematical derivation of problems $({\mathcal {P}})$ and $({\mathcal {P}^c})$ which is clearer than the one given in [32]. They also make explicit the construction of the velocity potential u.

By simply combining Theorems 1.5 and 1.7, see Sect. 5.6 below, one gets the relation between the Lagrangian model and the constrained Eulerian one.

Corollary 1.9

(The relation between $({\mathcal {E}^c})$ and $({\mathcal {L}})$) For any weak solution $(p,\textbf{v},v)$ of $({\mathcal {E}^c})$ and any $t\in \mathbb {R}$, there is unique $\textbf{r}(t)\in H^1(\Omega )^3$ solving the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -B{{\,\textrm{div}\,}}\textbf{r}(t)=p(t),\quad &{}\text {in }\Omega ,\\ \quad {{\,\textrm{curl}\,}}\textbf{r}(t)=0\quad &{}\text {in }\Omega ,\\ \quad \textbf{r}(t)\cdot \varvec{\nu }=-v(t)\quad &{}\text {on }\Gamma _1,\\ \quad \textbf{r}(t)\cdot \varvec{\nu }=0\quad &{}\text {on }\Gamma _0. \end{array}\right. } \end{aligned}$$

(1.26)

The couple $(\textbf{r},v)\in X_{\mathcal {L}}^1$ is a weak solution of $({\mathcal {L}})$ which also satisfies

$$\begin{aligned} \textbf{r}_t(t)=\textbf{v}(t)\qquad \text {for all }t\in \mathbb {R}. \end{aligned}$$

(1.27)

Hence, $\textbf{r}$ can also be expressed as

$$\begin{aligned} \textbf{r}(t)=\textbf{r}(0)+\int _0^t \textbf{v}(\tau )\,d\tau ,\quad \text {where } \textbf{r}(0)\text { solves problem }(1.26) \text { for }t=0. \end{aligned}$$

(1.28)

Moreover, the map $(p,\textbf{v},v)\mapsto (\textbf{r},v)$ defines a bijective isomorphism

$$\begin{aligned} \Psi _{_{{\mathcal {E}^c}\!{\mathcal {L}}}}={\dot{\Psi }}_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}\cdot {\dot{\Psi }}_{_{{\mathcal {E}^c}\!{\mathcal {P}^c}}}\in \mathcal {L}(\mathcal {S}^1_{\mathcal {E}^c};\mathcal {S}^1_{\mathcal {L}}), \end{aligned}$$

(1.29)

which inverse $\Psi _{_{{\mathcal {L}}{\mathcal {E}^c}}}={\dot{\Psi }}_{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}\cdot {\dot{\Psi }}_{_{{\mathcal {L}}{\mathcal {P}^c}}}\in \mathcal {L}(\mathcal {S}^1_{\mathcal {L}};\mathcal {S}^1_{\mathcal {E}^c})$ is given by

$$\begin{aligned} \Psi _{_{{\mathcal {L}}{\mathcal {E}^c}}}(\textbf{r},v)=(p,\textbf{v},v)=(-B{{\,\textrm{div}\,}}\textbf{r},\textbf{r}_t,v). \end{aligned}$$

(1.30)

Finally, when $(\textbf{r},v)$ and $(p,\textbf{v},v)$, respectively, denote the solutions of the Cauchy problems $({\mathcal {L}}_0)$ and $({\mathcal {E}}_0^c)$ given by Theorems 1.3 and 1.4, we have

$$\begin{aligned} (\textbf{r},v)=\Psi _{_{{\mathcal {E}^c}\!{\mathcal {L}}}}(p,\textbf{v},v)\qquad \Longleftrightarrow \qquad -B{{\,\textrm{div}\,}}\textbf{r}_0=p_0\,\,\text {and}\,\, \textbf{r}_1=\textbf{v}_0. \end{aligned}$$

(1.31)

The relations between the isomorphisms obtained in Theorems 1.5–1.7 and in Corollary 1.9 are illustrated by the commutative diagrams

which also show that a Lagrangian counterpart of problems $({\mathcal {P}})$, $({\mathcal {E}})$ is missing. These relations may probably be best understood at an abstract level, i.e., by considering the groups of linear operators associated to the various problems treated, see Theorems 4.6, 5.2 and 5.12 below.

Organization of the paper. The organization of the paper is simple: in Sect. 2 we give all preliminaries needed in the paper, while Sects. 3, 4 and 5 are, respectively, devoted to the potential, the Lagrangian and the Eulerian models.

2 Notation and preliminaries

2.1 Notation

We shall denote ${\widetilde{\mathbb {N}}}=\mathbb {N}\cup \{\infty \}$. Borrowing a convention in Physics, for vectors in $\mathbb {C}^3$ and vector-valued functions we shall use boldface. For $\textbf{x}=(x_1,x_2,x_3)$, $\textbf{y}=(y_1,y_2,y_3)\in \mathbb {C}^3$ we shall denote $\textbf{x}\cdot \textbf{y}=\sum _{i=1}^3 x_iy_i$ and by $\overline{\textbf{x}}$ the vector conjugated to $\textbf{x}$.

We shall use the standard notation for functions spaces on $\Omega$, referring to [1]. As already done in formula (1.7), where $H^n(\Omega )^3=H^n(\Omega ;\mathbb {C}^3)$, to simplify the notation we shall systematically identify the $\mathbb {C}^3$-valued versions of all spaces above with the Cartesian cubes of the corresponding scalar spaces. Moreover, $\Vert \cdot \Vert _p$, $1\le p\le \infty$, will denote the norm in $L^p(\Omega )$ and in $L^p(\Omega )^3$, since no confusion will arise.

Moreover, for any Fréchet space X, we shall denote by $X'$ its dual, by $\langle \cdot ,\cdot \rangle _X$ the duality product. When X is a Banach space we shall use the standard notation for Bochner–Lebesgue and Sobolev spaces of X-valued functions.

2.2 Function spaces and operators on $\Gamma$

The assumption made on $\Omega$, $\Gamma _0$ and $\Gamma _1$ assures that $\Gamma$ inherits from $\mathbb {R}^3$ the structure of a Riemannian surface of class $C^r$, so in the sequel we shall use some notation of geometric nature, quite common when $r=\infty$, see [35], which can be easily extended to $r<\infty$, see for example [32] or [39, 40].

Moreover, since $\overline{\Gamma _0}\cap \overline{\Gamma _1}=\emptyset$, both $\Gamma _0$ and $\Gamma _1$ are relatively open on $\Gamma$. Hence, all geometrical concepts apply to them as well. To avoid repetition, in the sequel we shall denote by $\Gamma '$ any relatively open subset of $\Gamma$.

We shall denote by $(\cdot ,\cdot )_\Gamma$ the Riemannian metric inherited from $\mathbb {R}^3$ and uniquely extended to an Hermitian metric on the complexified tangent bundle $T(\Gamma ')$, and also the associated bundles metric on the complexified cotangent bundle $T^*(\Gamma ')$. By $|\cdot |_\Gamma ^2= (\cdot ,\cdot )_\Gamma$ we shall denote the associated bundle norms.

The standard surface elements $\omega$ associated to $(\cdot ,\cdot )_\Gamma$ are then the density of the Lebesgue surface measure on $\Gamma$, coinciding with the restriction to $\Gamma$ of the Hausdorff measure $\mathcal {H}^2$, i.e., $\omega =d\mathcal {H}^2$. In the sequel $\Gamma$ will be equipped without further comments with this measure, with the corresponding notions of a.e. equivalence, integrals and Lebesgue spaces $L^p(\Gamma ')$, $1\le p\le \infty$. For simplicity we shall denote $\Vert \cdot \Vert _{p,\Gamma '}=\Vert \cdot \Vert _{L^p(\Gamma ')}$. Moreover, the notation $d\mathcal {H}^2$ will be dropped from boundary integrals, and a.e. equivalence on $\mathbb {R}\times \Gamma$ will be referred to the Hausdorff measure $\mathcal {H}^3$ in $\mathbb {R}^4$.

Sobolev spaces on $\Gamma '$ are treated in many textbooks when $r=\infty$, see for example [21, 28]. The case $r<\infty$ is treated in [19] and, when $\Gamma$ is possibly non-compact, in [32]. Here we shall refer, for simplicity, to [19], and we shall use the standard notation. Moreover, mainly to simplify the notation, since $\overline{\Gamma _0}\cap \overline{\Gamma _1}=\emptyset$, by identifying the elements of $W^{s,q}(\Gamma _i)$, $i=0,1$, with their trivial extensions to $\Gamma$, we have the splitting

$$\begin{aligned} W^{s,q}(\Gamma )=W^{s,q}(\Gamma _0)\oplus W^{s,q}(\Gamma _1)\qquad \text {for }s\in \mathbb {R}, -r\le s\le r, 1\le q<\infty . \end{aligned}$$

(2.1)

We refer to [32] for details on the Riemannian gradient operator $\nabla _\Gamma$ and on the Riemannian divergence operator ${{\,\textrm{div}\,}}_\Gamma$. Here we just recall that one gets the operator

$$\begin{aligned} {{\,\textrm{div}\,}}_\Gamma (\sigma \nabla _\Gamma )\in \mathcal {L}\left( H^m(\Gamma _1),H^{m-2}(\Gamma _1)\right) \qquad \text {for } m\in \mathbb {N}, 2\le m\le r, \end{aligned}$$

(2.2)

and that, $\Gamma _1$ being compact, one gets that

$$\begin{aligned} \int _{\Gamma _1} -{{\,\textrm{div}\,}}_\Gamma (\sigma \nabla _\Gamma u)v=\int _{\Gamma _1}\sigma (\nabla _\Gamma u,\nabla _\Gamma {\overline{v}})_\Gamma \end{aligned}$$

(2.3)

for all $u\in H^2(\Gamma _1)$, $v\in H^1(\Gamma _1)$. Finally, in the sequel we shall use the well-known Trace Theorem, i.e., the existence of the trace operator ${{\,\textrm{Tr}\,}}\in \mathcal {L}\left( H^m(\Omega ),H^{m-1/2}(\Gamma )\right)$, for $m\in \mathbb {N}$, $m\le r$. We shall denote, as usual, ${{\,\textrm{Tr}\,}}u=u_{|\Gamma }$. By $u_{|\Gamma _0}$ and $u_{|\Gamma _1}$ we shall denote the restrictions of $u_{|\Gamma }$ to $\Gamma _0$ and $\Gamma _1$ and, when clear, we shall omit trace related subscripts.

2.3 Quotient spaces and projections

Denoting by $\mathbb {C}_\Omega$ the space of constant functions in $\Omega$, since $\Omega$ is bounded, we can introduce the quotient Hilbert and Fréchet spaces³${\dot{H}}^n(\Omega )=H^n(\Omega )/\mathbb {C}_\Omega$, $n\in \mathbb {N}$, and ${\dot{C}}^\infty ({\overline{\Omega }})=C^\infty ({\overline{\Omega }})/\mathbb {C}_\Omega$, endowed with the standard quotient norm and pseudo-norms (see [34]). We denote by $\sim$ the equivalence relation defined on $H^1(\Omega )$ as follows: $u\sim v$ if and only if $u-v\in \mathbb {C}_\Omega$. We denote by $u +\mathbb {C}_\Omega$ the equivalence class of u with respect to $\sim$. Consequently, denoting $\pi _0 u=u+\mathbb {C}_\Omega$, we have $\pi _0 \in \mathcal {L}\left( H^n(\Omega ); {\dot{H}}^n(\Omega )\right) \cap \mathcal {L}\left( C^\infty ({\overline{\Omega }}); {\dot{C}}^\infty ({\overline{\Omega }})\right)$ for all $n\in \mathbb {N}$. We make the reader aware that in the last formula and (when useful) in the sequel we shall implicitly restrict linear operators.

2.4 Basic notions of semigroup theory

We now recall, since in the sequel we shall repetitively use them, some basic facts in semigroup theory. Given any unbounded linear operator $A:D(A)\subset H\rightarrow H$ on the Hilbert space H, we shall consider strong (or classical) and generalized (or mild) solutions of the equation $U'+AU=0$ in H, and of the Cauchy problem

$$\begin{aligned} U'+AU=0\qquad \hbox { in}\ H, \qquad U(0)=U_0\in H, \end{aligned}$$

(2.4)

in the standard semigroup sense, see [13]. Moreover, when $-A:D(A)\subset H\rightarrow H$ is the generator of a strongly continuous semigroup on the Hilbert space H, trivially adapting  [13, Chapter II, pp. 124–125] to semigroups of arbitrary growth bound, one can inductively set, for $n\in \mathbb {N}_0$, the Hilbert space

$$\begin{aligned} D(A^0)=H,\qquad D(A^n)=\{U\in D(A^{n-1}): AU\in D(A^{n-1})\}, \end{aligned}$$

(2.5)

endowed with the graph norm and, for $n\in \mathbb {N}$, the part ${_n}A$ of A in $D(A^n)$, that is the unbounded operator

$$\begin{aligned} {_n}A: D({_n}A)\subset D(A^n)\rightarrow D(A^n), \quad D({_n}A)=D(A^{n+1}),\quad {_n}A= A_{|D(A^n)}. \end{aligned}$$

(2.6)

We also set, for $n\in \mathbb {N}$, the Fréchet space $Y^n=\bigcap _{i=0}^{n-1} C^i(\mathbb {R}; D(A^{n-1-i}))$. By standard semigroup theory, one then gets the following result.

Lemma 2.1

Let $-A:D(A)\subset H\rightarrow H$ be the generator of a strongly continuous group $\{T^1(t),t\in \mathbb {R}\}$ on the Hilbert space H. Then

• (i) for all $U_0\in H$ problem (2.4) has a unique generalized solution $U\in Y^1$ given by $U(t)=T^1(t)[U_0]$ for all $t\in \mathbb {R}$, continuously depending on $U_0$ in the topologies of the respective spaces;

• (ii) for all $n\in \mathbb {N}$ the operator ${_n}A$ generates on $D(A^n)$ the strongly continuous group $\{T^{n+1}(t),t\in \mathbb {R}\}$, given by $T^{n+1}(t)=T^1(t)_{|D(A^n)}$ for all $t\in \mathbb {R}$;

• (iii) the following three facts are equivalent: U is a strong solution of (2.4), $U_0\in D(A)$, $U\in Y^2$;

• (iv) for all $n\in \mathbb {N}$ we have $U\in Y^n$ if and only if $U_0\in D(A^{n-1})$ and, in this case, U continuously depends on $U_0$ in the topologies of the respective spaces.

3 The potential models

3.1 Functional spaces

In addition to the main phase spaces $\mathcal {H}_{\mathcal {P}}^n$ and $\mathcal {H}_{\mathcal {P}^c}^n$, introduced in § 1, we shall also use (see Sect. 2.3) for $n\in \mathbb {N}$, $n\le r$, the Hilbert spaces

$$\begin{aligned} \dot{\mathcal {H}}_{\mathcal {P}}^n&= {\dot{H}}^n(\Omega )\times H^n(\Gamma _1)\times H^{n-1}(\Omega )\times H^{n-1}(\Gamma _1),\\ \dot{\mathcal {H}}_{\mathcal {P}^c}^n&= \left\{ (u,v,w,z)\in \dot{\mathcal {H}}_{\mathcal {P}}^n: \rho _0\int _\Omega w=B\int _{\Gamma _1}v\right\} , \end{aligned}$$

and

$$\begin{aligned} H_{\mathcal {P}^c}^n=\left\{ (v,w)\in H^n(\Gamma _1)\times H^{n-1}(\Omega ): \rho _0\int _\Omega w=B\int _{\Gamma _1}v\right\} . \end{aligned}$$

(3.1)

When needed we shall use the trivial identification $\dot{\mathcal {H}}_{\mathcal {P}^c}^n={\dot{H}}^n(\Omega )\times H_{\mathcal {P}^c}^n\times H^{n-1}(\Gamma _1)$. We recall the compatibility conditions for problem $({\mathcal {P}}_0)$ with $c^2=B/\rho _0$ (see [32, Theorem 1.2.3]) given for $n\in \mathbb {N}$, $2\le n\le r$ and data $U_0=(u_0,v_0, u_1,v_1)\in \mathcal {H}_{\mathcal {P}}^n$, by the equations

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{\varvec{\nu }} \Delta ^iu_0=0,\qquad \text {on }\Gamma _0 \qquad \text { for }i=0,\ldots , \lfloor n/2 \rfloor -1,\\&\partial _{\varvec{\nu }} \Delta ^iu_1=0,\qquad \text {on } \Gamma _0 \qquad \text { for }i=0,\ldots , \lfloor (n-1)/2\rfloor -1,\text { when } n\ge 3,\\&\partial _{\varvec{\nu }} u_0=v_1,\qquad \,\,\,\, \text {on }\Gamma _1,\\&\mu \partial _{\varvec{\nu }} u_1={{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v_0)-\delta \partial _{\varvec{\nu }} u_0-\kappa v_0-\rho _0 u_1,\qquad \text {on }\Gamma _1, \quad \text { when }n\ge 3,\\&\begin{aligned} B\mu \partial _{\varvec{\nu }} \Delta ^i\!u_0\!=\!\rho _0[{{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma \partial _{\varvec{\nu }}\Delta ^{i-1}\!u_0)\! -\!\delta \partial _{\varvec{\nu }}\Delta ^{i-1}\!u_1\!-\!\kappa \partial _{\varvec{\nu }}\Delta ^{i-1}\!u_0\!-\! B\Delta ^i \!u_0]\\ \text {on }\Gamma _1 \quad \text { for }i=1,\ldots , \lfloor n/2 \rfloor -1, \quad \text { when }n\ge 4,\qquad \,\,\,&\end{aligned}\\&\begin{aligned} B\mu \partial _{\varvec{\nu }} \Delta ^i\!u_1\!=-\!B\delta \partial _{\varvec{\nu }}\Delta ^iu_0\!\!+\!\!\rho _0[{{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma \partial _{\varvec{\nu }}\Delta ^{i-1}\!u_1)\! \!-\!\kappa \partial _{\varvec{\nu }}\Delta ^{i-1}\!u_1\!-\! B\Delta ^i \!u_1]\\ \text {on }\Gamma _1 \quad \text { for }i=1,\ldots , \lfloor (n-1)/2 \rfloor -1, \quad \text { when }n\ge 5.\,\,\,&\end{aligned} \end{aligned}\right. \end{aligned}$$

(3.2)

In (3.2) and in the sequel $\lfloor \cdot \rfloor$ stands for the integer part. Since the equations (3.2) make sense also for data $U_0\in \dot{\mathcal {H}}_{\mathcal {P}}^n$, i.e., for $u_0\in {\dot{H}}^n(\Omega )$, we can set for $n\in \mathbb {N}$, $2\le n\le r$,

$$\begin{aligned} D^{n-1}_{\mathcal {P}}=\{U_0\in \mathcal {H}_{\mathcal {P}}^n: (3.2) \text { hold}\},\quad \text {and}\quad {\dot{D}}^{n-1}_{\mathcal {P}}=\{U_0\in \dot{\mathcal {H}}_{\mathcal {P}}^n: (3.2)\text { hold}\}, \end{aligned}$$

(3.3)

which by (2.2) are closed subspaces of $\mathcal {H}_{\mathcal {P}}^n$ and $\dot{\mathcal {H}}_{\mathcal {P}}^n$, together with their subspaces

$$\begin{aligned} D^{n-1}_{\mathcal {P}^c}=D^{n-1}_{\mathcal {P}}\cap \mathcal {H}_{\mathcal {P}^c}^1=D^{n-1}_{\mathcal {P}}\cap \mathcal {H}_{\mathcal {P}^c}^n,\qquad {\dot{D}}^{n-1}_{\mathcal {P}^c}={\dot{D}}^{n-1}_{\mathcal {P}}\cap \dot{\mathcal {H}}_{\mathcal {P}^c}^1={\dot{D}}^{n-1}_{\mathcal {P}}\cap \dot{\mathcal {H}}_{\mathcal {P}^c}^n. \end{aligned}$$

(3.4)

All of them will be equipped with the norms inherited from $\mathcal {H}_{\mathcal {P}}^n$ and $\dot{\mathcal {H}}_{\mathcal {P}}^n$. When $r=\infty$ we shall also use the product Fréchet spaces

$$\begin{aligned} \mathcal {H}_{\mathcal {P}}^\infty&= C^\infty ({\overline{\Omega }})\times C^\infty (\Gamma _1)\times C^\infty ({\overline{\Omega }})\times C^\infty (\Gamma _1),\nonumber \\ \dot{\mathcal {H}}_{\mathcal {P}}^\infty&= {\dot{C}}^\infty ({\overline{\Omega }})\times C^\infty (\Gamma _1)\times C^\infty ({\overline{\Omega }})\times C^\infty (\Gamma _1), \end{aligned}$$

(3.5)

and their closed subspaces

Using Morrey’s Theorem, one easily gets that

(3.6)

Introducing the one-dimensional subspace $\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}=\mathbb {C}_\Omega \times \{0\}\times \{0\}\times \{0\}$ of $\mathcal {H}_{\mathcal {P}}^1$, in the sequel we shall isometrically identify the quotient space $\mathcal {H}^1_{\mathcal {P}}/\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}$ with $\dot{\mathcal {H}}^1_{\mathcal {P}}$. The associated quotient (and hence surjective) map is

$$\begin{aligned} Q\in \mathcal {L}(\mathcal {H}^1_{\mathcal {P}}; \dot{\mathcal {H}}^1_{\mathcal {P}}),\quad Q(u,v,w,z)=(\pi _0 u,v,w,z)\quad \forall (u,v,w,z)\in \mathcal {H}^1_{\mathcal {P}}. \end{aligned}$$

(3.7)

Since $\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}\subset \mathcal {H}^n_{\mathcal {P}^c}\subset \mathcal {H}^n_{\mathcal {P}}$ for $n\in {\widetilde{\mathbb {N}}}$, $n\le r$ and $\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}\subset D^{m-1}_{\mathcal {P}^c}\subset D^{m-1}_{\mathcal {P}}$ for $m\in {\widetilde{\mathbb {N}}}$, $2\le m \le r$, and since one trivially has

$$\begin{aligned} Q\mathcal {H}^n_{\mathcal {P}}=\dot{\mathcal {H}}^n_{\mathcal {P}}, \quad Q\mathcal {H}^n_{\mathcal {P}^c}=\dot{\mathcal {H}}^n_{\mathcal {P}^c},\quad QD^{m-1}_{\mathcal {P}}={\dot{D}}^{m-1}_{\mathcal {P}}, \quad QD^{m-1}_{\mathcal {P}^c}={\dot{D}}^{m-1}_{\mathcal {P}^c}, \end{aligned}$$

(3.8)

we can also make the isometric identifications

$$\begin{aligned} \mathcal {H}^n_{\mathcal {P}}/\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}\!\!=\!\!\dot{\mathcal {H}}^n_{\mathcal {P}},\,\, \mathcal {H}^n_{\mathcal {P}^c}/\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}\!\!=\!\!\dot{\mathcal {H}}^n_{\mathcal {P}^c}\, \, D^{m-1}_{\mathcal {P}}/\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}\!\!=\!\!{\dot{D}}^{m-1}_{\mathcal {P}}\!\!\!,\,\,\, D^{m-1}_{\mathcal {P}^c}/\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}\!\!=\!\!{\dot{D}}^{m-1}_{\mathcal {P}^c}, \end{aligned}$$

(3.9)

so getting the associated quotient (and hence surjective) maps

$$\begin{aligned} Q\in \mathcal {L}(\mathcal {H}^n_{\mathcal {P}}; \dot{\mathcal {H}}^n_{\mathcal {P}})\cap \mathcal {L}(\mathcal {H}^n_{\mathcal {P}^c}; \dot{\mathcal {H}}^n_{\mathcal {P}^c})\cap \mathcal {L}(D^{m-1}_{\mathcal {P}}; {\dot{D}}^{m-1}_{\mathcal {P}})\cap \mathcal {L}(D^{m-1}_{\mathcal {P}^c}; {\dot{D}}^{m-1}_{\mathcal {P}^c}). \end{aligned}$$

(3.10)

All of them are restrictions of Q defined in (3.7).

In addition to the spaces $X^n_{\mathcal {P}}$ and $X^n_{\mathcal {P}^c}$ defined in (1.4) and (1.10), we also set, for $n\in {\widetilde{\mathbb {N}}}$, $n\le r$, the further Fréchet spaces

(3.11)

We remark that, by Morrey’s Theorem, when $r=\infty$ one easily gets that

$$\begin{aligned} X^\infty _{\mathcal {P}}=C^\infty (\mathbb {R}\times {\overline{\Omega }})\times C^\infty (\mathbb {R}\times \Gamma _1). \end{aligned}$$

The quotient map Q in (3.7) clearly induces the “pointwise quotient” operator

$$\begin{aligned} \mathcal {Q}\in \mathcal {L}(Y^1_{\mathcal {P}}; {\dot{Y}}^1_{\mathcal {P}}),\qquad (\mathcal {Q}U)t)=Q(U(t))\quad \text {for all }U\in Y^1_{\mathcal {P}}\text { and } t\in \mathbb {R}, \end{aligned}$$

(3.12)

explicitly given, for any $U=(u,v,w,z)\in Y^1_{\mathcal {P}}$ and $t\in \mathbb {R}$, by

$$\begin{aligned} \mathcal {Q}U(t)=( \pi _0 u(t),v(t),w(t),z(t)). \end{aligned}$$

(3.13)

The choice of distinguishing Q and $\mathcal {Q}$ is motivated by the fact that their kernels are different. Indeed, introducing the subspace $\mathbb {C}_{Y_{\mathcal {P}}}=\mathbb {C}_{\mathbb {R}\times \Omega }\times \{0\}\times \{0\}\times \{0\}$ of $Y^n_{\mathcal {P}}$ and $Y^n_{\mathcal {P}^c}$, for $n\in {\widetilde{\mathbb {N}}}$, $n\le \mathbb {R}$, by (3.13) one easily gets that $\ker \mathcal {Q}=C(\mathbb {R}; \mathbb {C}_{Y_{\mathcal {P}}})\simeq C(\mathbb {R})$, while $\ker Q=\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}\simeq \mathbb {C}$. By (3.10) the operator $\mathcal {Q}$ trivially restricts, for $n\in {\widetilde{\mathbb {N}}}$, $n\le r$, to

$$\begin{aligned} \mathcal {Q}\in \mathcal {L}(Y^n_{\mathcal {P}}; {\dot{Y}}^n_{\mathcal {P}})\cap \mathcal {L}(Y^n_{\mathcal {P}^c}; {\dot{Y}}^n_{\mathcal {P}^c}). \end{aligned}$$

(3.14)

3.2 The problem $({\mathcal {P}}_0)$ and the groups $\mathbf {\{T^n(t), t\in \mathbb {R}\}}$

At first we make precise which types of solutions of $({\mathcal {P}})$ we shall consider in the sequel.

Definition 3.1

We say that

• (i) $(u,v)\in X^2_{\mathcal {P}}$ is a strong solution of $({\mathcal {P}})$ provided $({\mathcal {P}})_1$ holds a.e. in $\mathbb {R}\times \Omega$ and $({\mathcal {P}})_2$–$({\mathcal {P}})_3$ hold a.e. on $\mathbb {R}\times \Gamma _1$, where $u_t$ and $\partial _{\varvec{\nu }} u$ on $\mathbb {R}\times \Gamma _1$ are taken in the pointwise trace sense given in Sect. 2.2;

• (ii) $(u,v)\in X^1_{\mathcal {P}}$ is a generalized solution of $({\mathcal {P}})$ provided it is the limit in $X^1_{\mathcal {P}}$ of a sequence of strong solutions of it;

• (iii) $(u,v)\in X^1_{\mathcal {P}}$ is a weak solution of $({\mathcal {P}})$ provided the distributional identities

  $$\begin{aligned}{} & {} \int _{-\infty }^\infty \left[ -\rho _0\int _\Omega u_t\varphi _t+B\int _\Omega \nabla u\nabla \varphi -B\int _{\Gamma _1}v_t\varphi \right] =0, \end{aligned}$$

  (3.15)
  $$\begin{aligned}{} & {} \int _{-\infty }^\infty \int _{\Gamma _1}\left[ -\mu v_t\psi _t+\sigma (\nabla _\Gamma v,\nabla _\Gamma {\overline{\psi }})_\Gamma +\delta v_t\psi +\kappa v\psi -\rho _0 u\psi _t\right] =0, \end{aligned}$$

  (3.16)

  hold for all $\varphi \in C^\infty _c(\mathbb {R}\times \mathbb {R}^3)$ and $\psi \in C^r_c(\mathbb {R}\times \Gamma _1)$.

Moreover, solutions of $({\mathcal {P}})$ to the types i)–iii) above are said to be solutions of the same type of: j) problem $({\mathcal {P}^c})$ when also (1.2) holds; jj) problem $({\mathcal {P}}_0)$ when also $({\mathcal {P}}_0)_4$–$({\mathcal {P}}_0)_6$ hold in $X^1_{\mathcal {P}}$ and jjj) problem $({\mathcal {P}}_0^c)$ when both j) and jj) hold.

Remark 3.2

Due to standard density properties on $\mathbb {R}\times \Gamma _1$, Definition 3.1-iii) is independent on the actual value of r, i.e., on the regularity of the boundary of $\Omega$.

Remark 3.3

The definition of strong (and consequently also of generalized) solution of $({\mathcal {P}})$ given above looks different from the one given in [32, Chapter 4, §4.2]. In the sequel we shall show that they are actually equivalent since $r\ge 2$. Moreover, the definition of weak solution of $({\mathcal {P}})$ in Definition 3.1 is actually stronger than  [32, Definition 4.2.1, §4.2]. On the other hand, in view of Theorem 3.7 below, which asserts the existence of such a weak solution, the difference only concerns the more restrictive extent of the uniqueness property for this weak solution. Since this problem was exhaustively studied in the quoted paper, here we shall use Definition 3.1-iii) for the sake of simplicity.

The following result points out some trivial relations among the three types of solutions in Definition 3.1.

Lemma 3.4

Let $(u,v)\in X^1_{\mathcal {P}}$ be a solution of $({\mathcal {P}})$ according to Definition 3.1. Then strong $\Rightarrow$ generalized $\Rightarrow$ weak and, if $(u,v)\in X^2_{\mathcal {P}}$, weak $\Rightarrow$ strong.

Proof

Trivially strong $\Rightarrow$ generalized. Moreover, for any strong solution (u, v), integrating by parts in $\mathbb {R}\times \Omega$ and on $\mathbb {R}\times \Gamma _1$ one gets (3.15) and (3.16), so strong $\Rightarrow$ weak. Since the distributional identities (3.15)–(3.16) are stable with respect to the convergence in $X^1_{\mathcal {P}}$, by Definition 3.1-ii) we also get the implication generalized $\Rightarrow$ weak. To complete the proof let now $(u,v)\in X^2_{\mathcal {P}}$ be a weak solution. Backward performing in (3.15)–(3.16) the same operation done before to show that strong $\Rightarrow$ weak we get that equation $({\mathcal {P}})_2$ holds in the sense of distributions, and hence also a.e., in $\mathbb {R}\times \Omega$. Moreover, we also get that $\int _{-\infty }^\infty \int _\Gamma (\partial _{\varvec{\nu }}u-\widetilde{v_t})\varphi =0 \qquad \text {for all }\varphi \in \mathcal {D}(\mathbb {R}\times \mathbb {R}^3)$, where $\widetilde{v_t}$ is the trivial extension of $v_t$ to $\mathbb {R}\times \Gamma$. By density the last equation extends to test functions $\varphi \in C_c(\mathbb {R}\times \mathbb {R}^3)$. Since $\Gamma$ is compact and of class $C^r$ any $\varphi \in C_c(\mathbb {R}\times \Gamma _1)$ can be extended to $\varphi \in C_c(\mathbb {R}\times \mathbb {R}^3)$, so proving $({\mathcal {P}})_3$ a.e. on $\mathbb {R}\times \Gamma$. Moreover, by (3.16), using (2.3), we also get that $({\mathcal {P}})_2$ holds in $[C_c^r(\mathbb {R}\times \Gamma _1)]'$, and then a.e. on $\mathbb {R}\times \Gamma _1$, concluding the proof. $\square$

Remark 3.5

A trivial consequence of Lemma 3.4 is that solutions $(u,v)\in X^2_{\mathcal {P}}$ are equivalently strong, generalized or weak, and thus, for every such solution, we have $(u(t), v(t), u_t(t), v_t(t))\in D^1_{\mathcal {P}}$ for all $t\in \mathbb {R}$.

Problem $({\mathcal {P}}_0)$ was studied in [32, Chapters 4 and 5] by a semigroup approach. To recall it we introduce the unbounded operator $A_{\mathcal {P}}: D(A_{\mathcal {P}})\subset \mathcal {H}^1_{\mathcal {P}}\rightarrow \mathcal {H}^1_{\mathcal {P}}$ given by

$$\begin{aligned} D(A_{\mathcal {P}})=D^1_{\mathcal {P}},\qquad A_{\mathcal {P}}\begin{pmatrix}u\\ v\\ w\\ z\end{pmatrix} = \begin{pmatrix}-w\\ -z\\ -(B/\rho _0)\Delta u\\ \frac{1}{\mu }\left[ -{{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v)+\delta z+\kappa v+\rho _0 w_{|\Gamma _1}\right] \end{pmatrix}, \end{aligned}$$

(3.17)

together with the abstract equation and Cauchy problem

$$\begin{aligned} U_{\mathcal {P}}'+A_{\mathcal {P}}U_{\mathcal {P}}= & {} 0\qquad \text {in }\mathcal {H}^1_{\mathcal {P}}, \end{aligned}$$

(3.18)

$$\begin{aligned} U_{\mathcal {P}}'+A_{\mathcal {P}}U_{\mathcal {P}}= & {} 0\qquad \hbox { in}\ \mathcal {H}^1_{\mathcal {P}}, \qquad U_{\mathcal {P}}(0)=U_{0{\mathcal {P}}}\in \mathcal {H}^1_{\mathcal {P}}. \end{aligned}$$

(3.19)

We point out that $A_{\mathcal {P}}$ in (3.17) coincides with the operator A defined in [32, Chapter 4, (4.3)–(4.4)] although it domain $D(A_{\mathcal {P}})$ seems to be strictly contained in D(A) and the operators $\Delta$, ${{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma )$, $\partial _{\varvec{\nu }}$ in [32, Chapter 4, (4.3)–(4.4)] are taken in a distributional sense. Indeed, while in [32, Chapter 4] also the case $r=1$ was considered, in the present paper we have $r\ge 2$. Hence, we can use the characterization of D(A) given in [32, Chapter 5, Lemma 5.0.4], which gives $D(A)=D^1_{\mathcal {P}}$ as in (3.17) and (see [32, Chapter 3, §3.3.2 and §3.35]) the operators $\Delta$, ${{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma )$ can be taken in the a.e. sense and $\partial _{\varvec{\nu }}$ in the trace one, as done in (3.17).

The relation among problems $({\mathcal {P}})$, $({\mathcal {P}}_0)$ and their abstract versions (3.18)–(3.19) are given by the following result.

Lemma 3.6

The couple $(u,v)\in X^2_{\mathcal {P}}$ is a strong solution of $({\mathcal {P}})$ if and only if u and v are the first two components of a strong solution $U_{\mathcal {P}}=(u,v,w,z)$ of (3.18). Moreover, in this case $U_{\mathcal {P}}\in Y^2_{\mathcal {P}}$. The same relation occurs between generalized solutions $(u,v)\in X^1_{\mathcal {P}}$ of $({\mathcal {P}})$ and generalized solutions $U_{\mathcal {P}}\in Y^1_{\mathcal {P}}$ of (3.18). Finally, in previous statements, one can replace $({\mathcal {P}})$ with $({\mathcal {P}}_0)$ provided one also replaces (3.18) with (3.19).

Proof

For strong solutions the asserted relation essentially follows from [32, Chapter 5, Lemma 5.0.4]. For generalized solutions we then use a standard density argument, as in [32, § 4.2]. $\square$

Since generalized and strong solutions of $({\mathcal {P}})$ were defined in  [32, Chapter 4] as couples (u, v) constituted by the first two components of an homologous solution of (3.18), Lemma 3.6 also shows that in Definition 3.1 we just made more explicit the definition given in the quoted paper.

We now recall the combination of [32, Theorem 1.2.1 and Theorem 4.1.5] in a form which is most adequate for our purpose. Trivially Theorem 1.1 is just a simplified form of the following statement.

Theorem 3.7

(Well-posedness for $({\mathcal {P}}_0)$ and (3.19)) The operator $-A_{\mathcal {P}}$ is densely defined and it generates on $\mathcal {H}^1_{\mathcal {P}}$ a strongly continuous group $\{T^1(t), t\in \mathbb {R}\}$. Consequently, for any $U_{0{\mathcal {P}}}=(u_0,v_0,u_1, v_1)\in \mathcal {H}^1_{\mathcal {P}}$, problem (3.19) has a unique generalized solution $U_{\mathcal {P}}\in Y^1_{\mathcal {P}}$ defined by $U_{\mathcal {P}}(t)=T^1(t)[U_{0{\mathcal {P}}}]$ for all $t\in \mathbb {R}$ and problem $({\mathcal {P}}_0)$ has a unique generalized solution $(u,v)\in X^1_{\mathcal {P}}$. Moreover, $U_{\mathcal {P}}=(u,v,u_t,v_t)$. Next, the solutions $U_{\mathcal {P}}$ and (u, v) continuously depend on $U_{0{\mathcal {P}}}$ in the topologies of the respective spaces, (u, v) satisfies the energy identity (1.5) and it is unique also among weak solutions of $({\mathcal {P}}_0)$. Finally, for any $U_{0{\mathcal {P}}}\in \mathcal {H}^1_{\mathcal {P}}$, the following properties are equivalent: (i) $U_{0{\mathcal {P}}}\in D^1_{\mathcal {P}}$; (ii) $U_{\mathcal {P}}$ is a strong solution;    (iii) (u, v) is a strong solution;    (iv) $U_{\mathcal {P}}\in Y^2_{\mathcal {P}}$;    v) $(u,v)\in X^2_{\mathcal {P}}$.

Proof

The first two sentences follow from [32, Theorem 4.1.5] and Lemma 3.6, and the asserted continuous dependence follows by Lemma 2.1-i). Since our definition of weak solutions is more restrictive that the one given in the quoted paper, the uniqueness of (u, v) among weak solutions of $({\mathcal {P}}_0)$ follows by [32, Theorem 1.2.1 or Lemma 4.2.5]. The energy identity is asserted in [32, Theorem 1.2.1]. To prove the final asserted equivalence, we remark that by Lemma 2.1-iii) we get i) $\Leftrightarrow$ ii), while ii) $\Leftrightarrow$ iii) $\Leftrightarrow$ iv) follow from Lemma 3.6. Finally, iv) $\Leftrightarrow$ v) is trivial. $\square$

Theorem 3.7 also shows that solutions of $({\mathcal {P}})$ are equivalently weak or generalized. Combining this remark with Remark 3.5, we thus recognize that all types of solutions in Definition 3.1coincide, strong solutions being defined only in the class $X^2_{\mathcal {P}}$. Consequently, in the sequel we shall only deal with weak solutions of $({\mathcal {P}})$, i.e., with elements of one of the spaces $\mathcal {S}^n_{\mathcal {P}}$ defined in (1.14) for $n\in {\widetilde{\mathbb {N}}}$, $n\le r$. For each n as above we also set the Fréchet space

$$\begin{aligned} \mathcal {T}^n_{\mathcal {P}}=\{U_{\mathcal {P}}\in Y^n_{\mathcal {P}}: \quad U_{\mathcal {P}}\text { is a generalized solution of }(3.18)\}, \end{aligned}$$

(3.20)

endowed with the topology inherited from $Y^n_{\mathcal {P}}$. The connection between $\mathcal {T}^n_{\mathcal {P}}$ and $\mathcal {S}^n_{\mathcal {P}}$ is given by the following result, which trivially follows from Lemma 3.6, (1.4) and (3.11).

Lemma 3.8

The operator $\mathcal {I}\in \mathcal {L}(X^1_{\mathcal {P}};Y^1_{\mathcal {P}})$, defined by $\mathcal {I}(u,v)=(u,v,u_t,v_t)$, restricts for each $n\in {\widetilde{\mathbb {N}}}$, $n\le r$, to a bijective isomorphism $\mathcal {I}^n_{\mathcal {P}}\in \mathcal {L}(\mathcal {S}^n_{\mathcal {P}};\mathcal {T}^n_{\mathcal {P}})$. Its inverse $\left( \mathcal {I}^n_{\mathcal {P}}\right) ^{-1}\in \mathcal {L}(\mathcal {T}^n_{\mathcal {P}};\mathcal {S}^n_{\mathcal {P}})$ is simply given by $\left( \mathcal {I}^n_{\mathcal {P}}\right) ^{-1}(u,v,w,z)=(u,v)$.

To show that regularity classes $\mathcal {S}^n_{\mathcal {P}}$ and $\mathcal {T}^n_{\mathcal {P}}$ above contain physically significant solutions (also when $n\ge 3$), we now recall the regularity results  [32, Theorem 1.2.3 and Lemma 5.0.4], combining them as follows.

Theorem 3.9

For $n\in {\widetilde{\mathbb {N}}}$, $2\le n\le r$, we have $D(A^{n-1}_{\mathcal {P}})=D^{n-1}_{\mathcal {P}}$, the respective norms being equivalent. Consequently, the operator $\,\, {_n}A_{\mathcal {P}}$ given by (2.6) generates on $D^{n-1}_{\mathcal {P}}$ a strongly continuous group $\{T^n(t), t\in \mathbb {R}\}$, given by $T^n(t)=T^1(t)_{|D^{n-1}_{\mathcal {P}}}$ for all $t\in \mathbb {R}$. Finally, for any $U_{0{\mathcal {P}}}\in \mathcal {H}^1_{\mathcal {P}}$, denoting by $(u,v)\in X^1_{\mathcal {P}}$ the weak solution given in Theorem 3.7 and denoting $U_{\mathcal {P}}=(u,v,u_t,v_t)$, we have the equivalences

$$\begin{aligned} U_{0{\mathcal {P}}}\in D^{n-1}_{\mathcal {P}}\Leftrightarrow U_{\mathcal {P}}\in Y^n_{\mathcal {P}}\Leftrightarrow (u,v)\in X^n_{\mathcal {P}}\quad \text {for any } n\in {\widetilde{\mathbb {N}}}, 2\le n\le r. \end{aligned}$$

(3.21)

Moreover, when $U_{0{\mathcal {P}}}\in D^{n-1}_{\mathcal {P}}$, $U_{\mathcal {P}}$ and (u, v) continuously depend on it in the topologies of the respective spaces.

3.3 The quotient groups $\mathbf {\{\mathcal {P}^n(t), t\in \mathbb {R}\}}$

The group $\{T^1(t),t\in \mathbb {R}\}$ introduced in Theorem 3.7 and the induced groups $\{T^n(t),t\in \mathbb {R}\}$ introduced in Theorem 3.9 are not directly connected to problems $({\mathcal {E}})$, $({\mathcal {E}^c})$ and $({\mathcal {L}})$. Hence, we are now going to introduce their quotient groups with respect to the space $\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}$ defined in Sect. 3. Indeed, since $({\mathcal {P}})$ possesses the trivial solutions $u(t,x)\equiv c\in \mathbb {C}$, $v(t,x)\equiv 0$, the space $\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}$is invariant under the flow of all the groups $\{T^n(t),t\in \mathbb {R}\}$. The identifications (3.9) allow to make these quotient groups more concrete.

We preliminarily set the unbounded operator ${\dot{A}}_{\mathcal {P}}: D({\dot{A}}_{\mathcal {P}})\subset \dot{\mathcal {H}}^1_{\mathcal {P}}\rightarrow \dot{\mathcal {H}}^1_{\mathcal {P}}$, defined by

$$\begin{aligned}{} & {} D({\dot{A}}_{\mathcal {P}})={\dot{D}}^1_{\mathcal {P}}, \end{aligned}$$

(3.22)

$$\begin{aligned}{} & {} {\dot{A}}_{\mathcal {P}}\begin{pmatrix}\pi _0 u\\ v\\ w\\ z\end{pmatrix} = Q A_{\mathcal {P}}\begin{pmatrix}u\\ v\\ w\\ z\end{pmatrix} = \begin{pmatrix}-\pi _0 w\\ -z\\ -(B/\rho _0)\Delta u\\ \frac{1}{\mu }\left[ -{{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v)+\delta z+\kappa v+\rho _0 w_{|\Gamma _1}\right] \end{pmatrix},\nonumber \\ \end{aligned}$$

(3.23)

for all $(u,v,w,z)\in D^1_{\mathcal {P}}$, together with the abstract equation and Cauchy problem

$$\begin{aligned}{} & {} {\dot{U}}_{\mathcal {P}}'+{\dot{A}}_{\mathcal {P}}{\dot{U}}_{\mathcal {P}}=0\qquad \text {in } \dot{\mathcal {H}}^1_{\mathcal {P}}, \end{aligned}$$

(3.24)

$$\begin{aligned}{} & {} {\dot{U}}_{\mathcal {P}}'+{\dot{A}}_{\mathcal {P}}{\dot{U}}_{\mathcal {P}}=0\qquad \hbox { in}\ \dot{\mathcal {H}}^1_{\mathcal {P}}, \qquad {\dot{U}}_{\mathcal {P}}(0)={\dot{U}}_{0{\mathcal {P}}}\in \dot{\mathcal {H}}^1_{\mathcal {P}}. \end{aligned}$$

(3.25)

The following result is a direct consequence of Theorems 3.7 and 3.9.

Proposition 3.10

(Well-posedness and regularity for (3.25))

• I) The operator $-{\dot{A}}_{\mathcal {P}}$ is densely defined and it generates on $\dot{\mathcal {H}}^1_{\mathcal {P}}=\mathcal {H}^1_{\mathcal {P}}/\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}$ the quotient strongly continuous group $\{{\mathcal {P}}^1(t), t\in \mathbb {R}\}$ defined by

  $$\begin{aligned} {\mathcal {P}}^1(t)[QU_{0{\mathcal {P}}}]=QT^1(t)[U_{0{\mathcal {P}}}]\qquad \text {for all } U_{0{\mathcal {P}}}\in \mathcal {H}^1_{\mathcal {P}}\text { and }t\in \mathbb {R}. \end{aligned}$$

  (3.26)

  Consequently, for any ${\dot{U}}_{0{\mathcal {P}}}\in \dot{\mathcal {H}}^1_{\mathcal {P}}$, problem (3.25) has a unique generalized solution ${\dot{U}}_{\mathcal {P}}\in {\dot{Y}}_{\mathcal {P}}^1$ given by ${\dot{U}}_{\mathcal {P}}(t)={\mathcal {P}}^1(t)[{\dot{U}}_{0{\mathcal {P}}}]$, $t\in \mathbb {R}$, continuously depending on ${\dot{U}}_{0{\mathcal {P}}}$ in the topologies of the respective spaces.

  More explicitly, denoting by $U_{\mathcal {P}}\in Y^1_{\mathcal {P}}$ the unique generalized solution of (3.19) given by Theorem 3.7 with any initial datum $U_{0{\mathcal {P}}}\in Q^{-1}({\dot{U}}_{0{\mathcal {P}}})$ and denoting by (u, v) the corresponding weak solution of $({\mathcal {P}}_0)$, we have

  $$\begin{aligned} {\dot{U}}_{\mathcal {P}}(t)= Q U_{\mathcal {P}}(t)= (\pi _0 u(t), v(t), u_t(t), v_t(t))\qquad \text {for all }t\in \mathbb {R}. \end{aligned}$$

  (3.27)

  Moreover, ${\dot{U}}_{0{\mathcal {P}}}\in {\dot{D}}^1_{\mathcal {P}}\Leftrightarrow$ $U_{\mathcal {P}}$ is a strong solution $\Leftrightarrow {\dot{U}}_{\mathcal {P}}\in {\dot{Y}}^2_{\mathcal {P}}$.

• II) For $n\in {\widetilde{\mathbb {N}}}$, $2\le n\le r$, we have $D({\dot{A}}^{n-1}_{\mathcal {P}})={\dot{D}}^{n-1}_{\mathcal {P}}$, the respective norms being equivalent. Consequently, the operator $\,\, {_n}{\dot{A}}_{\mathcal {P}}$ given by (2.6) generates on ${\dot{D}}^{n-1}_{\mathcal {P}}$ the strongly continuous group $\{\mathcal {P}^n(t), t\in \mathbb {R}\}$ given by ${\mathcal {P}}^n(t)={\mathcal {P}}^1(t)_{|{\dot{D}}^{n-1}_{\mathcal {P}}}$ for all $t\in \mathbb {R}$. The group $\{\mathcal {P}^n(t), t\in \mathbb {R}\}$ can be equivalently defined as the quotient group on ${\dot{D}}^{n-1}_{\mathcal {P}}=D^{n-1}_{\mathcal {P}}$ given by

  $$\begin{aligned} {\mathcal {P}}^n(t)[QU_{0{\mathcal {P}}}]=QT^n(t)[U_{0{\mathcal {P}}}]\qquad \text {for all } U_{0{\mathcal {P}}}\in D^{n-1}_{\mathcal {P}}\text { and }t\in \mathbb {R}. \end{aligned}$$

  (3.28)

• III) For any ${\dot{U}}_{0{\mathcal {P}}}\in \mathcal {H}^1_{\mathcal {P}}$ and $n\in {\widetilde{\mathbb {N}}}$, $2\le n\le r$, we have ${\dot{U}}_{0{\mathcal {P}}}\in {\dot{D}}^{n-1}_{\mathcal {P}}$ if and only if ${\dot{U}}_{\mathcal {P}}\in {\dot{Y}}^n_{\mathcal {P}}$, and in this case ${\dot{U}}_{\mathcal {P}}$ continuously depends on ${\dot{U}}_{0{\mathcal {P}}}$ in the topologies of the respective spaces.

Proof

To prove part I), we first remark that, since by Theorem 3.7$D(A_{\mathcal {P}})=D^1_{\mathcal {P}}$ is dense in $\mathcal {H}^1_{\mathcal {P}}$, one immediately gets that $D({\dot{A}}_{\mathcal {P}})={\dot{D}}^1_{\mathcal {P}}=D^1_{\mathcal {P}}/\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}$ is dense in $\dot{\mathcal {H}}^1_{\mathcal {P}}=\mathcal {H}^1_{\mathcal {P}}/\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}$, where we used (3.9). Moreover, a standard construction in semigroup theory shows, using the already remarked invariance of $\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}$, that (3.26) defines on $\mathcal {H}^1_{\mathcal {P}}$ a (quotient) strongly continuous group. Using Theorem  3.7, its generator is given by the operator $-QA_{\mathcal {P}}$ with domain $QD(A_{\mathcal {P}})$. Hence, since $D(A_{\mathcal {P}})=D^1_{\mathcal {P}}$, by (3.9) we have $QD(A_{\mathcal {P}})={\dot{D}}^1_{\mathcal {P}}=D({\dot{A}}_{\mathcal {P}})$. Comparing (3.17) and (3.23), one easily gets $QA_{\mathcal {P}}={\dot{A}}_{\mathcal {P}}$. We then get the well-posedness of problem (3.25) asserted in the statement and, by Theorem 3.7 and (3.7), also (3.27). By Lemma 2.1-iii) then ${\dot{U}}_{0{\mathcal {P}}}\in {\dot{D}}^1_{\mathcal {P}}$ if and only if ${\dot{U}}_{\mathcal {P}}$ is a strong solution of (3.25). To complete the proof of part I), we first claim that the graph norm and the $\dot{\mathcal {H}}^2_{\mathcal {P}}$-norm are equivalent on ${\dot{D}}^1_{\mathcal {P}}$. By (3.23) one sees that ${\dot{A}}_{\mathcal {P}}\in \mathcal {L}({\dot{D}}^1_{\mathcal {P}}; \dot{\mathcal {H}}^1_{\mathcal {P}})$, so $\Vert \cdot \Vert _{D({\dot{A}}_{\mathcal {P}})}\le \text {Const.}\Vert \cdot \Vert _{\dot{\mathcal {H}}^2_{\mathcal {P}}}$ on ${\dot{D}}^1_{\mathcal {P}}$ and, since ${\dot{D}}^1_{\mathcal {P}}$ is complete with respect to both norms, our claim follows by the Two Norms Theorem. Hence, if ${\dot{U}}_{\mathcal {P}}\in {\dot{Y}}^2_{\mathcal {P}}$, by (3.11) we have ${\dot{U}}_{\mathcal {P}}(0)\in {\dot{H}}^1_{\mathcal {P}}$, which by the definition of domain yields ${\dot{U}}_{0{\mathcal {P}}}\in D({\dot{A}}_{\mathcal {P}})={\dot{D}}^1_{\mathcal {P}}$. Conversely, if ${\dot{U}}_{0{\mathcal {P}}}\in D({\dot{A}}_{\mathcal {P}})={\dot{D}}^1_{\mathcal {P}}$, by Lemma 2.1-iii) we get ${\dot{U}}_{\mathcal {P}}\in C(\mathbb {R}; D({\dot{A}}_{\mathcal {P}}))\cap C^1(\mathbb {R}: \dot{\mathcal {H}}_{\mathcal {P}})$, which by previous claim implies ${\dot{U}}_{\mathcal {P}}\in {\dot{Y}}^2_{\mathcal {P}}$, completing the proof of part I).

To prove part II), we first claim that $D({\dot{A}}^{n-1}_{\mathcal {P}})={\dot{D}}^{n-1}_{\mathcal {P}}$ for $n\in {\widetilde{\mathbb {N}}}$, $2\le n\le r$. Since $D({\dot{A}}_{\mathcal {P}})={\dot{D}}^1_{\mathcal {P}}$, by induction we suppose that $n\ge 3$ and $D({\dot{A}}^{n-2}_{\mathcal {P}})={\dot{D}}^{n-2}_{\mathcal {P}}$. Using (2.5), (3.9) and (3.23) we have $D({\dot{A}}^{n-1}_{\mathcal {P}})=\{QU_{0{\mathcal {P}}}: U_{0{\mathcal {P}}}\in D^{n-2}_{\mathcal {P}}\quad \text {and}\quad QA_{\mathcal {P}}U_{0{\mathcal {P}}}\in {\dot{D}}^{n-2}_{\mathcal {P}}\}$. Since $\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}\subset D^{n-2}_{\mathcal {P}}$ in the previous formula, we have $QA_{\mathcal {P}}U_{0{\mathcal {P}}}\in {\dot{D}}^{n-2}_{\mathcal {P}}$ if and only if $A_{\mathcal {P}}U_{0{\mathcal {P}}}\in D^{n-2}_{\mathcal {P}}$, so $D({\dot{A}}^{n-1}_{\mathcal {P}})=Q D^{n-1}_{\mathcal {P}}={\dot{D}}^{n-1}_{\mathcal {P}}$, proving our claim. The asserted norm equivalence is then proved by the argument already used to get it in the case $n=2$ above, provided we first recognize that ${\dot{A}}_{\mathcal {P}}\in \mathcal {L}({\dot{D}}^i_{\mathcal {P}};{\dot{D}}^{i-1}_{\mathcal {P}})$ for $i=1,\ldots , n-1$, where ${\dot{D}}^0_{\mathcal {P}}:={\dot{H}}^1_{\mathcal {P}}$. But, by Theorem 3.9, $A_{\mathcal {P}}\in \mathcal {L}(D^i_{\mathcal {P}};D^{i-1}_{\mathcal {P}})$. Then, by (3.10), $QA_{\mathcal {P}}\in \mathcal {L}(D^i_{\mathcal {P}};{\dot{D}}^{i-1}_{\mathcal {P}})$ and consequently, since ${\dot{D}}^i_{\mathcal {P}}=D^i_{\mathcal {P}}/\mathbb {C}_{\mathcal {H}_{\mathcal {P}}}$, we get ${\dot{A}}_{\mathcal {P}}\in \mathcal {L}({\dot{D}}^i_{\mathcal {P}};{\dot{D}}^{i-1}_{\mathcal {P}})$. By applying Lemma 2.1-ii), we then get the strongly continuous group $\{{\mathcal {P}}^n(t), t\in \mathbb {R}\}$ on ${\dot{D}}^{n-1}_{\mathcal {P}}$, given by ${\mathcal {P}}^n(t)={\mathcal {P}}^1(t)_{|{\dot{D}}^{n-1}_{\mathcal {P}}}$ for all $t\in \mathbb {R}$, which has generator ${_n}{\dot{A}}_{\mathcal {P}}$. Since $T^n(t)=T^1(t)_{|D^{n-1}_{\mathcal {P}}}$, (3.28) follows by (3.26).

To prove part III), we first consider $n\in \mathbb {N}$ and we remark that when ${\dot{U}}_{0{\mathcal {P}}}\in {\dot{D}}^{n-1}_{\mathcal {P}}$, by Lemma 2.1 we get ${\dot{U}}_{\mathcal {P}}\in {\dot{Y}}^n_{\mathcal {P}}$ while, when ${\dot{U}}_{\mathcal {P}}\in {\dot{Y}}^n_{\mathcal {P}}$ we have ${\dot{U}}_{\mathcal {P}}'(0)\in {\dot{H}}^{n-1}_{\mathcal {P}}$, so ${\dot{U}}_{0{\mathcal {P}}}\in D({_{n-2}}{\dot{A}}_{\mathcal {P}})={\dot{D}}^{n-1}_{\mathcal {P}})$. The case $n=\infty$ then follows by (3.6). $\square$

Remark 3.11

While the first two components u and v of a solution of (3.19) constitute a solution of $({\mathcal {P}})$, the first two components ${\dot{u}}$ and v of a solution of (3.24) do not enjoy the same property, in general. We recall the example given in [32, §6.1]. Let $\kappa (x)\equiv k_0\in \mathbb {R}{\setminus }\{0\}$ and the solutions $u(t,x)=u_1 t$, $v(t,x)\equiv -\rho _0u_1/k_0$. By (3.27) the strong solution ${\dot{U}}_{\mathcal {P}}$ of (3.25) with data ${\dot{U}}_{0{\mathcal {P}}}=(0,-\rho _0 u_1/k_0, u_1,0)$ is ${\dot{U}}_{\mathcal {P}}(t)={\dot{U}}_{0{\mathcal {P}}}$ for all $t\in \mathbb {R}$. Trivially $(0,-\rho _0 u_1/k_0)$ does not solve equation $({\mathcal {P}})_2$.

We now set, for $n\in {\widetilde{\mathbb {N}}}$, $n\le r$, the Fréchet space constituted by the trajectories of the group $\{{\mathcal {P}}^n(t),t\in \mathbb {R}\}$, i.e.,

$$\begin{aligned} \dot{\mathcal {T}}^n_{\mathcal {P}}=\{{\dot{U}}_{\mathcal {P}}\in {\dot{Y}}^n_{\mathcal {P}}: {\dot{U}}_{\mathcal {P}}\quad \text {is a generalized solution of }(3.24)\}, \end{aligned}$$

(3.29)

endowed with the topology inherited from ${\dot{Y}}^n_{\mathcal {P}}$. The following result relates solutions of $({\mathcal {P}})$, (3.18) and (3.24). It will be relevant in connection with problem $({\mathcal {E}})$.

Proposition 3.12

The pointwise quotient operator $\mathcal {Q}$ defined in (3.12) restricts to a surjective operator $\mathcal {Q}^1_{\mathcal {P}}\in \mathcal {L}(\mathcal {T}^1_{\mathcal {P}};\dot{\mathcal {T}}^1_{\mathcal {P}})$, so the operator $\mathcal {J}^1_{\mathcal {P}}=\mathcal {Q}^1_{\mathcal {P}}\cdot \mathcal {I}^1_{\mathcal {P}}\in \mathcal {L}(\mathcal {S}^1_{\mathcal {P}}; \dot{\mathcal {T}}^1_{\mathcal {P}})$ given by

$$\begin{aligned} {[}\mathcal {J}^1_{\mathcal {P}}(u,v)](t)=(\pi _0 u(t), v(t), u_t(t), v_t(t))\qquad \text {for all }t\in \mathbb {R}, \end{aligned}$$

(3.30)

is surjective as well. Moreover, for all $n\in {\widetilde{\mathbb {N}}}$, $n\le r$, the operators $\mathcal {Q}^1_{\mathcal {P}}$, $\mathcal {J}^1_{\mathcal {P}}$ restrict to surjective operators $\mathcal {Q}^n_{\mathcal {P}}\in \mathcal {L}(\mathcal {T}^n_{\mathcal {P}};\dot{\mathcal {T}}^n_{\mathcal {P}})$, $\mathcal {J}^n_{\mathcal {P}}\in \mathcal {L}(\mathcal {S}^n_{\mathcal {P}}; \dot{\mathcal {T}}^n_{\mathcal {P}})$ and we have

$$\begin{aligned} \ker \mathcal {Q}^n_{\mathcal {P}}=\mathbb {C}_{Y_{\mathcal {P}}}\qquad \text {and}\quad \ker \mathcal {J}^n_{\mathcal {P}}=\mathbb {C}_{X_{\mathcal {P}}}. \end{aligned}$$

(3.31)

Consequently, $\mathcal {J}^n_{\mathcal {P}}$ subordinates a bijective isomorphism $\dot{\mathcal {J}}^n_{\mathcal {P}}\in \mathcal {L}(\dot{\mathcal {S}}^n_{\mathcal {P}}; \dot{\mathcal {T}}^n_{\mathcal {P}})$.

Proof

Combining (3.12) with Proposition 3.10, we get the surjective operator $\mathcal {Q}^1_{\mathcal {P}}\in \mathcal {L}(\mathcal {T}^1_{\mathcal {P}};\dot{\mathcal {T}}^1_{\mathcal {P}})$, so by Lemma 3.8 the operator $\mathcal {J}^1_{\mathcal {P}}=\mathcal {Q}^1_{\mathcal {P}}\cdot \mathcal {I}^1_{\mathcal {P}}\in \mathcal {L}(\mathcal {S}^1_{\mathcal {P}}; \dot{\mathcal {T}}^1_{\mathcal {P}})$ is surjective and it satisfies (3.30). By (3.14) we get that $\mathcal {Q}^1_{\mathcal {P}}$ restricts to $\mathcal {Q}^n_{\mathcal {P}}\in \mathcal {L}(\mathcal {T}^n_{\mathcal {P}};\dot{\mathcal {T}}^n_{\mathcal {P}})$, so by Lemma 3.8 the operator $\mathcal {J}^1_{\mathcal {P}}$ restricts to $\mathcal {J}^n_{\mathcal {P}}\in \mathcal {L}(\mathcal {S}^n_{\mathcal {P}}; \dot{\mathcal {T}}^n_{\mathcal {P}})$. To show that $\mathcal {Q}^n_{\mathcal {P}}$ is surjective also when $n\ge 2$, let us take ${\dot{U}}_{\mathcal {P}}\in \dot{\mathcal {T}}^n_{\mathcal {P}}$. By Proposition 3.10-III) we then have ${\dot{U}}_{\mathcal {P}}(0)\in {\dot{D}}^{n-1}_{\mathcal {P}}$. By part I) of the same result we have ${\dot{U}}_{\mathcal {P}}=\mathcal {Q}U_{\mathcal {P}}$, where $U_{\mathcal {P}}$ is the solution of the Cauchy problem (3.19) corresponding to any datum $U_{0{\mathcal {P}}}\in Q^{-1}({\dot{U}}_{\mathcal {P}}(0))$. By (3.8), from ${\dot{U}}_{\mathcal {P}}(0)\in {\dot{D}}^{n-1}_{\mathcal {P}}$ it follows that $U_{0{\mathcal {P}}}\in D^{n-1}_{\mathcal {P}}$. Consequently, by Theorem 3.9, formula (3.21), we have $U_{\mathcal {P}}\in Y^n_{\mathcal {P}}$ and then ${\dot{U}}_{\mathcal {P}}=\mathcal {Q}^n_{\mathcal {P}}U_{\mathcal {P}}$, proving that $\mathcal {Q}^n_{\mathcal {P}}$ is surjective. The surjectivity of $\mathcal {J}^n_{\mathcal {P}}$ then follows by Lemma 3.8. To prove (3.31) we first remark that, by Lemma 3.8, we just have to prove that $\ker \mathcal {J}^n_{\mathcal {P}}=\mathbb {C}_{X_{\mathcal {P}}}$. Moreover, since $\mathbb {C}_{X_{\mathcal {P}}}\subset \mathcal {S}^n_{\mathcal {P}}$, we just have to prove that $\ker \mathcal {J}^1_{\mathcal {P}}=\mathbb {C}_{X_{\mathcal {P}}}$. The last identity is a trivial consequence of (3.30). $\square$

Proposition 3.12 shows that solutions of $({\mathcal {P}})$ corresponds, up to an additive space-time constants, to trajectories of $\{T^1(t), t\in \mathbb {R}\}$, regularity classes being preserved.

3.4 The problem $({\mathcal {P}}_0^c)$ and the groups $\mathbf {\{T_c^n(t), t\in \mathbb {R}\}}$, $\mathbf {\{\mathcal {P}_c^n(t), t\in \mathbb {R}\}}$

In this subsection we are going to show that results in Sects. 3.2–3.3 can be adapted to the restricted potential problem $({\mathcal {P}^c})$. This fact depends on the following easy result, which (trivially) generalizes [32, Lemma 6.2.1], with the same proof.

Lemma 3.13

For any $(u,v)\in \mathcal {S}^1_{\mathcal {P}}$ we have

$$\begin{aligned} \rho _0\int _\Omega u_t(t)-B\int _{\Gamma _1}v(t)=\rho _0\int _\Omega u_t(s)-B\int _{\Gamma _1}v(s)\qquad \text {for all }s,t\in \mathbb {R}. \end{aligned}$$

(3.32)

Consequently, the spaces $\mathcal {H}^1_{\mathcal {P}^c}$ and $\dot{\mathcal {H}}^1_{\mathcal {P}^c}$ are invariant with respect to the flows of the groups $\{T^1(t), t\in \mathbb {R}\}$ and $\{{\mathcal {P}}^1(t), t\in \mathbb {R}\}$. For $n\in \mathbb {N}$, $2\le n\le r$, the spaces $D^{n-1}_{\mathcal {P}^c}$ and ${\dot{D}}^{n-1}_{\mathcal {P}^c}$ are invariant with respect to the flows of the groups $\{T^n(t), t\in \mathbb {R}\}$ and $\{{\mathcal {P}}^n(t), t\in \mathbb {R}\}$. Moreover,

$$\begin{aligned} {{\,\textrm{Rg}\,}}(A_{\mathcal {P}})\subseteq \mathcal {H}^1_{\mathcal {P}^c}\qquad \text {and}\quad {{\,\textrm{Rg}\,}}({\dot{A}}_{\mathcal {P}})\subseteq \dot{\mathcal {H}}^1_{\mathcal {P}^c}. \end{aligned}$$

(3.33)

To state the extensions of the results in Sects. 3.2–3.3, we now set, thanks to (3.33), the unbounded operators $A_{\mathcal {P}^c}: D(A_{\mathcal {P}^c})\subset \mathcal {H}^1_{\mathcal {P}^c}\rightarrow \mathcal {H}^1_{\mathcal {P}^c}$ and ${\dot{A}}_{\mathcal {P}^c}: D({\dot{A}}_{\mathcal {P}^c})\subset \dot{\mathcal {H}}^1_{\mathcal {P}^c}\rightarrow \dot{\mathcal {H}}^1_{\mathcal {P}^c}$, defined by

$$\begin{aligned} D(A_{\mathcal {P}^c})=D^1_{\mathcal {P}^c}, \quad A_{\mathcal {P}^c}=A_{{\mathcal {P}}|D^1_{\mathcal {P}^c}}, \quad D({\dot{A}}_{\mathcal {P}^c})={\dot{D}}^1_{\mathcal {P}^c}, \quad {\dot{A}}_{\mathcal {P}^c}={\dot{A}}_{{\mathcal {P}}|{\dot{D}}^1_{\mathcal {P}^c}}. \end{aligned}$$

(3.34)

Moreover, we set, for $n\in {\widetilde{\mathbb {N}}}$, $n\le r$, the Fréchet spaces

$$\begin{aligned} \mathcal {S}^n_{\mathcal {P}^c}=\mathcal {S}^n_{\mathcal {P}}\cap X^n_{\mathcal {P}^c},\qquad \mathcal {T}^n_{\mathcal {P}^c}=\mathcal {T}^n_{\mathcal {P}}\cap Y^n_{\mathcal {P}^c}\qquad \text {and}\quad \dot{\mathcal {T}}^n_{\mathcal {P}^c}=\dot{\mathcal {T}}^n_{\mathcal {P}}\cap {\dot{Y}}^n_{\mathcal {P}^c}, \end{aligned}$$

(3.35)

and we point out the following extension of Lemma 3.8.

Corollary 3.14

For each $n\in {\widetilde{\mathbb {N}}}$, $n\le r$, the operator $\mathcal {I}^n_{\mathcal {P}}$ in Lemma 3.8 further restricts to a bijective isomorphism $\mathcal {I}^n_{\mathcal {P}^c}\in \mathcal {L}(\mathcal {S}^n_{\mathcal {P}^c};\mathcal {T}^n_{\mathcal {P}^c})$, with inverse given as in Lemma 3.8.

Proof

By (1.6), (3.11) and (3.35) one trivially gets $\mathcal {I}^1_{\mathcal {P}}\cdot \mathcal {S}^1_{\mathcal {P}^c}=\mathcal {T}^1_{\mathcal {P}^c}$. When $n>1$ the statement follows in the same way once one remarks that, differentiating in time, we have $X^n_{\mathcal {P}^c}=\left\{ (u,v)\in X^n_{\mathcal {P}}: \rho _0\int _\Omega \frac{\partial ^i u}{\partial t^i}=B\int _{\Gamma _1}\frac{\partial ^{i-1} v}{\partial t^i}\quad \hbox { for}\ i\in \mathbb {N}, i\le n\right\}$, from which one easily gets that $\mathcal {I}^n_{\mathcal {P}}\cdot \mathcal {S}^n_{\mathcal {P}^c}=\mathcal {T}^n_{\mathcal {P}^c}$. $\square$

Using Lemma 3.13 and Corollary 3.14, it is straightforward to get the following consequence of results in Sects. 3.2–3.3, which will be used in sequel. Corollary 1.2 is just a partial statement of it.

Corollary 3.15

(Problem ${({\mathcal {P}}_0^c)}$ and the groups ${\{T^n_c(t), t\in \mathbb {R}\}}$) The statements of Theorem 3.7 and 3.9 continue to hold when one, respectively, replaces (also in problem (3.19)): the spaces $\mathcal {H}^1_{\mathcal {P}}$, $D^{n-1}_{\mathcal {P}}$, $X^n_{\mathcal {P}}$, $Y^n_{\mathcal {P}}$ with the spaces $\mathcal {H}^1_{\mathcal {P}^c}$, $D^{n-1}_{\mathcal {P}^c}$, $X^n_{\mathcal {P}^c}$, $Y^n_{\mathcal {P}^c}$; the operators $A_{\mathcal {P}}$ and ${_n}A_{\mathcal {P}}$ with the operators $A_{\mathcal {P}^c}$ and ${_n}A_{\mathcal {P}^c}$; problem $({\mathcal {P}}_0)$ with problem $({\mathcal {P}}_0^c)$; the groups $\{T^n(t),t\in \mathbb {R}\}$ with the groups $\{T^n_c(t),t\in \mathbb {R}\}$ defined (denoting $D^0_{\mathcal {P}^c}=\mathcal {H}^1_{\mathcal {P}^c}$) by $T^n_c(t)=T^n(t)_{|D^{n-1}_{\mathcal {P}^c}}$ for $n\in \mathbb {N}$, $n\le r$.

Proof

Recalling (3.1) and (3.4), the statement follows by a standard construction in semigroup theory combined with Theorem 3.7, 3.9 and Corollary 3.14. $\square$

Using Lemma 3.13 one can also generalize Proposition 3.10 as follows.

Corollary 3.16

(Problem (3.25) in ${\dot{\mathcal {H}}^1_{\mathcal {P}^c}}$ and the groups ${\{\mathcal {P}_c^n(t), t\in \mathbb {R}\}}$) The statement of Proposition 3.10 continues to hold when one, respectively, replaces, also in problem (3.25): the spaces $\dot{\mathcal {H}}^1_{\mathcal {P}}$, ${\dot{D}}^{n-1}_{\mathcal {P}}$, ${\dot{Y}}^n_{\mathcal {P}}$ with the spaces $\dot{\mathcal {H}}^1_{\mathcal {P}^c}$, ${\dot{D}}^{n-1}_{\mathcal {P}^c}$, ${\dot{Y}}^n_{\mathcal {P}^c}$; the operators ${\dot{A}}_{\mathcal {P}}$ and ${_n}{\dot{A}}_{\mathcal {P}}$ with the operators ${\dot{A}}_{\mathcal {P}^c}$ and ${_n}{\dot{A}}_{\mathcal {P}^c}$; the group $\{{\mathcal {P}}^n(t),t\in \mathbb {R}\}$ with the subspace group $\{{\mathcal {P}}^n_c(t),t\in \mathbb {R}\}$ defined (denoting ${\dot{D}}^0_{\mathcal {P}^c}=\dot{\mathcal {H}}^1_{\mathcal {P}^c}$) by ${\mathcal {P}}^n_c(t)={\mathcal {P}}^n(t)_{|{\dot{D}}^{n-1}_{\mathcal {P}^c}}$ for $n\in \mathbb {N}$, $n\le r$.

Proof

We first claim that

$$\begin{aligned} {\dot{Y}}^n_{\mathcal {P}^c}={\dot{Y}}^1_{\mathcal {P}^c}\cap {\dot{Y}}^n_{\mathcal {P}}\qquad \text {for all }n\in {\widetilde{\mathbb {N}}}, n\le r. \end{aligned}$$

(3.36)

By (3.11) trivially ${\dot{Y}}^n_{\mathcal {P}^c}\subseteq {\dot{Y}}^1_{\mathcal {P}^c}\cap {\dot{Y}}^n_{\mathcal {P}}$ and, deriving in time as many times as needed, one also gets the reverse inclusion. By Proposition 3.10, using Lemma 3.13 and (3.36), one gets the assertion. $\square$

Also Proposition 3.12 extends to problem $({\mathcal {P}^c})$ as follows.

Corollary 3.17

For all $n\in {\widetilde{\mathbb {N}}}$, $n\le r$, the operators $\mathcal {Q}^n_{\mathcal {P}}$ and $\mathcal {J}^n_{\mathcal {P}}$ defined in Proposition 3.12 further restrict to surjective operators $\mathcal {Q}^n_{\mathcal {P}^c}\in \mathcal {L}(\mathcal {T}^n_{\mathcal {P}^c};\dot{\mathcal {T}}^n_{\mathcal {P}^c})$ and $\mathcal {J}^n_{\mathcal {P}^c}\in \mathcal {L}(\mathcal {S}^n_{\mathcal {P}^c}; \dot{\mathcal {T}}^n_{\mathcal {P}^c})$, still satisfying $\ker \mathcal {Q}^n_{\mathcal {P}^c}=\mathbb {C}_{Y_{\mathcal {P}}}$ and $\ker \mathcal {J}^n_{\mathcal {P}^c}=\mathbb {C}_{X_{\mathcal {P}}}$. Consequently, the bijective isomorphism $\dot{\mathcal {J}}^n_{\mathcal {P}}$ in Proposition 3.12 further restricts to a bijective isomorphism $\dot{\mathcal {J}}^n_{\mathcal {P}^c}\in \mathcal {L}(\dot{\mathcal {S}}^n_{\mathcal {P}^c}; \dot{\mathcal {T}}^n_{\mathcal {P}^c})$.

Proof

By Proposition 3.12 and (3.14) we have $\mathcal {Q}^n_{\mathcal {P}^c}\in \mathcal {L}(\mathcal {T}^n_{\mathcal {P}^c};\dot{\mathcal {T}}^n_{\mathcal {P}^c})$ and, since $\mathcal {Q}^n_{\mathcal {P}}$ is surjective and by (3.36) one has $\mathcal {Q}^n_{\mathcal {P}}\cdot \mathcal {T}^n_{\mathcal {P}^c}=\dot{\mathcal {T}}^n_{\mathcal {P}^c}$, also $\mathcal {Q}^n_{\mathcal {P}^c}$ is surjective. Moreover, since $\mathbb {C}_{Y_{\mathcal {P}}}\subset \mathcal {T}^n_{\mathcal {P}^c}$ and $\mathbb {C}_{X_{\mathcal {P}}}\subset \mathcal {S}^n_{\mathcal {P}^c}$, from (3.31) one gets $\ker \mathcal {Q}^n_{\mathcal {P}^c}=\mathbb {C}_{Y_{\mathcal {P}}}$ and $\ker \mathcal {J}^n_{\mathcal {P}^c}=\mathbb {C}_{X_{\mathcal {P}}}$. Consequently, $\mathcal {J}^n_{\mathcal {P}^c}$ subordinates a bijective isomorphism $\dot{\mathcal {J}}^n_{\mathcal {P}^c}\in \mathcal {L}(\dot{\mathcal {S}}^n_{\mathcal {P}^c}; \dot{\mathcal {T}}^n_{\mathcal {P}^c})$, which is the restriction of the operator $\dot{\mathcal {J}}^n_{\mathcal {P}}$ in Proposition 3.12. $\square$

4 The Lagrangian model

4.1 Preliminaries

We start by introducing all functional spaces needed to deal with problem $({\mathcal {L}})$, in addition to the ones already defined in Sect. 1.2. We start with the Hilbert space $L^2_{\mathcal {L}}=L^2(\Omega )^3\times L^2(\Gamma _1)$ endowed with the standard inner product. Moreover, recalling the spaces $H^n_{{{\,\textrm{curl}\,}}0}(\Omega )$, $H^n_{\mathcal {L}}$, $\mathbb {H}^n_{\mathcal {L}}$ and $\mathcal {H}^n_{\mathcal {L}}$ defined in (1.7)–(1.9), in the sequel we shall use, when useful, the trivial identifications

$$\begin{aligned} \mathcal {H}^n_{\mathcal {L}}\,=\,\mathbb {H}^n_{\mathcal {L}}\times H^{n-1}_{{{\,\textrm{curl}\,}}0}(\Omega )\times H^{n-1}(\Gamma _1)\,=\,\mathbb {H}^n_{\mathcal {L}}\times H^{n-1}_{\mathcal {L}}. \end{aligned}$$

(4.1)

To fix the notation, we anticipate the compatibility conditions for problem $({\mathcal {L}}_0)$. They are given, for $n\in \mathbb {N}$, $2\le n\le r$ and data $U_{0{\mathcal {L}}}=(\textbf{r}_0,v_0,\textbf{r}_1,v_1)\in \mathcal {H}^n_{\mathcal {L}}$, by

$$\begin{aligned} \left\{ \begin{aligned}&\textbf{r}_1\cdot \varvec{\nu }=-v_1,\qquad \text {on }\Gamma _1,\qquad \textbf{r}_1\cdot \varvec{\nu }=0,\quad \text {on }\Gamma _0, \\&\Delta ^i\textbf{r}_0\cdot \varvec{\nu } =0,\qquad \text {on }\Gamma _0 \qquad \text { for }i=1,\ldots , \lfloor (n-1)/2 \rfloor ,\text { when }n\ge 3,\\&\Delta ^i\textbf{r}_1\cdot \varvec{\nu } =0,\qquad \text {on }\Gamma _0 \qquad \text { for }i=1,\ldots , \lfloor n/2\rfloor -1, \text { when }n\ge 4,\\&\begin{aligned} \frac{B\mu }{\rho _0} \Delta ^i\!\textbf{r}_0\!\cdot \!\varvec{\nu }\!=\!{{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma (\Delta ^{i-1}\!\!\textbf{r}_0\!\cdot \!\varvec{\nu }))\! -\!\delta \Delta ^{i-1}\!\!\textbf{r}_1\!\cdot \!\varvec{\nu }\!-\!\kappa \Delta ^{i-1}\!\!\textbf{r}_0\!\cdot \!\varvec{\nu }\!-\! B{{\,\textrm{div}\,}}\Delta ^{i-1} \!\textbf{r}_0\\ \quad \text { on }\Gamma _1 \quad \text { for }i=1,\ldots , \lfloor (n-1)/2\rfloor , \quad \text { when }n\ge 3,\qquad \, \end{aligned}\\&\begin{aligned} \frac{B\mu }{\rho _0} \Delta ^i\!\textbf{r}_1\!\cdot \!\varvec{\nu }\!=\!{{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma (\Delta ^{i-1}\!\!\textbf{r}_1\!\cdot \!\varvec{\nu }))\! -\!\frac{B\delta }{\rho _0} \Delta ^i\textbf{r}_0\!\cdot \!\varvec{\nu }\!-\!\kappa \Delta ^{i-1}\!\!\textbf{r}_1\!\cdot \!\varvec{\nu }\!-\! B{{\,\textrm{div}\,}}\Delta ^{i-1} \!\!\textbf{r}_1\\ \quad \text { on }\Gamma _1 \quad \text { for }i=1,\ldots , \lfloor n/2\rfloor -1, \quad \text { when }n\ge 4.\qquad \quad \end{aligned}\end{aligned}\right. \end{aligned}$$

(4.2)

By (2.2) and the Trace Theorem, it is straightforward to check that

$$\begin{aligned} D^{n-1}_{\mathcal {L}}:=\{U_{0{\mathcal {L}}}\in \mathcal {H}^n_{\mathcal {L}}: \quad (4.2)\text { hold}\} \end{aligned}$$

(4.3)

is a closed subspace of $\mathcal {H}^n_{\mathcal {L}}$ and, hence, a Hilbert space. When $r=\infty$ we also set the Fréchet space $\mathcal {H}_{\mathcal {L}}^\infty = \left[ C^\infty ({\overline{\Omega }})^3\times C^\infty (\Gamma _1)\times C^\infty ({\overline{\Omega }})^3\times C^\infty (\Gamma _1)\right] \cap \mathcal {H}^1_{\mathcal {L}},$ endowed with the topology of the product space in square brackets, and its closed subspace

$$\begin{aligned} D_{\mathcal {L}}^\infty = \{U_{0{\mathcal {L}}}\in \mathcal {H}_{\mathcal {L}}^\infty : \quad (4.2)\hbox { hold for all}\ n\in \mathbb {N}\}. \end{aligned}$$

(4.4)

We remark that, by Morrey’s Theorem, $D_{\mathcal {L}}^\infty =\bigcap _{n=1}^\infty D^n_{\mathcal {L}}$. Recalling the space $X^n_{\mathcal {L}}$ defined in (1.10) for $n\in {\widetilde{\mathbb {N}}}$, $n\le r$, we remark that, when $r=\infty$,

$$\begin{aligned} X^\infty _{\mathcal {L}}= & {} \{(\textbf{r},v)\in C^\infty (\mathbb {R}\times {\overline{\Omega }})\times C^\infty (\mathbb {R}\times \Gamma _1): \quad {{\,\textrm{curl}\,}}\textbf{r}=0\quad \hbox { in}\ \mathbb {R}\times \Omega ,\\{} & {} \quad \textbf{r}\cdot \varvec{\nu }=-v\quad \text {on } \mathbb {R}\times \Gamma _1,\qquad \textbf{r}\cdot \varvec{\nu }=0\quad \text {on }\mathbb {R}\times \Gamma _0\}. \end{aligned}$$

For the same n’s we also set the Fréchet space

$$\begin{aligned} Y^n_{\mathcal {L}}=\bigcap _{i=0}^{n-1} C^i(\mathbb {R};\mathcal {H}^{n-i}_{\mathcal {L}}). \end{aligned}$$

(4.5)

At first we essentially recall a result in [12].

Lemma 4.1

The operator $G\in \mathcal {L}({\dot{H}}^1(\Omega ); H^0_{{{\,\textrm{curl}\,}}0}(\Omega ))$, defined by

$$\begin{aligned} G{\dot{u}}=-\nabla {\dot{u}} \qquad \text {for all }{\dot{u}}\in {\dot{H}}^1(\Omega ), \end{aligned}$$

(4.6)

restricts to a bijective isomorphism between ${\dot{H}}^n(\Omega )$ and $H^{n-1}_{{{\,\textrm{curl}\,}}0}(\Omega )$ for all $n\in \mathbb {N}$.

Proof

Since $\Gamma$ is compact, it has finitely many connected components. Moreover, $\Omega$ is bounded, simply connected and of class $C^r$, $r\ge 2$. Hence, it satisfies [12, Chapter IX, §1, assumption (1.45) p. 217]. In the authors’ notation $N=0$. Hence, one can apply [12, Chapter IX, §1, Proposition 2, p. 219] with $\mathbb {H}_1=\{0\}$, this space being isomorphic to the first cohomology space. We then get that $H^0_{{{\,\textrm{curl}\,}}0}(\Omega )=\{\nabla \varphi , \, \varphi \in H^1(\Omega )\}$, from which $H^{n-1}_{{{\,\textrm{curl}\,}}0}(\Omega )=\{\nabla \varphi , \, \varphi \in H^n(\Omega )\}$ trivially follows. Since $\ker \nabla =\mathbb {C}_\Omega$ in $H^1(\Omega )$, the proof is complete. $\square$

The key result to connect problems $({\mathcal {P}^c})$ and $({\mathcal {L}})$ is the following one.

Lemma 4.2

For all $w\in L^2(\Omega )$ and $v\in H^{1/2}(\Gamma _1)$ satisfying the compatibility condition

$$\begin{aligned} \rho _0\int _\Omega w=B\int _{\Gamma _1}v, \end{aligned}$$

(4.7)

the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -B{{\,\textrm{div}\,}}\textbf{r}=\rho _0 w,\quad &{}\text {in }\Omega ,\\ \quad {{\,\textrm{curl}\,}}\textbf{r}=0\quad &{}\text {in }\Omega ,\\ \quad \textbf{r}\cdot \varvec{\nu }=-v\quad &{}\text {on }\Gamma _1,\\ \quad \textbf{r}\cdot \varvec{\nu }=0\quad &{}\text {on }\Gamma _0, \end{array}\right. } \end{aligned}$$

(4.8)

has a unique solution $\textbf{r}\in H^1_{{{\,\textrm{curl}\,}}0}(\Omega )$. Consequently, recalling the spaces defined in (1.8) and (3.1), the formula $S(v,w)=(\textbf{r},v)$, for all $(v,w)\in H^1_{\mathcal {P}^c}$, defines a bijective isomorphism $S\in \mathcal {L}(H^1_{\mathcal {P}^c};\mathbb {H}^1_{\mathcal {L}})$, with inverse $S^{-1}\in \mathcal {L}(\mathbb {H}^1_{\mathcal {L}};H^1_{\mathcal {P}^c})$ given by $S^{-1}(\textbf{r},v)=\left( v, - B{{\,\textrm{div}\,}}\textbf{r}/\rho _0\right)$.

Moreover, for all $n\in \mathbb {N}$, $2\le n\le r$, and $q\in [2,\infty )$, there is a positive constant $c_{n,q}=c_{n,q}(\Omega )$ such that, for all $w\in W^{n-1,q}(\Omega )$ and $v\in W^{n-1/q,q}(\Gamma _1)$ satisfying (4.7), one has

$$\begin{aligned} \Vert \textbf{r}\Vert _{[W^{n,q}(\Omega )]^3}\le c_{n,q}\left( \Vert w\Vert _{W^{n-1,q}(\Omega )}+\Vert v\Vert _{v\in W^{n-1/q,q}(\Gamma _1)}\right) . \end{aligned}$$

(4.9)

Consequently, $S\in \mathcal {L}(H^n_{\mathcal {P}^c};\mathbb {H}^n_{\mathcal {L}})$ and $S^{-1}\in \mathcal {L}(\mathbb {H}^n_{\mathcal {L}};H^n_{\mathcal {P}^c})$. Finally, for all $w\in C^{r-1}({\overline{\Omega }})$ and $v\in C^{r-1}(\Gamma _1)$ satisfying (4.7), the solution $\textbf{r}$ of (4.8) can be extended to $\textbf{r}\in [C^{r-1}(\mathbb {R}^3)]^3$.

Proof

By Lemma 4.1 solving problem (4.8) in the space $H^1_{{{\,\textrm{curl}\,}}0}(\Omega )$ it is equivalent to solving the inhomogeneous Neumann problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -B\Delta \varphi =\rho _0 w,\quad &{}\text {in }\Omega ,\\ \quad \partial _{\varvec{\nu }}\varphi =-v\quad &{}\text {on }\Gamma _1,\\ \quad \partial _{\varvec{\nu }}\varphi =0\quad &{}\text {on }\Gamma _0, \end{array}\right. } \end{aligned}$$

(4.10)

with $\varphi \in {\dot{H}}^2(\Omega )$, and hence take $\textbf{r}=\nabla \varphi$. Using the splitting (2.1), problem (4.10) can be rewritten as $-B\Delta \varphi =\rho _0 w$ in $\Omega$, $\partial _{\varvec{\nu }}\varphi =-v$ on $\Gamma$, with $v\in H^{1/2}(\Gamma )$, and $v\in W^{n-1/q,q}(\Gamma _1)$ is rewritten as $v\in W^{n-1/q,q}(\Gamma )$. By standard elliptic theory this problem has a unique solution $\varphi \in {\dot{H}}^2(\Omega )$ and, moreover, if also $w\in W^{n-1,q}(\Omega )$ and $v\in W^{n-1/q,q}(\Gamma )$, one has $\varphi \in W^{n+1,q}(\Omega )/\mathbb {C}_\Omega$ and there is a positive constant $c_{n,q}=c_{n,q}(\Omega )$ such that $\Vert \varphi \Vert _{W^{n+1,q}(\Omega )/\mathbb {C}_\Omega }\le c_{n,q}\left( \Vert w\Vert _{W^{n-1,q}(\Omega )}+\Vert v\Vert _{v\in W^{n-1/q,q}(\Gamma _1)}\right)$.

We thus found the unique solution $\textbf{r}=\nabla \varphi \in H^1_{{{\,\textrm{curl}\,}}0}(\Omega )$ of (4.8) and (4.9) holds. Taking $q=2$ in (4.9) we then get that $S\in \mathcal {L}(H^n_{\mathcal {P}^c};\mathbb {H}^n_{\mathcal {L}})$ for $n\in \mathbb {N}$, $n\le r$, with inverse being trivially given as in the statement, where the Divergence Theorem shows that $S^{-1}\in \mathcal {L}(\mathbb {H}^n_{\mathcal {L}};H^n_{\mathcal {P}^c})$. Let now assume that $w\in C^{r-1}({\overline{\Omega }})$, $v\in C^{r-1}(\Gamma _1)$ and that (4.7) hold. When $r\in \mathbb {N}$, since $\Gamma _1$ is compact, by (4.9) we get $\textbf{r}\in \bigcap _{q\in [2,\infty )} W^{r,q}(\Omega )^3$. By Stein Extension Theorem (see [1, Chapter 5, Theorem 5.24, p. 154]) we can then extend $\textbf{r}$ to $\textbf{r}\in \bigcap _{q\in [2,\infty )} W^{r,q}(\mathbb {R}^3)^3$ and, by Morrey’s Theorem, $\textbf{r}\in C^{r-1}(\mathbb {R}^3)^3$. Since the extension operator in Stein Extension Theorem is independent on r, the case $r=\infty$ follows from the previous one. $\square$

As a byproduct of Lemma 4.2, we can now derive the following non-trivial density result. It will be useful in the sequel.

Lemma 4.3

The space $\mathbb {Y}=\{(\textbf{r}_{|\Omega },v)\in \mathbb {H}^1_{\mathcal {L}}: \textbf{r}\in C^{r-1}(\mathbb {R}^3)^3\}$ is dense in $\mathbb {H}^1_{\mathcal {L}}$.

Proof

Using Lemma 4.2 we can endow $\mathbb {H}^1_{\mathcal {L}}$ with the inner product $((\textbf{r},v), (\varvec{\phi },\psi ))_{\mathbb {H}^1_{\mathcal {L}}}:=\int _\Omega {{\,\textrm{div}\,}}\textbf{r}{{\,\textrm{div}\,}}{\overline{\varvec{\phi }}} + \int _{\Gamma _1} (\nabla _\Gamma v,\nabla _\Gamma \psi )_\Gamma +\int _{\Gamma _1} v{\overline{\psi }}$, which induces a norm equivalent to the one inherited from $H^1(\Omega )^3\times H^1(\Gamma _1)$. To prove our assertion we then take $(\textbf{r}_0,v_0)\in \mathbb {H}^1_{\mathcal {L}}$ such that

$$\begin{aligned} \int _\Omega {{\,\textrm{div}\,}}\textbf{r}_0{{\,\textrm{div}\,}}{\overline{\varvec{\phi }}} + \int _{\Gamma _1} (\nabla _\Gamma v_0,\nabla _\Gamma \psi )_\Gamma +\int _{\Gamma _1} v_0{\overline{\psi }}=0\quad \text {for all }(\varvec{\phi },\psi )\in \mathbb {Y}, \end{aligned}$$

(4.11)

and we claim that $(\textbf{r}_0,v_0)=0$. We now take $\varvec{\phi }_1\in \mathcal {D}(\Omega )^3$, so $\int _\Omega {{\,\textrm{div}\,}}\varvec{\phi }_1=0$. By Lemma 4.2 then there is $\varvec{\phi }\in C^{r-1}(\mathbb {R}^3)^3$ such that ${{\,\textrm{div}\,}}\varvec{\phi }={{\,\textrm{div}\,}}\varvec{\phi }_1$ and ${{\,\textrm{curl}\,}}\varvec{\phi }=0$ in $\Omega$, with $\varvec{\phi }\cdot \varvec{\nu }=0$ on $\Gamma$. Consequently, $(\varvec{\phi }_{|\Omega },0)\in \mathbb {Y}$ and, by (4.11), we have

$$\begin{aligned} \int _\Omega {{\,\textrm{div}\,}}\textbf{r}_0{{\,\textrm{div}\,}}\overline{\varvec{\phi }_1} =\int _\Omega {{\,\textrm{div}\,}}\textbf{r}_0{{\,\textrm{div}\,}}{\overline{\varvec{\phi }}}= 0. \end{aligned}$$

(4.12)

Since $\varvec{\phi }_1\in \mathcal {D}(\Omega )^3$ is arbitrary, taking $\varvec{\phi }_1=(\varphi ,0,0), (0,\varphi ,0), (0,0,\varphi )$ with $\varphi \in \mathcal {D}(\Omega )$ we then get that ${{\,\textrm{div}\,}}\textbf{r}_0\in H^1(\Omega )$ and $\nabla {{\,\textrm{div}\,}}\textbf{r}_0=0$ in $\Omega$ in the sense of distributions. $\Omega$ being connected we thus have ${{\,\textrm{div}\,}}\textbf{r}_0\in \mathbb {C}_\Omega$. Moreover, since $(\textbf{r}_0,v_0)\in \mathbb {H}^1_{\mathcal {L}}$, by (1.8) and the Divergence Theorem we have

$$\begin{aligned} {{\,\textrm{div}\,}}\textbf{r}_0=-\tfrac{1}{|\Omega |}\int _{\Gamma _1}v_0\qquad \text {in }\Omega , \end{aligned}$$

(4.13)

where $|\Omega |$ denotes the volume of $\Omega$. Plugging (4.13) into (4.11), since $(\varvec{\phi },\psi )\in \mathbb {H}^1_{\mathcal {L}}$, using the Divergence Theorem again we get

$$\begin{aligned} \frac{1}{|\Omega |}\int _{\Gamma _1}v_0\int _{\Gamma _1}{\overline{\psi }} + \int _{\Gamma _1} (\nabla _\Gamma v_0,\nabla _\Gamma \psi )_\Gamma +\int _{\Gamma _1} v_0{\overline{\psi }}=0\quad \text {for all } (\varvec{\phi },\psi )\in \mathbb {Y}. \end{aligned}$$

(4.14)

We now claim that (4.14) actually holds for all $\psi \in H^1(\Gamma _1)$. To prove our claim we can assume, by density, that $\psi \in C^r(\Gamma _1)$. Applying Lemma 4.2 again with $w=\tfrac{B}{\rho _0|\Omega |}\int _{\Gamma _1}\psi$ and $v=\psi$, so $w\in C^r({\overline{\Omega }})$, $v\in C^r(\Gamma _1)$ and $\rho _0\int _\Omega w-B\int _{\Gamma _1}v=0$, we then get that there is $\varvec{\phi }\in [C^{r-1}(\mathbb {R}^3)]^3$ such that ${{\,\textrm{curl}\,}}\varvec{\phi }=0$ in $\Omega$, $\varvec{\phi }\cdot \varvec{\nu }=-\psi$ on $\Gamma _1$ and $\varvec{\phi }\cdot \varvec{\nu }=0$ on $\Gamma _0$, so that $(\varvec{\phi },\psi )\in \mathbb {Y}$ and consequently (4.14) holds, proving our claim. Using it we can take $\psi =v_0$ in (4.14), getting $v_0=0$ and, by (4.13), ${{\,\textrm{div}\,}}\textbf{r}_0=0$. Then $((\textbf{r}_0,v_0),(\textbf{r}_0,v_0))_{\mathbb {H}^1_{\mathcal {L}}}=0$ and consequently $(\textbf{r}_0,v_0)=0$. $\square$

Our final preliminary result is a second non-trivial density result. To state it we introduce the Hilbert space

$$\begin{aligned} V_{\mathcal {L}}=\{(\textbf{r},v)\in H^1(\Omega )^3\times H^1(\Gamma _1):\quad \textbf{r}\cdot \varvec{\nu }=-v\quad \text {on }\Gamma _1, \quad \textbf{r}\cdot \varvec{\nu }=0\quad \hbox { on}\ \Gamma _0\}, \end{aligned}$$

(4.15)

equipped with the norm inherited from the product. Trivially $V_{\mathcal {L}}\hookrightarrow L^2_{\mathcal {L}}$.

Lemma 4.4

The embedding $V_{\mathcal {L}}\hookrightarrow L^2_{\mathcal {L}}$ is dense.

Proof

Using the standard inner product of $L^2_{\mathcal {L}}$, to prove our assertion we take $(\textbf{r}_0,v_0)\in L^2_{\mathcal {L}}$ such that

$$\begin{aligned} \int _\Omega \textbf{r}_0\cdot {\overline{\varvec{\phi }}} + \int _{\Gamma _1} v_0{\overline{\psi }}=0\quad \text {for all }(\varvec{\phi },\psi )\in V_{\mathcal {L}}, \end{aligned}$$

(4.16)

and we claim that $(\textbf{r}_0,v_0)=0$. First taking in (4.16) test functions $(\varvec{\phi },\psi )=(\varvec{\phi }_1,0)$, with $\varvec{\phi }_1\in \mathcal {D}(\Omega )^3$, so trivially $(\varvec{\phi },\psi )\in V_{\mathcal {L}}$, we get $\int _\Omega \textbf{r}_0\cdot \varvec{\phi }_1=0$ and consequently, $\varvec{\phi }_1$ being arbitrary, $\textbf{r}_0=0$. Hence, (4.16) reads as

$$\begin{aligned} \int _{\Gamma _1} v_0{\overline{\psi }}=0\quad \hbox { for all}\ (\varvec{\phi },\psi )\in V_{\mathcal {L}}\end{aligned}$$

(4.17)

We now claim that (4.17) actually holds for all $\psi \in H^1(\Gamma _1)$, from which one immediately gets $v_0=0$ and concludes the proof. To prove our claim, we apply Lemma 4.2 once again with $w=\tfrac{B}{\rho _0|\Omega |}\int _{\Gamma _1}\psi$ and $v=\psi$. It follows that there is $\varvec{\phi }\in H^1_{{{\,\textrm{curl}\,}}0}(\Omega )$ such that $(\varvec{\phi },\psi )\in V_{\mathcal {L}}$, so (4.17) holds true and our claim is proved. $\square$

4.2 The main isomorphism

To deal with problem $({\mathcal {L}})$, we first construct the main isomorphism between its phase space $\mathcal {H}^1_{\mathcal {L}}$ and the phase space $\mathcal {H}^1_{\mathcal {P}^c}$. We set

$$\begin{aligned} F_{_{{\mathcal {L}}{\mathcal {P}^c}}}(\textbf{r},v,\textbf{s},z)=\left( {\dot{u}}, v, -B{{\,\textrm{div}\,}}\textbf{r}/{\rho _0},z\right) \qquad \text {for all } (\textbf{r},v,\textbf{s},z)\in \mathcal {H}^1_{\mathcal {L}}, \end{aligned}$$

(4.18)

where ${\dot{u}}\in {\dot{H}}^1(\Omega )$ is the unique solution of the equation

$$\begin{aligned} -\nabla {\dot{u}}=\textbf{s}\end{aligned}$$

(4.19)

given by Lemma 4.1. By (1.6), (1.9) and the Divergence Theorem we have $F_{_{{\mathcal {L}}{\mathcal {P}^c}}}\in \mathcal {L}(\mathcal {H}^1_{\mathcal {L}}; \dot{\mathcal {H}}^1_{\mathcal {P}^c})$. We also set

$$\begin{aligned} F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}({\dot{u}}, v,w,z)=(\textbf{r},v,-\nabla {\dot{u}},z)\qquad \text {for all } ({\dot{u}},v,w,z)\in \mathcal {H}^1_{\mathcal {P}^c}, \end{aligned}$$

(4.20)

where $\textbf{r}\in H^1_{{{\,\textrm{curl}\,}}0}(\Omega )$ is the unique solution of problem (4.8) given by Lemma 4.2. Since for any ${\dot{u}}\in {\dot{H}}^1(\Omega )$ we have ${{\,\textrm{curl}\,}}\nabla {\dot{u}}=0$ in $\mathcal {D}'(\Omega )$, see [12, Chapter 3, p. 202], one trivially gets $F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}\in \mathcal {L}(\dot{\mathcal {H}}^1_{\mathcal {P}^c};\mathcal {H}^1_{\mathcal {L}})$. By (3.11) and (4.5) the operators $F_{_{{\mathcal {L}}{\mathcal {P}^c}}}$ and $F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}$ trivially induce the operators $\Phi _{_{{\mathcal {L}}{\mathcal {P}^c}}}\in \mathcal {L}(Y^1_{\mathcal {L}}; {\dot{Y}}^1_{\mathcal {P}^c})$ and $\Phi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}\in \mathcal {L}({\dot{Y}}^1_{\mathcal {P}^c}; Y^1_{\mathcal {L}})$ given by

$$\begin{aligned} (\Phi _{_{{\mathcal {L}}{\mathcal {P}^c}}}U_{\mathcal {L}})(t)=F_{_{{\mathcal {L}}{\mathcal {P}^c}}}U_{\mathcal {L}}(t), \quad (\Phi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}{\dot{U}}_{\mathcal {P}})(t)=F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}{\dot{U}}_{\mathcal {P}}(t)\quad \text {for all } t\in \mathbb {R}. \end{aligned}$$

(4.21)

The following result points out all properties of these operators needed in the sequel.

Proposition 4.5

The operator $F_{_{{\mathcal {L}}{\mathcal {P}^c}}}$ is a bijective isomorphism between $\mathcal {H}^1_{\mathcal {L}}$ and $\dot{\mathcal {H}}^1_{\mathcal {P}^c}$ having $F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}$ as inverse. Consequently, $\Phi _{_{{\mathcal {L}}{\mathcal {P}^c}}}$ is a bijective isomorphism between $Y^1_{\mathcal {L}}$ and ${\dot{Y}}^1_{\mathcal {P}^c}$ having $\Phi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}$ as inverse. Moreover, for all $n\in \mathbb {N}$, $2\le n\le r$, they restrict to

$$\begin{aligned} \begin{aligned} F_{_{{\mathcal {L}}{\mathcal {P}^c}}}\in \mathcal {L}(\mathcal {H}^n_{\mathcal {L}}; \dot{\mathcal {H}}^n_{\mathcal {P}^c}), \qquad&F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}\in \mathcal {L}(\dot{\mathcal {H}}^n_{\mathcal {P}^c};\mathcal {H}^n_{\mathcal {L}}),\\ \Phi _{_{{\mathcal {L}}{\mathcal {P}^c}}}\in \mathcal {L}(Y^n_{\mathcal {L}}; {\dot{Y}}^n_{\mathcal {P}^c}),\qquad&\Phi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}\in \mathcal {L}({\dot{Y}}^n_{\mathcal {P}^c}; Y^n_{\mathcal {L}}), \end{aligned} \end{aligned}$$

(4.22)

and, since

$$\begin{aligned} F_{_{{\mathcal {L}}{\mathcal {P}^c}}}D^{n-1}_{\mathcal {L}}={\dot{D}}^{n-1}_{\mathcal {P}^c}, \end{aligned}$$

(4.23)

$F_{_{{\mathcal {L}}{\mathcal {P}^c}}}$ and $F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}$ also restrict to $F_{_{{\mathcal {L}}{\mathcal {P}^c}}}\in \mathcal {L}(D^{n-1}_{\mathcal {L}}; {\dot{D}}^{n-1}_{\mathcal {P}^c})$ and $F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}\in \mathcal {L}({\dot{D}}^{n-1}_{\mathcal {P}^c};D^{n-1}_{\mathcal {L}})$.

Proof

To recognize that $F_{_{{\mathcal {L}}{\mathcal {P}^c}}}$ and $F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}$ are the inverse of each other, we simply recall that the identifications (4.1) and $\dot{\mathcal {H}}_{\mathcal {P}^c}^n={\dot{H}}^n(\Omega )\times H_{\mathcal {P}^c}^n\times H^{n-1}(\Gamma _1)$, when $n=1$, read as $\mathcal {H}^1_{\mathcal {L}}=\mathbb {H}^1_{\mathcal {L}}\times H^0_{{{\,\textrm{curl}\,}}0}(\Omega )\times L^2(\Gamma _1)$ and $\dot{\mathcal {H}}_{\mathcal {P}^c}^1={\dot{H}}^1(\Omega )\times H_{\mathcal {P}^c}^1\times L^2(\Gamma _1)$. Using them together with (4.18) and (4.20), we can represent the operators $F_{_{{\mathcal {L}}{\mathcal {P}^c}}}$ and $F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}$ in the matrix form

$$\begin{aligned} F_{_{{\mathcal {L}}{\mathcal {P}^c}}}= \begin{pmatrix}0 &{}\quad G^{-1} &{}\quad 0\\ S^{-1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\text {Id}\end{pmatrix}\qquad \text {and}\quad F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}= \begin{pmatrix}0 &{}\quad S &{}\quad 0\\ G &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \text {Id}\end{pmatrix}, \end{aligned}$$

(4.24)

where G and S are the bijective isomorphism introduced in Lemmas 4.1 and 4.2, so trivially $F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}=F_{_{{\mathcal {L}}{\mathcal {P}^c}}}^{-1}$. The asserted properties of $\Phi _{_{{\mathcal {L}}{\mathcal {P}^c}}}$ and $\Phi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}$ then follow trivially. Moreover, for $n\in \mathbb {N}$, $2\le n\le r$, using (4.24), Lemmas 4.1–4.2 and recalling (3.11), (4.5), we get (4.22). Next, from (4.22) and (4.23) it trivially follow that $F_{_{{\mathcal {L}}{\mathcal {P}^c}}}\in \mathcal {L}(D^{n-1}_{\mathcal {L}}; {\dot{D}}^{n-1}_{\mathcal {P}^c})$ and $F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}\in \mathcal {L}({\dot{D}}^{n-1}_{\mathcal {P}^c};D^{n-1}_{\mathcal {L}})$, so we just have to prove (4.23) to complete the proof.

By (3.3), (3.4), (4.3) and (4.18)–(4.19) to prove (4.23) reduces to prove that for all $(\textbf{r}_0,v_0,\textbf{r}_1,v_1)\in \mathcal {H}^n_{\mathcal {L}}$ and $(u_0,v_0,u_1,v_1)\in \dot{\mathcal {H}}^n_{\mathcal {P}^c}$ such that

$$\begin{aligned} \rho _0u_1=-B{{\,\textrm{div}\,}}\textbf{r}_0\qquad \text {and}\quad \nabla u_0=-\textbf{r}_1 \end{aligned}$$

(4.25)

the compatibility conditions (3.2) and (4.2) are equivalent. To prove this fact, we preliminarily remark that one easily gets that for all $\textbf{r}\in H^0_{{{\,\textrm{curl}\,}}0}(\Omega )$ and $i\in \mathbb {N}$ ${{\,\textrm{curl}\,}}\Delta ^i\textbf{r}=0$ and $\Delta ^i\textbf{r}=(\nabla {{\,\textrm{div}\,}})^i\textbf{r}$ in $\mathcal {D}'(\Omega )^3$, Then, starting from (4.25) and using them one gets that, in $\Omega$, one has $\Delta ^iu_0=-{{\,\textrm{div}\,}}\Delta ^{i-1}\textbf{r}_1$ for $i=1,\ldots ,[n/2]$, $\nabla \Delta ^iu_0=-\Delta ^i\textbf{r}_1$ for $i=0,\ldots ,[(n-1)/2]$, $\rho _0\Delta ^iu_1=-B{{\,\textrm{div}\,}}\Delta ^i\textbf{r}_0$, for $i=0,\ldots ,[(n-1)/2]$ and $\rho _0\nabla \Delta ^iu_1=-B\Delta ^{i+1}\textbf{r}_0$ for $i=0,\ldots ,[n/2]-1$. Passing these relations to traces one easily checks that (3.2) and (4.2) are equivalent. $\square$

4.3 Abstract analysis of problems $({\mathcal {L}})$ and $({\mathcal {L}}_0)$

To deal with problem $({\mathcal {L}}_0)$ in a semigroup setting, we reduce it to the first-order problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \textbf{r}_t=\textbf{s}\qquad &{}\text {in }\mathbb {R}\times \Omega ,\\ v_t =z\qquad &{}\text {on }\mathbb {R}\times \Gamma _1,\\ \rho _0\textbf{s}_t-B\nabla {{\,\textrm{div}\,}}\textbf{r}=0\qquad &{}\text {in }\mathbb {R}\times \Omega ,\\ {{\,\textrm{curl}\,}}\textbf{r}=0\qquad &{}\text {in }\mathbb {R}\times \Omega ,\\ \mu z_t- {{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v)+\delta z+\kappa v-B{{\,\textrm{div}\,}}\textbf{r}=0\qquad &{}\text {on }\mathbb {R}\times \Gamma _1,\\ \textbf{r}\cdot {\varvec{\nu }} =0 \quad \text {on }\mathbb {R}\times \Gamma _0,\qquad \textbf{r}\cdot {\varvec{\nu }} =-v\qquad &{}\text {on }\mathbb {R}\times \Gamma _1,\\ \textbf{r}(0,x)=\textbf{r}_0(x),\quad \textbf{s}(0,x)=\textbf{r}_1(x) &{} \text {in }\Omega ,\\ v(0,x)=v_0(x),\quad z(0,x)=v_1(x) &{} \text {on }\Gamma _1. \end{array}\right. } \end{aligned}$$

(4.26)

More formally, working in the phase space $\mathcal {H}^1_{\mathcal {L}}$, in which equations (4.26)$_4$ and (4.26)$_6$ implicitly hold, we introduce the unbounded operator $A_{\mathcal {L}}: D(A_{\mathcal {L}})\subset \mathcal {H}^1_{\mathcal {L}}\rightarrow \mathcal {H}^1_{\mathcal {L}}$ given by

$$\begin{aligned} D(A_{\mathcal {L}})=D^1_{\mathcal {L}}=\{(\textbf{r},v,\textbf{s},z)\in \mathcal {H}^2_{\mathcal {L}}: (\textbf{s},z)\in \mathbb {H}^1_{\mathcal {L}}\},\end{aligned}$$

(4.27)

$$\begin{aligned} A_{\mathcal {L}}\begin{pmatrix}\textbf{r}\\ v\\ \textbf{s}\\ z\end{pmatrix} = \begin{pmatrix}-\textbf{s}\\ -z\\ -(B/\rho _0)\nabla {{\,\textrm{div}\,}}\textbf{r}\\ \frac{1}{\mu }\left[ -{{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v)+\delta z+\kappa v-B{{\,\textrm{div}\,}}\textbf{r}_{|\Gamma _1}\right] \end{pmatrix}, \end{aligned}$$

(4.28)

together with the abstract equation and Cauchy problem

$$\begin{aligned} U_{\mathcal {L}}'+A_{\mathcal {L}}U_{\mathcal {L}}=0\qquad \text {in }\mathcal {H}^1_{\mathcal {L}}, \end{aligned}$$

(4.29)

$$\begin{aligned} U_{\mathcal {L}}'+A_{\mathcal {L}}U_{\mathcal {L}}=0\qquad \hbox { in}\ \mathcal {H}^1_{\mathcal {L}}, \qquad U_{\mathcal {L}}(0)=U_{0{\mathcal {L}}}\in \mathcal {H}^1_{\mathcal {L}}. \end{aligned}$$

(4.30)

The following result shows that the restriction of problem (4.30) to $\dot{\mathcal {H}}^1_{\mathcal {P}^c}$ is essentially equivalent to problem (3.25), and hence, it is well-posed.

Theorem 4.6

(Well-posedness for (4.30)) The operator $-A_{\mathcal {L}}$ is densely defined and it generates on $\mathcal {H}^1_{\mathcal {L}}$ the strongly continuous group $\{{\mathcal {L}}^1(t),t\in \mathbb {R}\}$ given by

$$\begin{aligned} {\mathcal {L}}^1(t)=F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}{\mathcal {P}}_c^1(t) F_{_{{\mathcal {L}}{\mathcal {P}^c}}}\qquad \text {for all }t\in \mathbb {R}, \end{aligned}$$

(4.31)

and hence similar to the group $\{{\mathcal {P}}_c^1(t),t\in \mathbb {R}\}$ in Corollary 3.16. Consequently, for any $U_{0{\mathcal {L}}}\in \mathcal {H}^1_{\mathcal {L}}$, problem (4.30) has a unique generalized solution $U_{\mathcal {L}}\in Y^1_{\mathcal {L}}$ given by $U_{\mathcal {L}}(t)={\mathcal {L}}^1(t)[U_{0{\mathcal {L}}}]$ for all $t\in \mathbb {R}$ and hence continuously depending on $U_{0{\mathcal {L}}}$ in the topologies of the respective spaces. Moreover, $U_{\mathcal {L}}$ is a strong solution if and only if $U_{0{\mathcal {L}}}\in D(A_{\mathcal {L}})$. Finally, if ${\dot{U}}_{\mathcal {P}}\in {\dot{Y}}^1_{\mathcal {P}^c}$ denotes the unique generalized solution of problem (3.25) with data ${\dot{U}}_{0{\mathcal {P}}}=F_{_{{\mathcal {L}}{\mathcal {P}^c}}}U_{0{\mathcal {L}}}$, one has $U_{\mathcal {L}}=\Phi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}{\dot{U}}_{\mathcal {P}^c}$.

Proof

By standard semigroup theory formula (4.31) defines on $\mathcal {H}^1_{\mathcal {L}}$ a strongly continuous group $\{{\mathcal {L}}^1(t),t\in \mathbb {R}\}$, similar to $\{{\mathcal {P}}_c^1(t),t\in \mathbb {R}\}$. Its generator is given by the operator $-B_1$ defined by $D(B_1)=F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}D({\dot{A}}_{\mathcal {P}^c})$ (so $B_1$ is densely defined) and $B_1=F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}{\dot{A}}_{\mathcal {P}^c}F_{_{{\mathcal {L}}{\mathcal {P}^c}}}$. We now claim that $B_1=A_{\mathcal {L}}$. By combining (3.34), (4.27) and (4.23) we get $D(B_1)=D^1_{\mathcal {L}}=D(A_{\mathcal {L}})$. Moreover, using (4.18) and (4.19), for any $U=(\textbf{r},v,\textbf{s},z)\in D(A_{\mathcal {L}})$ we have

$$\begin{aligned} B_1U=F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}{\dot{A}}_{\mathcal {P}}F_{_{{\mathcal {L}}{\mathcal {P}^c}}}U=F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}{\dot{A}}_{\mathcal {P}}({\dot{u}},v,-B{{\,\textrm{div}\,}}\textbf{r}/\rho _0,z)^t \end{aligned}$$

where ${\dot{u}}\in {\dot{H}}^2(\Omega )$ is the unique solution of $-\nabla {\dot{u}}=\textbf{s}$. Consequently, using also (3.23), (3.34) and (4.20), one gets

$$\begin{aligned} B_1 U=\begin{pmatrix}\textbf{t}\\ -z\\ -\tfrac{B}{\rho _0}\nabla {{\,\textrm{div}\,}}\textbf{r}\\ \frac{1}{\mu }\left[ -{{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v)+\delta z+\kappa v-B{{\,\textrm{div}\,}}\textbf{r}_{|\Gamma _1}\right] \end{pmatrix} \end{aligned}$$

(4.32)

where $\textbf{t}\in H^1_{{{\,\textrm{curl}\,}}0}(\Omega )$ is the unique solution of the problem $-B{{\,\textrm{div}\,}}\textbf{t}=B{{\,\textrm{div}\,}}\textbf{s}$ in $\Omega$, ${{\,\textrm{curl}\,}}\textbf{t}=0$ in $\Omega$, $\textbf{t}\cdot \varvec{\nu }=z$ on $\Gamma _1$, $\textbf{t}\cdot \varvec{\nu }=0$ on $\Gamma _0$. Since, by (4.27), $(\textbf{s},z)\in \mathbb {H}^1_{\mathcal {L}}$, using Lemma 4.2 we get $\textbf{t}=-\textbf{s}$, so by (4.28) and (4.32) we have $B_1U=A_{\mathcal {L}}U$, proving our claim. The proof can then be completed by using Lemma 2.1-i) and iii). $\square$

We now set the Fréchet spaces of generalized solutions of (4.29) that is, for $n\le r$,

$$\begin{aligned} \mathcal {T}^n_{\mathcal {L}}=\{U_{\mathcal {L}}\in Y^n_{\mathcal {L}}: \quad U_{\mathcal {L}}\text { is a generalized solution of }(4.29)\}, \end{aligned}$$

(4.33)

endowed with the topology inherited from $Y^n_{\mathcal {L}}$. We point out that, as a trivial consequence of Theorem 4.6, the following result holds.

Corollary 4.7

The operators $\Phi _{_{{\mathcal {L}}{\mathcal {P}^c}}}$ and $\Phi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}$ restrict to bijective isomorphisms between $\mathcal {T}^n_{\mathcal {L}}$ and $\dot{\mathcal {T}}^n_{\mathcal {P}^c}$ for all $n\in {\widetilde{\mathbb {N}}}$, $n\le r$, being the inverse of each other.

Abstract regularity properties of solutions of (4.30) are then given as follows.

Theorem 4.8

(Regularity for (4.30)) For all $n\in {\widetilde{\mathbb {N}}}$, $2\le n\le r$, one has $D(A^{n-1}_{\mathcal {L}})=D^{n-1}_{\mathcal {L}}$, the respective norms being equivalent, so the operator $\,\, {_n}A_{\mathcal {L}}$ given by (2.6) generates on $D^{n-1}_{\mathcal {L}}$ the strongly continuous group $\{{\mathcal {L}}^n(t), t\in \mathbb {R}\}$ given by

$$\begin{aligned} {\mathcal {L}}^n(t)={\mathcal {L}}^1(t)_{|D^{n-1}_{\mathcal {L}}}= F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}{\mathcal {P}}^n_c(t)F_{_{{\mathcal {L}}{\mathcal {P}^c}}}\qquad \text {for all }t\in \mathbb {R}, \end{aligned}$$

(4.34)

and hence similar to the group $\{{\mathcal {P}}^n_c(t),t\in \mathbb {R}\}$ in Corollary 3.16.

Moreover, for any $U_{0{\mathcal {L}}}\in \mathcal {H}^1_{\mathcal {L}}$, denoting by $U_{\mathcal {L}}$ the generalized solution of (4.30), for any $n\in {\widetilde{\mathbb {N}}}$, $2\le n\le r$, one has $U_{0{\mathcal {L}}}\in D^{n-1}_{\mathcal {L}}$ if and only if $U_{\mathcal {L}}\in Y^n_{\mathcal {L}}$. In this case $U_{\mathcal {L}}$ continuously depends on $U_{0{\mathcal {L}}}$ in the topologies of the respective spaces.

Proof

In the proof of Theorem 4.6 we got that $A_{\mathcal {L}}=F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}{\dot{A}}_{\mathcal {P}^c}F_{_{{\mathcal {L}}{\mathcal {P}^c}}}$. Since, by Corollary 3.16, $D({\dot{A}}^{n-1}_{\mathcal {P}^c})={\dot{D}}^{n-1}_{\mathcal {P}^c}$, by Proposition 4.5 and (2.5) one easily gets by induction that $D(A^{n-1}_{\mathcal {L}})=D^{N-1}_{\mathcal {L}}$, with equivalence of norms. By Lemma 2.1-ii) then ${_n}A_{\mathcal {L}}$ generates on $D^{n-1}_{\mathcal {L}}$ the strongly continuous group $\{{\mathcal {L}}^n(t), t\in \mathbb {R}\}$ given by ${\mathcal {L}}^n(t)={\mathcal {L}}^1(t)_{|D^{n-1}_{\mathcal {L}}}$ for all $t\in \mathbb {R}$. By (4.31) and Proposition 4.5 then, since ${\mathcal {P}}^n_c(t)={\mathcal {P}}^1_c(t)_{|{\dot{D}}^{n-1}_{\mathcal {P}^c}}$, we get formula (4.34) and the group similarity. The proof is then completed by using Corollary 4.7 and Lemma 2.1-iv) to get the asserted continuous dependence when $n\in \mathbb {N}$, while the case $n=\infty$ follows from the case $n\in \mathbb {N}$ and (4.5). $\square$

4.4 Solutions of $({\mathcal {L}})$ and $({\mathcal {L}}_0)$

To apply the abstract results in Sect. 4.3 to problems $({\mathcal {L}})$ and $({\mathcal {L}}_0)$, we first make precise which type of solutions we shall consider in the sequel. We recall that $({\mathcal {L}})_2$ and $({\mathcal {L}})_4$ are implicit in $X^1_{\mathcal {L}}$.

Definition 4.9

We say that

• i) $(\textbf{r},v)\in X^2_{\mathcal {L}}$ is a strong solution of $({\mathcal {L}})$ provided $({\mathcal {L}})_1$ holds a.e. in $\mathbb {R}\times \Omega$ and $({\mathcal {L}})_3$ holds a.e. on $\mathbb {R}\times \Gamma _1$, where ${{\,\textrm{div}\,}}\textbf{r}$ on $\mathbb {R}\times \Gamma _1$ is taken in the pointwise trace sense given in Sect. 2.2;

• ii) $(\textbf{r},v)\in X^1_{\mathcal {L}}$ is a generalized solution of $({\mathcal {L}})$ provided it is the limit in $X^1_{\mathcal {L}}$ of a sequence of strong solutions of it;

• iii) $(\textbf{r},v)\in X^1_{\mathcal {L}}$ is a weak solution of $({\mathcal {L}})$ provided the distributional identity

  $$\begin{aligned}{} & {} \int _{-\infty }^\infty \int _\Omega [\rho _0\textbf{r}_t\cdot \varvec{\phi }_t-B{{\,\textrm{div}\,}}\textbf{r}{{\,\textrm{div}\,}}\varvec{\phi }]\nonumber \\{} & {} \quad + \int _{-\infty }^\infty \int _{\Gamma _1}\left[ \mu v_t\psi _t-\sigma (\nabla _\Gamma v,\nabla _\Gamma {\overline{\psi }})_\Gamma -\delta v_t\psi -\kappa v\psi \right] =0 \end{aligned}$$

  (4.35)

  holds for all $\varvec{\phi }\in C^{r-1}_c(\mathbb {R}\times \mathbb {R}^3)^3$ such that $\varvec{\phi }\cdot \varvec{\nu }=0$ on $\mathbb {R}\times \Gamma _0$, where $\psi =-\varvec{\phi }\cdot \varvec{\nu }$ on $\mathbb {R}\times \Gamma _1$;

• iv) $(\textbf{r},v)$ is a strong, generalized or weak solution of problem $({\mathcal {L}}_0)$ provided it is a solution of $({\mathcal {L}})$ of the same type and $({\mathcal {L}})_5$–$({\mathcal {L}})_6$ hold in the space $X^1_{\mathcal {L}}$.

Remark 4.10

Definition 4.9-iii) seems to depend on r (i.e., on the regularity of $\Gamma$), but its independence on it is a trivial consequence of Theorem 1.3.

Trivially strong solutions in Definition 4.9 are also generalized ones. Moreover, strong and generalized solutions correspond to the homologous solutions of (4.29) or (4.3), as the following results shows.

Lemma 4.11

The couple $(\textbf{r},v)$ is a strong or generalized solution of $({\mathcal {L}})$ if and only if $\textbf{r}$ and v are the first two components of a solution $U_{\mathcal {L}}=(\textbf{r},v,\textbf{s},z)$ of (4.29) of the same type. In this case $\textbf{r}_t=\textbf{s}$ and $v_t=z$. The same relation occurs between solutions of $({\mathcal {L}}_0)$ and (4.30).

Proof

Since, by Theorems 4.6 and 4.8, strong solutions of (4.28) belong to $Y^2_{\mathcal {L}}$, using (4.28), the relation between strong solutions of $({\mathcal {L}})$ and (4.29) is trivial. The relation then extends to generalized solutions by a standard density argument. $\square$

By Lemma 4.11 generalized and strong solutions of $({\mathcal {L}})$ naturally arise from Theorem 4.6. While strong solutions are a.e. classical solutions, generalized solutions solve $({\mathcal {L}})_1$ and $({\mathcal {L}})_3$ in a quite indirect sense. Since couples $(\textbf{r},v)\in X^1_{\mathcal {L}}$ are not regular enough to be a.e. solutions, a classical way to characterize them would be to consider equations $({\mathcal {L}})_1$ and $({\mathcal {L}})_3$ in a distributional sense, that is in $\mathcal {D}'(\mathbb {R}\times \Omega )^3$ and in $[C^r_c(\mathbb {R}\times \Gamma _1)]'$. On the other hand, while for $(\textbf{r},v)\in X^1_{\mathcal {L}}$ equation $({\mathcal {L}})_1$ has in $\mathcal {D}'(\mathbb {R}\times \Omega )^3$ the natural form

$$\begin{aligned} \int _{-\infty }^\infty \int _\Omega \rho _0\textbf{r}_t\cdot \varvec{\phi }_t-B{{\,\textrm{div}\,}}\textbf{r}{{\,\textrm{div}\,}}\varvec{\phi }=0\qquad \text {for all } \varvec{\phi }\in [\mathcal {D}(\mathbb {R}\times \Omega )]^3, \end{aligned}$$

(4.36)

equation $({\mathcal {L}})_3$ cannot be written in a distributional sense unless the term ${{\,\textrm{div}\,}}\textbf{r}$ in it, merely belonging to $C(\mathbb {R};L^2(\Omega ))$, has some trace sense on $\mathbb {R}\times \Gamma _1$. This type of difficulty also arises when dealing with the somehow related wave equation with hyperbolic dynamical boundary conditions, studied by the author in a series of paper, see [39,40,41]. The natural solution of this difficulty consists in combining equations $({\mathcal {L}})_1$ and $({\mathcal {L}})_3$ in a single distributional identity, as done in Definition 4.9.

The following result, which will be also useful in the sequel, shows that Definition 4.9-iii) is the closest possible approximation of the notion of distributional solutions of $({\mathcal {L}})_1$ and $({\mathcal {L}})_3$. Its straightforward proof is omitted. The interested reader can find it in the preprint version of the paper, see https://arxiv.org/abs/2307.07775.

Proposition 4.12

Let $(\textbf{r},v)\in X^1_{\mathcal {L}}$ be such that ${{\,\textrm{div}\,}}\textbf{r}\in L^1_{\text {loc}}(\mathbb {R}; H^1(\Omega ))$. Then $(\textbf{r},v)$ is a weak solution of $({\mathcal {L}})$ if and only if it satisfies (4.36) and the further distributional identity

$$\begin{aligned} \int _{-\infty }^\infty \int _{\Gamma _1}\mu v_t\psi _t-\sigma (\nabla _\Gamma v,\nabla _\Gamma {\overline{\psi }})_\Gamma -\delta v_t\psi -\kappa v\psi +B{{\,\textrm{div}\,}}\textbf{r}\psi =0 \end{aligned}$$

(4.37)

for all $\psi \in C^r_c(\mathbb {R}\times \Gamma _1)$.

Proposition 4.12 allows to point out the trivial relations among the three types of solutions of $({\mathcal {L}})$ introduced in Definition 4.9.

Lemma 4.13

Let $(\textbf{r},v)\in X^1_{\mathcal {L}}$ be a solution of $({\mathcal {L}})$ according to Definition 4.9. Then strong $\Rightarrow$ generalized $\Rightarrow$ weak and, if $(\textbf{r},v)\in X^2_{\mathcal {L}}$, weak $\Rightarrow$ strong.

Proof

Strong solutions are also generalized ones and, by Proposition 4.12, also weak. Since (4.35) is stable with respect to the convergence in $X^1_{\mathcal {L}}$, we then get that generalized $\Rightarrow$ weak. To prove the final conclusion let $(\textbf{r},v)\in X^2_{\mathcal {L}}$ be a weak solution. By Proposition 4.12 it also satisfies $({\mathcal {L}})_1$ and $({\mathcal {L}})_3$ in a distributional sense. Being regular enough we can integrate by parts to show that it also satisfies them a.e., so $(\textbf{r},v)$ is a strong solution. $\square$

Remark 4.14

By the last result the same conclusions in Remark 3.5 apply as well to problem $({\mathcal {L}})$, so solutions $(\textbf{r},v)\in X^2_{\mathcal {L}}$ are equivalently strong, generalized or weak.

In view of Theorem 4.6 and Lemmas 4.11, 4.13, which provide the existence of a generalized and thus weak solution of $({\mathcal {L}}_0)$, a natural question arising is the uniqueness of weak solutions. To positively answer the question, it is useful to characterize weak solutions of $({\mathcal {L}})$ as solutions of an abstract second order ODE.

Specifically, for this purpose we shall identify in the sequel $L^2(\Omega )^3$ and $L^2(\Gamma _1)$ with their duals $[L^2(\Omega )^3]'$ and $[L^2(\Gamma _1)]'$, coherently with the distributional identification. Moreover, we shall also identify $L^2(\Omega )^3$ and $L^2(\Gamma _1)$ with their isometric copies $L^2(\Omega )^3\times \{0\}$ and $\{0\}\times L^2(\Gamma _1)$ contained in $L^2_{\mathcal {L}}=L^2(\Omega )^3\times L^2(\Gamma _1)$. As a consequence, we shall identify $L^2_{\mathcal {L}}$ with its dual ${L^2_{\mathcal {L}}}'$ according to the identity $\langle \textbf{u},\textbf{w}\rangle _{L^2_{\mathcal {L}}}=(\textbf{u},\overline{\textbf{w}})_{L^2_{\mathcal {L}}}$ for all $\textbf{u,w}\in L^2_{\mathcal {L}}$. By Lemma 4.4 we can introduce the chain of dense embeddings, or Gel’fand triple, $V_{\mathcal {L}}\hookrightarrow L^2_{\mathcal {L}}\simeq {L^2_{\mathcal {L}}}'\hookrightarrow V_{\mathcal {L}}'$, in view of which the last identity particularizes to

$$\begin{aligned} \langle \textbf{u},\textbf{w}\rangle _{V_{\mathcal {L}}}=(\textbf{u},\overline{\textbf{w}})_{L^2_{\mathcal {L}}}\qquad \text {for all } \textbf{u}\in L^2_{\mathcal {L}}\text { and }\textbf{w}\in V_{\mathcal {L}}. \end{aligned}$$

(4.38)

We also introduce the multiplication operator $R\in \mathcal {L}(L^2_{\mathcal {L}})$, the projection operator $\Pi _{\Gamma _1}\in \mathcal {L}(L^2_{\mathcal {L}}; L^2(\Gamma _1))$ and $M\in \mathcal {L}(V_{\mathcal {L}};V_{\mathcal {L}}')$ given by

$$\begin{aligned} R(\textbf{r},v)=(\rho _0 \textbf{r},\mu v),\qquad \Pi _{\Gamma _1}(\textbf{r},v)=(0,v)\qquad \text {for all }(\textbf{r},v)\in L^2_{\mathcal {L}}, \end{aligned}$$

(4.39)

and, for all $(\textbf{r},v), (\varvec{\phi },\psi )\in V_{\mathcal {L}}$, by

$$\begin{aligned} \langle M(\textbf{r},v),(\varvec{\phi },\psi )\rangle _{V_{\mathcal {L}}}=-B\int _\Omega {{\,\textrm{div}\,}}\textbf{r}{{\,\textrm{div}\,}}\varvec{\phi }-\int _{\Gamma _1}\sigma (\nabla _\Gamma v,\nabla _\Gamma {\overline{\psi }})_\Gamma -\int _{\Gamma _1}kv\psi . \end{aligned}$$

(4.40)

We then get the following result.

Proposition 4.15

For any $\textbf{u}=(\textbf{r},v)\in X^1_{\mathcal {L}}$ the following properties are equivalent:

• i) $(\textbf{r},v)$ is a weak solution of $({\mathcal {L}})$;

• ii) $R\textbf{u}'\in C^1(\mathbb {R};V_{\mathcal {L}}')$ and

  $$\begin{aligned} (R\textbf{u}')' +\delta \Pi _{\Gamma _1}\textbf{u}'-M\textbf{u}=0\qquad \text {in } C(\mathbb {R};V_{\mathcal {L}}'); \end{aligned}$$

  (4.41)

• iii) $(\textbf{r},v)$ satisfies the generalized distributional identity

  $$\begin{aligned}{} & {} \int _s^t\left\{ \int _\Omega \rho _0\textbf{r}_t\cdot \varvec{\phi }_t-B{{\,\textrm{div}\,}}\textbf{r}{{\,\textrm{div}\,}}\varvec{\phi }+ \int _{\Gamma _1}\mu v_t\psi _t-\sigma (\nabla _\Gamma v,\nabla _\Gamma {\overline{\psi }})_\Gamma \right. \nonumber \\{} & {} \quad \left. -\int _{\Gamma _1}\delta v_t\psi +\kappa v\psi \right\} =\left[ \rho _0\int _\Omega \textbf{r}_t\cdot \varvec{\phi }+\int _{\Gamma _1}\mu v_t\psi \right] _s^t \end{aligned}$$

  (4.42)

  for all $s,t\in \mathbb {R}$ and $(\varvec{\phi },\psi )\in C(\mathbb {R};V_{\mathcal {L}})\cap C^1(\mathbb {R}; L^2_{\mathcal {L}})$.

Proof

We shall prove the implications i)$\Rightarrow$ ii)$\Rightarrow$ iii) $\Rightarrow$ i), starting from the trivial one. Indeed (4.42) reduces to (4.35) when taking test functions as in Definition 4.9-iii), hence iii) $\Rightarrow$ i).

Moreover, ii) $\Rightarrow$ iii). Indeed, by ii), we have $R\textbf{u}'\in C(\mathbb {R};V_{\mathcal {L}})\cap C^1(\mathbb {R};V_{\mathcal {L}}')$. By a trivial extension of the Leibnitz rule then, for any $\varvec{\psi }=(\varvec{\phi },\psi )\in C(\mathbb {R};V_{\mathcal {L}})\cap C^1(\mathbb {R};L^2_{\mathcal {L}})$ we have $\frac{d}{dt}(R\textbf{u}',\overline{\varvec{\psi }})_{L^2_{\mathcal {L}}}=\langle (R\textbf{u}')',\varvec{\psi }\rangle _{V_{\mathcal {L}}} +(R\mathbf {u'},\overline{\varvec{\psi }}')_{L^2_{\mathcal {L}}}\in C(\mathbb {R})$, so $(R\textbf{u}',\overline{\varvec{\psi }})_{L^2_{\mathcal {L}}}\in C^1(\mathbb {R})$. Consequently, integrating from s to t and using (4.41), we get $\left[ (R\textbf{u}',\overline{\varvec{\psi }})_{L^2_{\mathcal {L}}}\right] _s^t=\int _s^t -(\delta \Pi _{\Gamma _1}\textbf{u}',\overline{\varvec{\psi }})_{L^2_{\mathcal {L}}}+\langle M\textbf{u},\varvec{\psi }\rangle _{V_{\mathcal {L}}} +(R\textbf{u}', \overline{\varvec{\psi }}')_{L^2_{\mathcal {L}}}$ which, using (4.39)–(4.40), is (4.42).

To complete the proof, we then just have to prove that i)$\Rightarrow$ ii). Let $(\textbf{r},v)$ be a weak solution of $({\mathcal {L}})$. Taking in (4.35) test functions $\varvec{\phi }(t,x)=\varphi (t)\textbf{w}(x)$, with $\varphi \in \mathcal {D}(\mathbb {R})$ and $\textbf{w}\in C^{r-1}_c(\mathbb {R}^3)^3$ such that $\textbf{w}\cdot \varvec{\nu }=0$ on $\Gamma _0$, and denoting $\chi =-\textbf{w}\cdot \varvec{\nu }_{|\Gamma _1}$, we get

$$\begin{aligned}{} & {} \rho _0\int _{-\infty }^\infty \varphi '\int _\Omega \textbf{r}_t\cdot \textbf{w}-B \int _{-\infty }^\infty \varphi \int _\Omega {{\,\textrm{div}\,}}\textbf{r}{{\,\textrm{div}\,}}\textbf{w}+ \int _{-\infty }^\infty \varphi '\int _{\Gamma _1} \mu v_t\chi \nonumber \\{} & {} \quad -\int _{-\infty }^\infty \varphi \int _{\Gamma _1}\sigma (\nabla _\Gamma v,\nabla _\Gamma {\overline{\chi }})_\Gamma +(\delta v_t+\kappa v)\chi =0, \end{aligned}$$

(4.43)

and we remark that $(\textbf{w},\chi )\in V_{\mathcal {L}}$.

We claim that (4.43) actually holds for all $(\textbf{w},\chi )\in V_{\mathcal {L}}$. Fixing such a $(\textbf{w},\chi )$ we remark that, by (4.15) and the Divergence Theorem, the couple $\left( -\frac{B}{\rho _0}{{\,\textrm{div}\,}}\textbf{w},\chi \right)$ satisfies (4.7), so by Lemma 4.2 there is a unique $\textbf{w}_1\in H^1_{{{\,\textrm{curl}\,}}0}(\Omega )$ such that ${{\,\textrm{div}\,}}\textbf{w}_1={{\,\textrm{div}\,}}\textbf{w}$ in $\Omega$ and $(\textbf{w}_1,\chi )\in \mathbb {H}^1_{\mathcal {L}}$. We now set $\textbf{w}_2=\textbf{w}-\textbf{w}_1$, so

$$\begin{aligned} \textbf{w}= \textbf{w}_1+\textbf{w}_2,\quad {{\,\textrm{div}\,}}\textbf{w}_2=0\quad \text {in } \Omega ,\text { and}\quad \textbf{w}_2\cdot \varvec{\nu }=0\quad \text {on }\Gamma . \end{aligned}$$

(4.44)

By linearity we shall prove (4.43) for $\textbf{w}=\textbf{w}_i$, $i=1,2$. When $\textbf{w}=\textbf{w}_2$, by (4.44) we can rewrite (4.43) as

$$\begin{aligned} \rho _0\int _{-\infty }^\infty \varphi '\int _\Omega \textbf{r}_t\cdot \textbf{w}_2=0. \end{aligned}$$

(4.45)

Now, since $(\textbf{r},v)\in X^1_{\mathcal {L}}$, by (1.7)–(1.10) and Lemma 4.1 for any $t\in \mathbb {R}$ there is $\varphi _2(t)\in {\dot{H}}^2(\Omega )$ such that $\textbf{r}(t)=-\nabla \varphi _2(t)$. Consequently, by (4.44) and the Divergence Theorem, we have

$$\begin{aligned} \int _\Omega \textbf{r}(t)\cdot \textbf{w}_2\!=\!-\int _\Omega \nabla \varphi _2(t)\cdot \textbf{w}_2\!=\!\int _\Omega \varphi _2(t)\!{{\,\textrm{div}\,}}\textbf{w}_2\!-\!\int _\Gamma \varphi _2(t)\,\textbf{w}_2\cdot \varvec{\nu }\!=\!0\!\!\quad \text {for all }t\!\in \!\mathbb {R}, \end{aligned}$$

from which (4.45) trivially follows.

When $\textbf{w}=\textbf{w}_1$, since $(\textbf{w}_1,\chi )\in \mathbb {H}^1_{\mathcal {L}}$, using Lemma 4.3 we can suppose by density that $(\textbf{w}_1,\chi )\in \mathbb {Y}$, i.e., that $\textbf{w}_1$ extends to $\textbf{w}_1\in C^{r-1}(\mathbb {R}^3)^3$. Moreover, since $\Omega$ is bounded, a standard truncation argument allows also to suppose that $\textbf{w}_1\in C^{r-1}_c(\mathbb {R}^3)^3$. For such a $\textbf{w}_1$ we can take, as a test function in (4.35), $\varvec{\phi }(t,x)=\varphi (t)\textbf{w}_1(x)$, from which (4.43) trivially follows when $\textbf{w}=\textbf{w}_1$, completing the proof of our claim.

Recalling (4.39) and (4.40) we can then rewrite (4.43), for test functions $\varvec{\phi }_0=(\textbf{w},\chi )\in V_{\mathcal {L}}$, as $\int _{-\infty }^\infty \varphi '(R\textbf{u}',\overline{\varvec{\phi }_0})_{L^2_{\mathcal {L}}}+\varphi \left[ \langle M\textbf{u},\varvec{\phi }_0\rangle _{V_{\mathcal {L}}}-(\delta \Phi _{\Gamma _1}\textbf{u}',\overline{\varvec{\phi }_0})_{L^2_{\mathcal {L}}}\right] =0$ or, using (4.38) and Bochner integrals, $\int _{-\infty }^\infty \left[ \varphi 'R\textbf{u}'+\varphi \left( M\textbf{u}-\delta \Phi _{\Gamma _1}\textbf{u}'\right) \right] =0$ in $V_{\mathcal {L}}'$, for all $\varphi \in \mathcal {D}(\mathbb {R})$. Consequently, $\textbf{u}$ solves equation (4.41) in $\mathcal {D}'(\mathbb {R})$. Since $M\textbf{u}-\delta \Phi _{\Gamma _1}\textbf{u}'\in C(\mathbb {R};V_{\mathcal {L}}')$, we consequently get that $R\textbf{u}'\in C^1(\mathbb {R};V_{\mathcal {L}}')$ and (4.41) holds. $\square$

We can give the promised uniqueness result.

Theorem 4.16

(Uniqueness) Weak solutions of $({\mathcal {L}}_0)$ are unique.

Proof

By linearity it is enough to prove that $\textbf{r}_0=\textbf{r}_1=0$ and $v_0=v_1=0$ yield $(\textbf{r},v)\equiv 0$ in $\mathbb {R}$. Moreover, since the couple $({\widetilde{\textbf{r}}},{\widetilde{v}})$ defined by ${\widetilde{\textbf{r}}}(t)=\textbf{r}(-t)$ and ${\widetilde{v}}(t)=v(-t)$ for $t\in \mathbb {R}$ is still a weak solution of $({\mathcal {L}}_0)$ provided $\delta$ is replaced by$-\delta$, we just have to prove that $(\textbf{r},v)\equiv 0$ in $[0,\infty )$.

Adapting the arguments in the proof of [32, Chapter 4, Lemma 4.2.5], we fix $t>0$ and test functions $(\varvec{\phi },\psi )\in C^1(\mathbb {R};V_{{\mathcal {L}}})$, depending on t, given by

$$\begin{aligned} \varvec{\phi }(s)= {\left\{ \begin{array}{ll} \int _s^t {\overline{\textbf{r}}}(\tau )\,d\tau \quad &{}\hbox { if}\ s\le t,\\ {\overline{\textbf{r}}}(t)(t-s) &{}\hbox { if}\ s\ge t, \end{array}\right. } \qquad \psi (s)= {\left\{ \begin{array}{ll} \int _s^t {\overline{v}}(\tau )\,d\tau \quad &{}\hbox { if}\ s\le t,\\ {\overline{v}}(t)(t-s) &{}\hbox { if}\ s\ge t. \end{array}\right. } \end{aligned}$$

(4.46)

They trivially verify

$$\begin{aligned} \varvec{\phi }(t)=0,\quad \psi (t)=0,\quad \text {and}\quad \varvec{\phi }_t=-{\overline{\textbf{r}}},\quad \psi _t=-{\overline{v}}\quad \text {in }[0,t]. \end{aligned}$$

(4.47)

By Proposition 4.15 we can take $(\varvec{\phi },\psi )$ as a test function in (4.42), writing it for $s=0$. Since $\textbf{r}_1=0$ and $v_0=v_1=0$, also using (4.47), we consequently get

$$\begin{aligned}{} & {} \int _0^t\left\{ \int _\Omega [\rho _0\textbf{r}_t\cdot {\overline{\textbf{r}}}-B{{\,\textrm{div}\,}}{\overline{\varvec{\phi }}}_t{{\,\textrm{div}\,}}\varvec{\phi }]+\int _{\Gamma _1} \left[ \mu v_t{\overline{v}}-\sigma (\nabla _\Gamma ({\overline{\psi }}_t),\nabla _\Gamma {\overline{\psi }})_\Gamma -\kappa \psi {\overline{\psi }}_t]\right] \right\} \\{} & {} \quad =-\int _0^t\int _{\Gamma _1}\delta v_t\psi =\int _0^t \int _{\Gamma _1}\delta v\psi _t=-\int _0^t\int _{\Gamma _1} \delta |v|^2, \end{aligned}$$

where we also integrated by parts in time the right-hand side. Taking the real part, we then get

$$\begin{aligned} \int _0^t\frac{1}{2}\frac{d}{dt}\left[ \rho _0\Vert \textbf{r}\Vert _2^2-B\Vert {{\,\textrm{div}\,}}\varvec{\phi }\Vert _2^2+\!\!\int _{\Gamma _1}\mu |v|^2\!\!-\!\! \int _{\Gamma _1}\sigma |\nabla _\Gamma \psi |_\Gamma ^2-\!\!\int _{\Gamma _1}\kappa |\psi |^2\right] \! =\! -\int _0^t\int _{\Gamma _1}\delta |v|^2. \end{aligned}$$

Since $\textbf{r}_0=\varvec{\phi }(t)=0$ and $v_0=\psi (t)=0$ we can rewrite it as

$$\begin{aligned} \tfrac{\rho _0}{2}\Vert \textbf{r}(t)\Vert _2^2+\tfrac{B}{2}\Vert {{\,\textrm{div}\,}}\varvec{\phi }(0)\Vert _2^2+\int _{\Gamma _1}\mu |v|^2+ \int _{\Gamma _1}\sigma |\nabla _\Gamma \psi (0)|_\Gamma ^2=-\int _{\Gamma _1}\kappa |\psi (0)|^2 -\int _0^t\int _{\Gamma _1}\delta |v|^2. \end{aligned}$$

(4.48)

We now estimate the right-hand side of (4.48) by using assumption (A), (4.46) and Hölder inequality. We get $\Vert v(t)\Vert _{2,\Gamma _1}^2\le \tfrac{1}{\mu _0}(\Vert \delta \Vert _{\infty ,\Gamma _1}+\Vert \kappa \Vert _{\infty ,\Gamma _1} t) \int _0^t\Vert v\Vert _{2,\Gamma _1}^2$ for all $t\ge 0$. Denoting $\Upsilon (t)=\int _0^t\Vert v\Vert _{2,\Gamma _1}^2$ for $t\in [0,\infty )$, we can rewrite the last estimate as $\Upsilon '(t)\le \tfrac{1}{\mu _0}(\Vert \delta \Vert _{\infty ,\Gamma _1}+\Vert \kappa \Vert _{\infty ,\Gamma _1} t)\Upsilon (t)$ for all $t\ge 0$. Since $\Upsilon (0)=0$, a standard integration then yields $\Upsilon \equiv 0$ in $[0,\infty )$, from which $v\equiv 0$ in $[0,\infty )$.

Fixing $t>0$ again, and recalling the test functions $(\varvec{\phi },\psi )$ in (4.46), we then get $\psi (0)=0$. Plugging it in the consequent identity (4.48), also recalling that $v\equiv 0$, by assumption (A) we then get $\textbf{r}(t)=0$. Being $t>0$ arbitrary we then have $(\textbf{r},v)\equiv 0$ in $[0,\infty )$, concluding the proof. $\square$

4.5 Main results for problems $({\mathcal {L}})$ and $({\mathcal {L}}_0)$

We can finally prove Theorems 1.3 and 1.5 and deal with optimal regularity issues.

Proof of Theorem 1.3

By combining Theorem 4.6 and Lemma 4.11, we get that for all $U_{0{\mathcal {L}}}\in \mathcal {H}^1_{\mathcal {L}}$ problem $({\mathcal {L}}_0)$ has a unique generalized solution $(\textbf{r},v)\in X^1_{\mathcal {L}}$, continuously depending on $U_{0{\mathcal {L}}}$ in the topologies of the respective spaces. By Lemma 4.13 the couple $(\textbf{r},v)$ is also a weak solution of $({\mathcal {L}}_0)$ which, by Theorem 4.16, is unique among weak solutions. By Theorems 4.6–4.8 and Lemma 4.11 the couple $(\textbf{r},v)$ is a strong solution if and only if $U_{0{\mathcal {L}}}\in D(A_{\mathcal {L}})$, which by (4.27) exactly reads as in the statement, the continuous dependence following by Theorem 4.8.

To complete the proof, we just have to get the energy identity (1.11), which can be obtained either by re-deriving it for strong solutions and hence using a density argument or deducing it from the energy identity (1.5) for problem $({\mathcal {P}}_0)$. Choosing the second alternative, we remark that by Lemma 4.11 and Theorem 4.6, $\textbf{r}$ and v are the first two components of $\Phi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}{\dot{U}}_{\mathcal {P}}$, where ${\dot{U}}_{\mathcal {P}}$ is the solution of (3.25) corresponding to ${\dot{U}}_{0{\mathcal {P}}}=F_{_{{\mathcal {L}}{\mathcal {P}^c}}}U_{0{\mathcal {L}}}$. By (3.27) we have ${\dot{U}}_{\mathcal {P}}=(\pi _0 u,v,u_t,v_t)$ for the solution (u, v) of $({\mathcal {P}}_0)$ corresponding to $U_{0{\mathcal {P}}}\in Q^{-1}{\dot{U}}_{0{\mathcal {P}}}$. Using (4.20) and (4.21) we have $-\nabla u=\textbf{r}_t$ and $-B{{\,\textrm{div}\,}}\textbf{r}=\rho _0u_t$. Plugging them into (1.5) we then get (1.11) and complete the proof. $\square$

The uniqueness of weak solutions of $({\mathcal {L}}_0)$ in Theorem 4.16, together with the existence of a generalized and hence weak solution of $({\mathcal {L}}_0)$, shows that solutions of $({\mathcal {L}})$ are equivalently weak or generalized. Combining this remark with Remark 4.14, we thus obtain that, as for problem $({\mathcal {P}})$, all types of solutions in Definition 4.9coincide, strong solutions being defined only in the class $X^2_{\mathcal {L}}$. Consequently, in the sequel, we shall only deal with weak solutions of $({\mathcal {L}})$, i.e., with elements of the spaces $\mathcal {S}^n_{\mathcal {L}}$ defined in (1.14). Recalling the spaces $\mathcal {T}_{\mathcal {L}}^n$ defined in (4.33), we point out the following result, which plays the role of Lemma 3.8 for problem $({\mathcal {L}})$.

Lemma 4.17

The operator $\mathcal {I}'\in \mathcal {L}(X^1_{\mathcal {L}};Y^1_{\mathcal {L}})$, defined by $\mathcal {I}'(\textbf{r},v)=(\textbf{r},v,\textbf{r}_t,v_t)$, restricts for each $n\in {\widetilde{\mathbb {N}}}$, $n\le r$, to a bijective isomorphism $\mathcal {I}^n_{\mathcal {L}}\in \mathcal {L}(\mathcal {S}^n_{\mathcal {L}};\mathcal {T}^n_{\mathcal {L}})$ with inverse $\left( \mathcal {I}^n_{\mathcal {L}}\right) ^{-1}\in \mathcal {L}(\mathcal {T}^n_{\mathcal {L}};\mathcal {S}^n_{\mathcal {L}})$ given by $\left( \mathcal {I}^n_{\mathcal {L}}\right) ^{-1}(\textbf{r},v,\textbf{s},z)=(\textbf{r},v)$.

Proof

It follows by (1.7)–(1.10), Lemma 4.11 and the remarked equivalence. $\square$

Proof of Theorem 1.5

By Proposition 3.12, Corollary 3.17 and Corollary 4.7, for any $(u,v)\in \mathcal {S}^1_{\mathcal {P}^c}$ we have $[\Phi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}\mathcal {J}^1_{\mathcal {P}^c}(u,v)](t)= (\textbf{r}(t),v(t),-\nabla u(t),v_t(t))$ for all $t\in \mathbb {R}$, where, by using Lemma 4.2, $\textbf{r}(t)$ is defined by (1.15), and (1.16) holds, so by Lemma 4.17 we get the operator $\Psi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}$ as in the statement, that is

$$\begin{aligned} \Psi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}=(\mathcal {I}_{\mathcal {L}}^1)^{-1}\cdot \Phi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}\cdot \mathcal {J}^1_{\mathcal {P}^c}\in \mathcal {L}(\mathcal {S}^1_{\mathcal {P}^c}; \mathcal {S}^1_{\mathcal {L}}). \end{aligned}$$

(4.49)

By Proposition 3.12, Corollary 3.17 and Corollary 4.7 we also get that $\text {Ker } \Psi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}= \mathbb {C}_{X_{\mathcal {P}}}$, and the bijective isomorphisms

$$\begin{aligned} {\dot{\Psi }}_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}=(\mathcal {I}_{\mathcal {L}}^1)^{-1}\!\!\!\cdot \Phi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}\!\!\!\cdot \dot{\mathcal {J}}^1_{\mathcal {P}^c}\!\in \mathcal {L}(\dot{\mathcal {S}}^1_{\mathcal {P}^c}; \mathcal {S}^1_{\mathcal {L}}),\quad {\dot{\Psi }}_{_{{\mathcal {L}}{\mathcal {P}^c}}}=(\dot{\mathcal {J}}^1_{\mathcal {P}^c})^{-1}\!\!\cdot \Phi _{_{{\mathcal {L}}{\mathcal {P}^c}}}\!\!\!\cdot \mathcal {I}_{\mathcal {L}}^1 \!\in \mathcal {L}(\mathcal {S}^1_{\mathcal {L}};\dot{\mathcal {S}}^1_{\mathcal {P}^c}), \end{aligned}$$

(4.50)

trivially being the inverse of each other. Moreover, by (4.21) and (4.20), for any $(\textbf{r},v)\in \mathcal {S}^1_{\mathcal {L}}$ we have

$$\begin{aligned}{}[\Phi _{_{{\mathcal {L}}{\mathcal {P}^c}}}\mathcal {I}^1_{\mathcal {L}}(\textbf{r},v)](t)= F_{_{{\mathcal {L}}{\mathcal {P}^c}}}(\textbf{r}(t),v(t),\textbf{r}_t(t),v_t(t))=\left( {\dot{u}}(t),v(t),-\tfrac{B}{\rho _0}{{\,\textrm{div}\,}}\textbf{r}(t),v_t(t)\right) \end{aligned}$$

for all $t\in \mathbb {R}$, where ${\dot{u}}(t)\in {\dot{H}}^1(\Omega )$ is the unique solution of $-\nabla {\dot{u}}(t)=\textbf{r}_t(t)$. Consequently, since $\mathcal {J}^1_{\mathcal {P}^c}$ is surjective, there is $(u,v)\in \mathcal {S}^1_{\mathcal {P}^c}$ such that

$$\begin{aligned} \left( {\dot{u}}(t), v(t), -\tfrac{B}{\rho _0}{{\,\textrm{div}\,}}\textbf{r}(t),v_t(t\right) =(\pi _0 u(t), v(t), u_t(t), v_t(t))\quad \text {for all }t\in \mathbb {R}, \end{aligned}$$

(4.51)

that is

$$\begin{aligned} \pi _0 u(t)= {\dot{u}}(t)\quad \text {and}\quad -B{{\,\textrm{div}\,}}\textbf{r}(t)=\rho _0 u_t(t)\qquad \text {for all }t\in \mathbb {R}. \end{aligned}$$

(4.52)

Since, by (1.16), we have $-\nabla {\dot{u}}(0)=\textbf{r}_t(0)$, from (4.52) we get (1.17). Moreover, trivially (1.17) defines u up to a space-time constant and then also completely defines ${\dot{\Psi }}_{_{{\mathcal {L}}{\mathcal {P}^c}}}(\textbf{r},v)$. Finally, to prove (1.18), we remark that, as we just proved, $(\textbf{r},v)=\Psi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}(u,v)$ is equivalent to (4.51), that is (4.52). By uniqueness then it is equivalent to (4.52) at $t=0$, that is to $-\nabla u_0=\textbf{r}_1$ and $-B{{\,\textrm{div}\,}}\textbf{r}_0=\rho _0u_1$. $\square$

We now give an optimal regularity result for problem $({\mathcal {L}}_0)$, and we show that the operators in Theorem 1.5 have a good behavior with respect to regularity classes.

Theorem 4.18

(Optimal regularity for $({\mathcal {L}}_0)$ and $({\mathcal {P}^c}) \rightleftarrows ({\mathcal {L}})$) For all data $U_{0{\mathcal {L}}}=(\textbf{r}_0, v_0,\textbf{r}_1,v_1)\in \mathcal {H}^1_{\mathcal {L}}$ and $n\in {\widetilde{\mathbb {N}}}$, $2\le n\le r$, the weak solution $(\textbf{r},v)$ given in Theorem 1.3 belongs to $X_{\mathcal {L}}^n$ if and only if $U_{0{\mathcal {L}}}\in D^{n-1}_{\mathcal {L}}$. In this case $(\textbf{r},v)$ continuously depends on the data $U_{0{\mathcal {L}}}$ in the topologies of the respective spaces. Moreover, the operator $\Psi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}$ in Theorem 1.5 restricts to a surjective operator $\Psi ^n_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}\in \mathcal {L}(\mathcal {S}^n_{\mathcal {P}^c};\mathcal {S}^n_{\mathcal {L}})$ such that $\ker \Psi ^n_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}=\mathbb {C}_{X_{\mathcal {P}}}$. Consequently, the bijective isomorphisms ${\dot{\Psi }}_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}$ and ${\dot{\Psi }}_{_{{\mathcal {L}}{\mathcal {P}^c}}}$ in the same result, respectively, restrict to bijective isomorphisms

$$\begin{aligned} {\dot{\Psi }}_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}^n\in \mathcal {L}(\dot{\mathcal {S}}^n_{\mathcal {P}^c};\mathcal {S}^n_{\mathcal {L}}),\quad \text {and}\quad {\dot{\Psi }}_{_{{\mathcal {L}}{\mathcal {P}^c}}}^n\in \mathcal {L}(\mathcal {S}^n_{\mathcal {L}};\dot{\mathcal {S}}^n_{\mathcal {P}^c}). \end{aligned}$$

(4.53)

Proof

The optimal regularity for problem $({\mathcal {L}}_0)$ immediately follows by combining Theorem 4.8 and Lemma 4.17. Moreover, by formulas (4.49) and (4.50), the restriction properties of the operators $\Psi _{_{{\mathcal {P}^c}\!{\mathcal {L}}}}$, ${\dot{\Psi }}_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}$ and ${\dot{\Psi }}_{_{{\mathcal {L}}{\mathcal {P}^c}}}$ are simple consequences of Corollaries 3.17, 4.7 and Lemma 4.17. $\square$

5 The Eulerian models

5.1 Preliminaries

To address problems $({\mathcal {E}})$ and $({\mathcal {E}^c})$, it is convenient to introduce some functional spaces in addition to the spaces already defined in § 1. We anticipate the compatibility conditions for problem $({\mathcal {E}}_0)$ needed in the sequel, that is, for $n\in \mathbb {N}$, $2\le n\le r$ and data $U_{0{\mathcal {E}}}=(p_0,\textbf{v}_0,v_0,v_1)\in \mathcal {H}^n_{\mathcal {E}}$,

$$\begin{aligned} \left\{ \begin{aligned}&\textbf{v}_0\cdot \varvec{\nu }=-v_1,\,\,\qquad \text {on }\Gamma _1,\qquad \textbf{v}_0\cdot \varvec{\nu }=0,\quad \text {on }\Gamma _0, \\&\partial _{\varvec{\nu }}\Delta ^i{{\,\textrm{div}\,}}\textbf{v}_0 =0,\quad \text {on } \Gamma _0 \qquad \text { for }i=0,\ldots , \lfloor n/2\rfloor -2, \,\,\,\,\qquad \text { when }n\ge 4,\\&\partial _{\varvec{\nu }}\Delta ^i p_0 =0,\quad \text {on } \Gamma _0 \qquad \text { for }i=0,\ldots , \lfloor (n-1)/2\rfloor -1, \qquad \text { when }n\ge 3,\\&\mu \partial _{\varvec{\nu }}p_0=\rho _0[\,{{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v_0)-\delta v_1-\kappa v_0-p_0\,],\quad \text {on }\Gamma _1, \qquad \,\,\, \text { when }n\ge 3,\\&\tfrac{B\mu }{\rho _0}\partial _{\varvec{\nu }}{{\,\textrm{div}\,}}\textbf{v}_0\!\!=\!-{{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v_1)\!+\!\tfrac{\delta }{\rho _0}\partial _{\varvec{\nu }}p_0\!+\!\kappa v_1\!+\!\tfrac{B}{\rho _0}\Delta p_0,\,\text {on }\Gamma _1,\text { when } n=4,\\&\begin{aligned} \tfrac{B\mu }{\rho _0} \partial _{\varvec{\nu }}{{\,\textrm{div}\,}}\Delta ^i\textbf{v}_0\,\,\,=\,\,\,{{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma {{\,\textrm{div}\,}}\Delta ^{i-1}\textbf{v}_0)+\tfrac{\delta }{\rho _0} \partial _{\varvec{\nu }}\Delta ^i p_0-\kappa \partial _{\varvec{\nu }}{{\,\textrm{div}\,}}\Delta ^{i-1}\textbf{v}_0\\ \qquad B{{\,\textrm{div}\,}}\Delta ^i \textbf{v}_0 \quad \text {on }\Gamma _1, \quad \, \text { for }i=1,\ldots , \lfloor n/2\rfloor -2, \quad \text { when }n\ge 6, \end{aligned}\\&\begin{aligned} \tfrac{B\mu }{\rho _0}\partial _{\varvec{\nu }}\Delta ^i p_0\quad \,=\,\,{{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma \partial _{\varvec{\nu }}\Delta ^{i-1}p_0) +B\delta \partial _{\varvec{\nu }}{{\,\textrm{div}\,}}\Delta ^{i-1}\textbf{v}_0-\kappa \partial _{\varvec{\nu }}\Delta ^{i-1}p_0\quad \\ \qquad - B{{\,\textrm{div}\,}}\Delta ^i p_0 \quad \text {on }\Gamma _1, \quad \text { for } i=1,\ldots , \lfloor (n-1)/2\rfloor -1, \text { when }n\ge 5. \end{aligned}\end{aligned}\right. \end{aligned}$$

(5.1)

By (2.2) and the Trace Theorem one gets that $D^{n-1}_{\mathcal {E}}:=\{U_{0{\mathcal {E}}}\in \mathcal {H}^n_{\mathcal {E}}: \quad (5.1)\text { hold}\}$ and $D^{n-1}_{\mathcal {E}^c}:=D^{n-1}_{\mathcal {E}}\cap \mathcal {H}^n_{\mathcal {E}^c}$ are closed subspaces, respectively, of $\mathcal {H}^n_{\mathcal {E}}$ and $\mathcal {H}^n_{\mathcal {E}^c}$, and then Hilbert spaces. When $r=\infty$ we also set the Fréchet spaces

$$\begin{aligned} \mathcal {H}_{\mathcal {E}}^\infty = \left[ C^\infty ({\overline{\Omega }})\times C^\infty ({\overline{\Omega }})^3\times C^\infty (\Gamma _1)\times C^\infty (\Gamma _1)\right] \cap \mathcal {H}^1_{\mathcal {E}},\quad \mathcal {H}_{\mathcal {E}^c}^\infty =\mathcal {H}_{\mathcal {E}}^\infty \cap \mathcal {H}^1_{\mathcal {E}^c}\end{aligned}$$

and their closed subspaces

$$\begin{aligned} D^\infty _{\mathcal {E}}=\{U_{0{\mathcal {E}}}\in \mathcal {H}^n_{\mathcal {E}}: \quad (5.1)\hbox { hold for all}\ n\in \mathbb {N}\}, \qquad D^\infty _{\mathcal {E}^c}=D^\infty _{\mathcal {E}}\cap \mathcal {H}^n_{\mathcal {E}^c}. \end{aligned}$$

Moreover, for $n\in {\widetilde{N}}$, $n\le r$, we set the Fréchet spaces

$$\begin{aligned} Y^n_{\mathcal {E}}=\bigcap _{i=0}^{n-1} C^i(\mathbb {R};\mathcal {H}^{n-i}_{\mathcal {E}})\quad \text {and}\quad Y^n_{\mathcal {E}^c}=\bigcap _{i=0}^{n-1} C^i(\mathbb {R};\mathcal {H}^{n-i}_{\mathcal {E}^c}). \end{aligned}$$

(5.2)

In this section we shall also use the Hilbert space

$$\begin{aligned} H({{\,\textrm{div}\,}},\Omega )=\{\textbf{v}\in L^2(\Omega )^3: {{\,\textrm{div}\,}}\textbf{v}\in L^2(\Omega )\}, \end{aligned}$$

(5.3)

where ${{\,\textrm{div}\,}}\textbf{v}$ is taken in the distributional sense, introduced in [12, Chapter IX, §2, p. 203], equipped with the norm $\Vert \cdot \Vert _{H({{\,\textrm{div}\,}},\Omega )}=\left( \Vert \cdot \Vert _2^2+\Vert {{\,\textrm{div}\,}}(\cdot )\Vert _2^2\right) ^{1/2}$. We recall, see [12, Chapter IX, §2, Theorem 1, p. 204 and (1.1), p. 206], that the map $\textbf{v}\mapsto \textbf{v}\cdot \varvec{\nu }$ defined from $C^\infty ({\overline{\Omega }})^3$ to $C^1(\Gamma )$ extends by density to a bounded linear operator from $H({{\,\textrm{div}\,}},\Omega )$ to $H^{-1/2}(\Gamma )$, for simplicity denoted by the same symbol, and that

$$\begin{aligned} \int _\Omega \textbf{v}\cdot \nabla \varphi +\int _\Omega {{\,\textrm{div}\,}}\textbf{v}\varphi =\langle \textbf{v}\cdot \varvec{\nu },\varphi _{|\Gamma }\rangle _{H^{1/2}(\Gamma )}\quad \text {for all }\textbf{v}\in H({{\,\textrm{div}\,}},\Omega ),\, \varphi \in H^1(\Omega ). \end{aligned}$$

(5.4)

5.2 The main isomorphisms

To deal with problems $({\mathcal {E}})$ and $({\mathcal {E}^c})$ we first introduce the main isomorphisms between the phase space of problem $({\mathcal {E}})$ and the space $\dot{\mathcal {H}}^1_{\mathcal {P}}$. We then set, for all $({\dot{u}},v,w,z)\in \dot{\mathcal {H}}^1_{\mathcal {P}}$ and $(p,\textbf{v},v,z)\in \mathcal {H}^1_{\mathcal {E}}$,

$$\begin{aligned} F_{_{{\mathcal {P}}{\mathcal {E}}}}({\dot{u}},v,w,z)=(\rho _0,w,-\nabla {\dot{u}}, v,z),\qquad F_{_{{\mathcal {E}}{\mathcal {P}}}}(p,\textbf{v},v,z)=\left( {\dot{u}}, v, p/\rho _0,z\right) , \end{aligned}$$

(5.5)

where ${\dot{u}}\in {\dot{H}}^1(\Omega )$ is the unique solution of $-\nabla {\dot{u}}=\textbf{v}$ given in Lemma 4.1.

Trivially $F_{_{{\mathcal {P}}{\mathcal {E}}}}\in \mathcal {L}(\dot{\mathcal {H}}^1_{\mathcal {P}};\mathcal {H}^1_{\mathcal {E}})$ and $F_{_{{\mathcal {E}}{\mathcal {P}}}}\in \mathcal {L}(\mathcal {H}^1_{\mathcal {E}};\dot{\mathcal {H}}^1_{\mathcal {P}})$, so they induce the operators $\Phi _{_{{\mathcal {P}}{\mathcal {E}}}}\in \mathcal {L}({\dot{Y}}^1_{\mathcal {P}}; Y^1_{\mathcal {E}})$ and $\Phi _{_{{\mathcal {E}}{\mathcal {P}}}}\in \mathcal {L}(Y^1_{\mathcal {E}}; {\dot{Y}}^1_{\mathcal {P}})$ given by

$$\begin{aligned} (\Phi _{_{{\mathcal {P}}{\mathcal {E}}}}{\dot{U}}_{\mathcal {P}})(t)=F_{_{{\mathcal {P}}{\mathcal {E}}}}{\dot{U}}_{\mathcal {P}}(t),\quad (\Phi _{_{{\mathcal {E}}{\mathcal {P}}}}U_{\mathcal {E}})(t)=F_{_{{\mathcal {E}}{\mathcal {P}}}}{U}_{\mathcal {E}}(t)\quad \text {for all }t\in \mathbb {R}. \end{aligned}$$

(5.6)

The following result points out all properties of these operators needed in the sequel

Lemma 5.1

The operator $F_{_{{\mathcal {P}}{\mathcal {E}}}}$ is a bijective isomorphism between $\dot{\mathcal {H}}^1_{\mathcal {P}}$ and $\mathcal {H}^1_{\mathcal {E}}$ having $F_{_{{\mathcal {E}}{\mathcal {P}}}}$ as inverse. Consequently, $\Phi _{_{{\mathcal {P}}{\mathcal {E}}}}$ is a bijective isomorphism between ${\dot{Y}}^1_{\mathcal {P}}$ and $Y^1_{\mathcal {E}}$ with inverse $\Phi _{_{{\mathcal {E}}{\mathcal {P}}}}$. They restrict to operators

$$\begin{aligned} \begin{aligned} F_{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}\in \mathcal {L}(\dot{\mathcal {H}}^1_{\mathcal {P}^c};\mathcal {H}^1_{\mathcal {E}^c}), \qquad&F_{_{{\mathcal {E}^c}\!{\mathcal {P}^c}}}\in \mathcal {L}(\mathcal {H}^1_{\mathcal {E}^c};\dot{\mathcal {H}}^1_{\mathcal {P}^c}),\\ \Phi _{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}\in \mathcal {L}({\dot{Y}}^1_{\mathcal {P}^c};Y^1_{\mathcal {E}^c}),\qquad&\Phi _{_{{\mathcal {E}^c}\!{\mathcal {P}^c}}}\in \mathcal {L}(Y^1_{\mathcal {E}^c}; {\dot{Y}}^1_{\mathcal {P}^c}), \end{aligned} \end{aligned}$$

(5.7)

and, for $n\in \mathbb {N}$, $2\le n\le r$, to operators

$$\begin{aligned} \begin{aligned} F_{_{{\mathcal {P}}{\mathcal {E}}}}\in \mathcal {L}(\dot{\mathcal {H}}^n_{\mathcal {P}};\mathcal {H}^n_{\mathcal {E}}), \qquad&F_{_{{\mathcal {E}}{\mathcal {P}}}}\in \mathcal {L}(\mathcal {H}^n_{\mathcal {E}};\dot{\mathcal {H}}^n_{\mathcal {P}}),\\ \Phi _{_{{\mathcal {P}}{\mathcal {E}}}}\in \mathcal {L}({\dot{Y}}^n_{\mathcal {P}};Y^n_{\mathcal {E}}),\qquad&\Phi _{_{{\mathcal {E}}{\mathcal {P}}}}\in \mathcal {L}(Y^n_{\mathcal {E}}; {\dot{Y}}^n_{\mathcal {P}}),\\ F_{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}\in \mathcal {L}(\dot{\mathcal {H}}^n_{\mathcal {P}^c};\mathcal {H}^n_{\mathcal {E}^c}), \qquad&F_{_{{\mathcal {E}^c}\!{\mathcal {P}^c}}}\in \mathcal {L}(\mathcal {H}^n_{\mathcal {E}^c};\dot{\mathcal {H}}^n_{\mathcal {P}^c}),\\ \Phi _{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}\in \mathcal {L}({\dot{Y}}^n_{\mathcal {P}^c};Y^n_{\mathcal {E}^c}),\qquad&\Phi _{_{{\mathcal {E}^c}\!{\mathcal {P}^c}}}\in \mathcal {L}(Y^n_{\mathcal {E}^c}; {\dot{Y}}^n_{\mathcal {P}^c}), \end{aligned} \end{aligned}$$

(5.8)

enjoying the same properties. Moreover, for $n\in \mathbb {N}$, $2\le n\le r$, we have

$$\begin{aligned} F_{_{{\mathcal {P}}{\mathcal {E}}}}{\dot{D}}^{n-1}_{\mathcal {P}}=D^{n-1}_{\mathcal {E}},\qquad \text {and}\quad F_{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}{\dot{D}}^{n-1}_{\mathcal {P}^c}=D^{n-1}_{\mathcal {E}^c}, \end{aligned}$$

(5.9)

so $F_{_{{\mathcal {P}}{\mathcal {E}}}}$, $F_{_{{\mathcal {E}}{\mathcal {P}}}}$, $F_{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}$ and $F_{_{{\mathcal {E}^c}\!{\mathcal {P}^c}}}$ further restrict to bijective isomorphisms between them.

Proof

The first two sentences in the statements, including (5.7) and (5.8), trivially follow from (1.3), (1.6), (1.12), (3.11), (5.2), (5.5), (5.6) and Lemma 4.1. Moreover, the last restriction property trivially follows from (5.7)–(5.9). To complete the proof we then have to show that (5.9) holds and, in particular, that $F_{_{{\mathcal {P}}{\mathcal {E}}}}{\dot{D}}^{n-1}_{\mathcal {P}}=D^{n-1}_{\mathcal {E}}$, since the second identity in (5.9) follows from the first one by (5.7). To prove that $F_{_{{\mathcal {P}}{\mathcal {E}}}}{\dot{D}}^{n-1}_{\mathcal {P}}=D^{n-1}_{\mathcal {E}}$ we just have to prove that, for all $(u_0,v_0,u_1,v_1)\in \mathcal {H}^n_{\mathcal {P}}$ and $(p_0,\textbf{v}_0,v_0,v_1)\in \mathcal {H}^1_{\mathcal {E}}$ such that

$$\begin{aligned} p_0=\rho _0 u_1\qquad \text {and}\quad -\nabla u_0=\textbf{v}_0, \end{aligned}$$

(5.10)

the compatibility conditions (3.2) and (5.1) are equivalent. Using (5.10) one immediately gets that $\textbf{v}_0\cdot \varvec{\nu }=-v_1$ is equivalent to $\partial _{\varvec{\nu }}u_0=v_1$ on $\Gamma _1$. It is then straightforward to get the required equivalence. $\square$

5.3 Abstract analysis of problems $({\mathcal {E}})$, $({\mathcal {E}}_0)$, $({\mathcal {E}^c})$ and $({\mathcal {E}}_0^c)$

To address problem $({\mathcal {E}}_0)$ in a semigroup setting, we reduce it to the first-order problem

$$\begin{aligned} {\left\{ \begin{array}{ll} p_t+B {{\,\textrm{div}\,}}\textbf{v}=0\qquad &{}\text {in }\mathbb {R}\times \Omega ,\\ \rho _0\textbf{v}_t+\nabla p=0\qquad &{}\text {in }\mathbb {R}\times \Omega ,\\ {{\,\textrm{curl}\,}}\textbf{v}=0\qquad &{}\text {in }\mathbb {R}\times \Omega ,\\ v_t =z\qquad &{}\text {on }\mathbb {R}\times \Gamma _1,\\ \mu z_t- {{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v)+\delta z+\kappa v+p =0\qquad &{}\text {on }\mathbb {R}\times \Gamma _1,\\ \textbf{v}\cdot {\varvec{\nu }} =0 \quad \text {on }\mathbb {R}\times \Gamma _0,\qquad \textbf{v}\cdot {\varvec{\nu }} =-z\qquad &{}\text {on }\mathbb {R}\times \Gamma _1,\\ p(0,x)=p_0(x),\quad \textbf{v}(0,x)=\textbf{v}_0(x) &{} \text {in }\Omega ,\\ v(0,x)=v_0(x),\quad z(0,x)=v_1(x) &{} \text {on }\Gamma _1. \end{array}\right. } \end{aligned}$$

(5.11)

More formally, working in the phase space $\mathcal {H}^1_{\mathcal {E}}$, in which (5.11)$_3$ holds, we introduce the unbounded operator $A_{\mathcal {L}}: D(A_{\mathcal {E}})\subset \mathcal {H}^1_{\mathcal {E}}\rightarrow \mathcal {H}^1_{\mathcal {E}}$ given by

$$\begin{aligned} D(A_{\mathcal {E}})=D^1_{\mathcal {E}}=\{(p,\textbf{v},v,z)\in \mathcal {H}^2_{\mathcal {E}}: (\textbf{v},z)\in \mathbb {H}^1_{\mathcal {L}}\}, \end{aligned}$$

(5.12)

$$\begin{aligned} A_{\mathcal {E}}\begin{pmatrix}p\\ \textbf{v}\\ v\\ z\end{pmatrix} = \begin{pmatrix}B{{\,\textrm{div}\,}}\textbf{v}\\ \tfrac{1}{\rho _0}\nabla p\\ -z\\ \frac{1}{\mu }\left[ -{{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v)+\delta z+\kappa v+p_{|\Gamma _1}\right] \end{pmatrix}, \end{aligned}$$

(5.13)

together with the abstract equation and Cauchy problem

$$\begin{aligned} U_{\mathcal {E}}'+A_{\mathcal {E}}U_{\mathcal {E}}=0\qquad \text {in }\mathcal {H}^1_{\mathcal {E}}, \end{aligned}$$

(5.14)

$$\begin{aligned} U_{\mathcal {E}}'+A_{\mathcal {E}}U_{\mathcal {E}}=0\qquad \hbox { in}\ \mathcal {H}^1_{\mathcal {E}}, \qquad U_{\mathcal {E}}(0)=U_{0{\mathcal {E}}}\in \mathcal {H}^1_{\mathcal {E}}. \end{aligned}$$

(5.15)

The following result shows that problems (3.25) and (5.15) are essentially equivalent, simultaneously restricting to the spaces $\dot{\mathcal {H}}^1_{\mathcal {P}^c}$ and $\mathcal {H}^1_{\mathcal {E}^c}$.

Theorem 5.2

(Well-posedness for (5.15) and its restriction to $\mathcal {H}^1_{\mathcal {E}^c}$)

• I) The operator $-A_{\mathcal {E}}$ is densely defined and it generates on $\mathcal {H}^1_{\mathcal {E}}$ the strongly continuous group $\{{\mathcal {E}}^1(t),t\in \mathbb {R}\}$ given by

  $$\begin{aligned} {\mathcal {E}}^1(t)=F_{_{{\mathcal {P}}{\mathcal {E}}}}{\mathcal {P}}^1(t) F_{_{{\mathcal {E}}{\mathcal {P}}}}\qquad \text {for all }t\in \mathbb {R}, \end{aligned}$$

  (5.16)

  and hence similar to the group $\{{\mathcal {P}}^1(t),t\in \mathbb {R}\}$ in Proposition 3.10. Consequently, for any $U_{0{\mathcal {E}}}\in \mathcal {H}^1_{\mathcal {E}}$, problem (5.15) has a unique generalized solution $U_{\mathcal {E}}\in Y^1_{\mathcal {E}}$, given by $U_{\mathcal {E}}(t)={\mathcal {E}}^1(t)[U_{0{\mathcal {E}}}]$ for all $t\in \mathbb {R}$ and hence continuously depending on $U_{0{\mathcal {E}}}$ in the topologies of the respective spaces. Moreover, $U_{\mathcal {E}}$ is a strong solution if and only if $U_{0{\mathcal {E}}}\in D(A_{\mathcal {E}})$. Next, if ${\dot{U}}_{\mathcal {P}}\in {\dot{Y}}^1_{\mathcal {P}}$ denotes the unique generalized solution of problem (3.25) with data ${\dot{U}}_{0{\mathcal {P}}}=F_{_{{\mathcal {E}}{\mathcal {P}}}}U_{0{\mathcal {E}}}$, one has $U_{\mathcal {E}}=\Phi _{_{{\mathcal {P}}{\mathcal {E}}}}{\dot{U}}_{\mathcal {P}}$.

• II) The subspace $\mathcal {H}^1_{\mathcal {E}^c}$ is invariant under the flow of $\{{\mathcal {E}}^1(t),t\in \mathbb {R}\}$, so the unbounded operator $-A_{\mathcal {E}^c}: D(A_{\mathcal {E}^c})\subset \mathcal {H}^1_{\mathcal {E}^c}\rightarrow \mathcal {H}^1_{\mathcal {E}^c}$ defined by

  $$\begin{aligned} D(A_{\mathcal {E}^c})=D(A_{\mathcal {E}})\cap \mathcal {H}^1_{\mathcal {E}^c},\qquad A_{\mathcal {E}^c}U=A_{\mathcal {E}}U\quad \text {for all }U\in D(A_{\mathcal {E}^c}), \end{aligned}$$

  (5.17)

  generates on $\mathcal {H}^1_{\mathcal {E}^c}$ the subspace group $\{{\mathcal {E}}_c^1(t),t\in \mathbb {R}\}$ given by ${\mathcal {E}}_c^1(t)={\mathcal {E}}^1(t)_{|\mathcal {H}^1_{\mathcal {E}^c}}$ for all $t\in \mathbb {R}$. This last group is similar to the group $\{{\mathcal {P}}_c^1(t),t\in \mathbb {R}\}$ in Corollary 3.16, since we have

  $$\begin{aligned} {\mathcal {E}}_c^1(t)=F_{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}{\mathcal {P}}_c^1(t) F_{_{{\mathcal {E}^c}\!{\mathcal {P}^c}}}\qquad \text {for all }t\in \mathbb {R}. \end{aligned}$$

  (5.18)

  Consequently, part I) continues to hold when replacing the spaces $\mathcal {H}^1_{\mathcal {E}}$, $Y^1_{\mathcal {E}}$, ${\dot{Y}}^1_{\mathcal {P}}$ with the spaces $\mathcal {H}^1_{\mathcal {E}^c}$, $Y^1_{\mathcal {E}^c}$, ${\dot{Y}}^1_{\mathcal {P}^c}$, the operator $A_{\mathcal {E}}$ with $A_{\mathcal {E}^c}$ and restricting problems (3.25) and (5.15) to the spaces $\dot{\mathcal {H}}^1_{\mathcal {P}^c}$ and $\mathcal {H}^1_{\mathcal {E}^c}$.

Proof

To prove part I), we use arguments similar to those in the proof of Theorem 4.6. Indeed, by standard semigroup theory, (5.16) defines on $\mathcal {H}^1_{\mathcal {E}}$ the strongly continuous group $\{{\mathcal {E}}^1(t),t\in \mathbb {R}\}$, similar to $\{{\mathcal {P}}^1(t),t\in \mathbb {R}\}$. Its generator is the operator $-B_2$ defined by $D(B_2)=F_{_{{\mathcal {P}}{\mathcal {E}}}}D(\dot{A}_{\mathcal {P}})$ and $B_2=F_{_{{\mathcal {P}}{\mathcal {E}}}}\dot{A}_{\mathcal {P}}F_{_{{\mathcal {E}}{\mathcal {P}}}}$. By (3.21), (5.12) and Lemma 5.1, we then get $D(B_2)=F_{_{{\mathcal {P}}{\mathcal {E}}}}{\dot{D}}^1_{\mathcal {P}}=D^1_{\mathcal {E}}=D(A_{\mathcal {E}})$, dense in $\mathcal {H}^1_{\mathcal {E}}$. Moreover, for any $U=(p,\textbf{v},v,z)\in D(A_{\mathcal {E}})$, using (3.21) and (5.5), one has

$$\begin{aligned} B_2U=F_{_{{\mathcal {P}}{\mathcal {E}}}}{\dot{A}}_{\mathcal {P}}\begin{pmatrix}{\dot{u}}\\ v\\ \tfrac{1}{\rho _0}p\\ z\end{pmatrix} =\begin{pmatrix}-\tfrac{1}{\rho _0}\pi _0 p\\ -z\\ \tfrac{B}{\rho _0}{{\,\textrm{div}\,}}\textbf{v}\\ \frac{1}{\mu }\left[ -{{\,\mathrm{div_{_\Gamma }}\,}}(\sigma \nabla _\Gamma v)+\delta z+\kappa v+p_{|\Gamma _1}\right] \end{pmatrix}=A_{\mathcal {E}}U, \end{aligned}$$

where ${\dot{u}}\in {\dot{H}}^2(\Omega )$ is the unique solution of $-\nabla {\dot{u}}=\textbf{v}$. Hence, $A_{\mathcal {E}}=B_2$, so $-A_{\mathcal {E}}$ generates the group $\{{\mathcal {E}}^1(t),t\in \mathbb {R}\}$. The proof of part I) can then be completed using Lemma 2.1-i) and iii).

To prove part II), we first remark that, by Corollary 3.16 and (5.7), formula (5.18) defines on $\mathcal {H}^1_{\mathcal {E}^c}$ the group $\{{\mathcal {E}}_c^1(t),t\in \mathbb {R}\}$ similar to the group $\{{\mathcal {P}}_c^1(t),t\in \mathbb {R}\}$. So by (5.16) and the invariance of $\dot{\mathcal {H}}^1_{\mathcal {P}^c}$ with respect to the group $\{{\mathcal {P}}^1(t),t\in \mathbb {R}\}$ asserted in Lemma 3.13, we get that $\mathcal {H}^1_{\mathcal {E}^c}$ is invariant with respect to $\{{\mathcal {E}}^1(t),t\in \mathbb {R}\}$, so $\{{\mathcal {E}}_c^1(t),t\in \mathbb {R}\}$ is the subspace group of $\{{\mathcal {E}}^1(t),t\in \mathbb {R}\}$ which has -$A_{\mathcal {E}^c}$ as generator. To complete the proof of part II) is then trivial. $\square$

We now set, for $n\in {\widetilde{\mathbb {N}}}$, $n\le r$, the Fréchet spaces

$$\begin{aligned} \mathcal {T}^n_{\mathcal {E}}=\{U_{\mathcal {E}}\in Y^n_{\mathcal {E}}: U_{\mathcal {E}}\text { is a generalized solution of }(5.14)\}, \quad \mathcal {T}^n_{\mathcal {E}^c}=\mathcal {T}^n_{\mathcal {E}}\cap Y^n_{\mathcal {E}^c}, \end{aligned}$$

(5.19)

both endowed with the topology inherited from $Y^n_{\mathcal {E}}$. The following result is a trivial consequence of formulas (3.11), (5.2), Lemma 5.1 and Theorem 5.2.

Corollary 5.3

For all $n\in {\widetilde{\mathbb {N}}}$, $n\le r$, the operators $\Phi _{_{{\mathcal {P}}{\mathcal {E}}}}$ and $\Phi _{_{{\mathcal {E}}{\mathcal {P}}}}$ in (5.6) restrict to bijective isomorphisms between $\dot{\mathcal {T}}^n_{\mathcal {P}}$ and $\mathcal {T}^n_{\mathcal {E}}$, being the inverse of each other, and also to bijective isomorphisms between $\dot{\mathcal {T}}^n_{\mathcal {P}^c}$ and $\mathcal {T}^n_{\mathcal {E}^c}$.

Abstract regularity properties of solutions of (5.15) are then given as follows.

Theorem 5.4

(Regularity for (5.15))

• I) For all $n\in {\widetilde{\mathbb {N}}}$, $2\le n\le r$, one has $D(A^{n-1}_{\mathcal {E}})=D^{n-1}_{\mathcal {E}}$, the respective norms being equivalent. So the operator $\,\, {_n}A_{\mathcal {E}}$ given by (2.6) generates on $D^{n-1}_{\mathcal {E}}$ the strongly continuous group $\{{\mathcal {E}}^n(t), t\in \mathbb {R}\}$ given by

  $$\begin{aligned} {\mathcal {E}}^n(t)= F_{_{{\mathcal {P}}{\mathcal {E}}}}{\mathcal {P}}^n(t)F_{_{{\mathcal {E}}{\mathcal {P}}}}\qquad \text {for all }t\in \mathbb {R}, \end{aligned}$$

  (5.20)

  and hence similar to the group $\{{\mathcal {P}}^n(t),t\in \mathbb {R}\}$ in Proposition 3.10-II). Consequently, for any $U_{0{\mathcal {E}}}\in \mathcal {H}^1_{\mathcal {E}}$, denoting by $U_{\mathcal {E}}$ the generalized solution of (5.15), for any $n\in {\widetilde{\mathbb {N}}}$, $2\le n\le r$, one has $U_{0{\mathcal {E}}}\in D^{n-1}_{\mathcal {E}}$ if and only if $U_{\mathcal {E}}\in Y^n_{\mathcal {E}}$. In this case $U_{\mathcal {E}}$ continuously depends on it $U_{0{\mathcal {E}}}$ in the topologies of the respective spaces.

• II) All assertions in part I) continue to hold when restricting problem (5.15) to $\mathcal {H}^1_{\mathcal {E}^c}$, provided one replaces the spaces $\mathcal {H}^1_{\mathcal {E}}$, $D^{n-1}_{\mathcal {E}}$, $Y^n_{\mathcal {E}}$ with the spaces $\mathcal {H}^1_{\mathcal {E}^c}$, $D^{n-1}_{\mathcal {E}^c}$, $Y^n_{\mathcal {E}^c}$, the operators $A_{\mathcal {E}}$ and ${_n}A_{\mathcal {E}}$ with $A_{\mathcal {E}^c}$ and ${_n}A_{\mathcal {E}^c}$ and, finally, the group $\{{\mathcal {E}}^n(t),t\in \mathbb {R}\}$ with the group $\{{\mathcal {E}^c}^n(t),t\in \mathbb {R}\}$ defined by ${\mathcal {E}}_c^n(t)={\mathcal {E}}^n(t)_{|D^{n-1}_{\mathcal {E}^c}}$ for $t\in \mathbb {R}$.

Proof

To prove part I), we remark that, since $A_{\mathcal {E}}=F_{_{{\mathcal {P}}{\mathcal {E}}}}\dot{A}_{\mathcal {P}}F_{_{{\mathcal {E}}{\mathcal {P}}}}$ (see the proof of Theorem 5.2), using Lemma 5.1 and (2.5) one gets, by induction, that $D(A^{n-1}_{\mathcal {E}})=D^{n-1}_{\mathcal {E}}$, the respective norms being equivalent, and that (5.20) defines on it the strongly continuous group $\{{\mathcal {E}}^n(t), t\in \mathbb {R}\}$, which has $\,{_n}A_{\mathcal {E}}$ as generator. The proof of part I) can then be completed by using Corollary 5.3 and Lemma 5.1, since they yield the implications $U_{0{\mathcal {E}}}\in D^{n-1}_{\mathcal {E}}\Leftrightarrow F_{_{{\mathcal {E}}{\mathcal {P}}}}U_{0,{\mathcal {E}}}\in {\dot{D}}^{n-1}_{\mathcal {P}}\Leftrightarrow {\dot{U}}_{\mathcal {P}}\in {\dot{Y}}^n_{\mathcal {P}}\Leftrightarrow U_{\mathcal {E}}\in Y^n_{\mathcal {E}}$, where ${\dot{U}}_{\mathcal {P}}=\Phi _{_{{\mathcal {E}}{\mathcal {P}}}}U_{\mathcal {E}}$. The proof of part II) uses similar arguments. $\square$

5.4 Solutions of $({\mathcal {E}})$, $({\mathcal {E}^c})$, $({\mathcal {E}}_0)$ and $({\mathcal {E}}_0^c)$

To apply the abstract results in Sect. 5.3 to all problems related with $({\mathcal {E}})$, we make precise, at first, which types of solutions of them we shall consider. We recall that $({\mathcal {E}})_3$ is implicit in the definition of $X^1_{\mathcal {E}}$.

Definition 5.5

We say that

• i) $(p,\textbf{v},v)\in X^2_{\mathcal {E}}$ is a strong solution of $({\mathcal {E}})$ provided $({\mathcal {E}})_1$–$({\mathcal {E}})_2$ hold a.e. in $\mathbb {R}\times \Omega$ and $({\mathcal {E}})_4$–$({\mathcal {E}})_5$ hold a.e. on $\mathbb {R}\times \Gamma _1$, where p and $\textbf{v}$ on $\mathbb {R}\times \Gamma _1$ are taken in the pointwise trace sense given in Sect. 2.2;

• ii) $(p,\textbf{v},v)\in X^1_{\mathcal {E}}$ is a generalized solution of $({\mathcal {E}})$ provided it is the limit in $X^1_{\mathcal {E}}$ of a sequence of strong solutions of it;

• iii) $(p,\textbf{v},v)\in X^1_{\mathcal {E}}$ is a weak solution of $({\mathcal {E}})$ provided the distributional identities

  $$\begin{aligned}&\int _{-\infty }^\infty \int _\Omega [p\varphi _t+B\textbf{v}\cdot \nabla \varphi ]+\int _{-\infty }^\infty \int _{\Gamma _1}v_t\varphi =0, \end{aligned}$$

  (5.21)
  $$\begin{aligned}&\int _{-\infty }^\infty \int _\Omega [\rho _0\textbf{v}\cdot \varvec{\phi }_t +p{{\,\textrm{div}\,}}\varvec{\phi }] \nonumber \\&\quad + \int _{-\infty }^\infty \int _{\Gamma _1}\left[ \mu v_t\psi _t-\sigma (\nabla _\Gamma v,\nabla _\Gamma {\overline{\psi }})_\Gamma -\delta v_t\psi -\kappa v\psi \right] =0, \end{aligned}$$

  (5.22)

  hold for all $\varphi \in C^r_c(\mathbb {R}\times \mathbb {R}^3)$ and $\varvec{\phi }\in C^{r-1}_c(\mathbb {R}\times \mathbb {R}^3)^3$ such that $\varvec{\phi }\cdot \varvec{\nu }=0$ on $\mathbb {R}\times \Gamma _0$, where $\psi =-\varvec{\phi }\cdot \varvec{\nu }$ on $\mathbb {R}\times \Gamma _1$.

Moreover, solutions of $({\mathcal {E}})$ of the types i)–iii) above are said to be solutions of the same type of: j) problem $({\mathcal {E}^c})$ when also (1.1) holds; jj) problem $({\mathcal {E}}_0)$ when also $({\mathcal {E}}_0)_6$–$({\mathcal {E}}_0)_7$ hold in $X^1_{\mathcal {E}}$; jjj) problem $({\mathcal {E}}_0^c)$ when both j) and jj) hold.

Trivially strong solutions in the definition above are also generalized ones. Moreover, strong and generalized solutions correspond to the homologous ones of (5.14) and (5.15), as the following result shows.

Lemma 5.6

The triple $(p,\textbf{v},v)$ is a strong or generalized solution of $({\mathcal {E}})$ if and only if p, $\textbf{v}$ and v are the first three components of a solution $U_{\mathcal {E}}=(p,\textbf{v},v,z)$ of (5.14) of the same type, and in this case $v_t=z$. The same relation occurs between: solutions of $({\mathcal {E}^c})$ and solutions of the restriction of (5.14) to $\mathcal {H}^1_{\mathcal {E}^c}$; solutions of $({\mathcal {E}}_0)$ and solutions of (5.15); solutions of $({\mathcal {E}}_0^c)$ and solutions of the restriction of (5.15) to $\mathcal {H}^1_{\mathcal {E}^c}$.

Proof

By Definition 5.5 only the first assertion needs a proof. Since, by Theorem 5.4-I), strong solutions of (5.14) belong to $Y^2_{\mathcal {E}}$, using (5.13) the assertion is trivial for strong solutions. Using this fact, given any generalized solution $(p,\textbf{v},v)\in X^1_{\mathcal {E}}$ of $({\mathcal {E}})$, the quadruple $(p,\textbf{v},v,v_t)\in Y^1_{\mathcal {E}}$ is a generalized solution of (5.14). The converse implication, with the identity $v_t=z$, is proved as in the proof of Lemma 4.11. $\square$

By Lemma 5.6, generalized and strong solutions of problems related with $({\mathcal {E}})$ naturally arise from Theorems 5.2 and 5.4. While strong solutions are a.e. classical solutions, generalized solutions of $({\mathcal {E}})$ solve all equations (but $({\mathcal {E}})_3$) in a quite indirect sense. Since, also in this case, solutions in $X^1_{\mathcal {E}}$ are not regular enough to be a.e. solutions, it is (again) natural to consider $({\mathcal {E}})_1$–$({\mathcal {E}})_2$ and $({\mathcal {E}})_4$–$({\mathcal {E}})_5$ in a distributional sense. On the other hand, while for $(p,\textbf{v},v)\in X^1_{\mathcal {E}}$ equations $({\mathcal {E}})_1$ and $({\mathcal {E}})_2$ have the natural distributional forms

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}\times \Omega } p\varphi _t+B\textbf{v}\cdot \nabla \varphi =0{} & {} \qquad \text {for all } \varphi \in \mathcal {D}(\mathbb {R}\times \Omega ), \end{aligned} \end{aligned}$$

(5.23)

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}\times \Omega }\rho _0\textbf{v}\cdot \varvec{\phi }_t+p {{\,\textrm{div}\,}}\varvec{\phi }=0{} & {} \qquad \text {for all }\varvec{\phi }\in \mathcal {D}(\mathbb {R}\times \Omega )^3, \end{aligned} \end{aligned}$$

(5.24)

equations $({\mathcal {E}})_4$ and $({\mathcal {E}})_5$ cannot be written in the sense of distributions unless the terms p and $\textbf{v}\cdot \varvec{\nu }$ in them have some trace sense on $\mathbb {R}\times \Gamma _1$. This type of difficulty, also arising for problem $({\mathcal {L}})$, was solved in Definition 5.5-iii) by combining equations $({\mathcal {E}})_1$ and $({\mathcal {E}})_5$ in the single distributional identity (5.21) and equations $({\mathcal {E}})_2$ and $({\mathcal {E}})_4$ in the single distributional identity (5.22). Of course one can be more precise when solutions are more regular. As to the term p one can simply ask that $p\in L^1_{\text {loc}}(\mathbb {R}; H^1(\Omega ))$. In this case equation $({\mathcal {E}})_4$, formally identical to $({\mathcal {L}}')_4$, has the distributional form

$$\begin{aligned} \int _{-\infty }^\infty \int _{\Gamma _1}\mu v_t\psi _t-\sigma (\nabla _\Gamma v,\nabla _\Gamma {\overline{\psi }})_\Gamma -\delta v_t\psi -\kappa v\psi -p\psi =0 \end{aligned}$$

(5.25)

for all $\psi \in C^r_c(\mathbb {R}\times \Gamma _1)$. As for the term $\textbf{v}\cdot \varvec{\nu }$, recalling (5.3), one can give a sense to it by asking that $\textbf{v}\in L^1_{\text {loc}}(\mathbb {R}; H({{\,\textrm{div}\,}},\Omega ))$. Indeed, using (5.4), in this case one has $\textbf{v}\cdot \varvec{\nu }\in L^1_{\text {loc}}(\mathbb {R}; H^{-1/2}(\Gamma ))$ and equation $({\mathcal {E}})_5$ has the natural distributional form

$$\begin{aligned} \int _{-\infty }^\infty \langle \textbf{v}\cdot \varvec{\nu }, \varphi _{|\Gamma }\rangle _{H^{1/2}(\Gamma )} +\int _{\mathbb {R}\times \Gamma _1}v_t\varphi =0\qquad \text {for all }\varphi \in C^r_c(\mathbb {R}\times \Gamma ). \end{aligned}$$

(5.26)

After these preliminary remarks, we can finally state the following result, which shows that Definition 5.5-iii) is the closest possible approximation of the notion of distributional solution of $({\mathcal {E}})$. It will be also useful in the sequel. Its straightforward proof is omitted. The interested reader can find it in the preprint version of the paper, see https://arxiv.org/abs/2307.07775.

Proposition 5.7

Let $(p,\textbf{v},v)\in X^1_{\mathcal {E}}$ be such that $p\in L^1_{\text {loc}}(\mathbb {R}; H^1(\Omega ))$ and $\textbf{v}\in L^1_{\text {loc}}(\mathbb {R}; H({{\,\textrm{div}\,}},\Omega ))$. Then $(p,\textbf{v},v)$ is a weak solution of $({\mathcal {E}})$ if and only if it satisfies the distributional equations (5.23)–(5.26).

Beside its independent interest, Proposition 5.7 also allows to point out the trivial relations occurring among the three types of solutions of $({\mathcal {E}})$ introduced in Definition 5.5. They are analogous to the ones occurring for problems $({\mathcal {P}})$ and $({\mathcal {L}})$.

Lemma 5.8

Let $(p,\textbf{v},v)\in X^1_{\mathcal {E}}$ be a solution of $({\mathcal {E}})$ according to Definition 5.5. Then strong $\Rightarrow$ generalized $\Rightarrow$ weak and, if $(p,\textbf{v},v)\in X^2_{\mathcal {E}}$, weak $\Rightarrow$ strong.

Proof

Strong solutions are also generalized ones and, by Proposition 5.7, they are also weak. Since (5.21) and (5.22) are stable with respect to the convergence in $X^1_{\mathcal {E}}$, we then get that generalized $\Rightarrow$ weak. To prove the final conclusion, let $(p,\textbf{v},v)\in X^2_{\mathcal {E}}$ be a weak solution. By Proposition 5.7 it also satisfies $({\mathcal {E}})_1$, $({\mathcal {E}})_2$ and $({\mathcal {E}})_4$ in a distributional sense. Being regular enough, it also satisfies them a.e., so it is a strong solution. $\square$

In view of Theorem 5.2, which by Lemmas 5.6 and 5.8 provides the existence of generalized and hence weak solutions of $({\mathcal {E}})$, it is natural to pose the question of uniqueness of weak solutions of $({\mathcal {E}}_0)$. The answer is given by the following result.

Theorem 5.9

(Uniqueness) Weak solutions of $({\mathcal {E}}_0)$ are unique.

Proof

By linearity it is enough to prove that $p_0=0$, $\textbf{v}_0=0$ and $v_0=v_1=0$ yield $(p,\textbf{v},v)\equiv 0$ in $\mathbb {R}$. At first we remark that, taking in (5.21) test functions $\varphi (t,x)=\varphi _1(t)\varphi _0(x)$, $\varphi _1\in C^r_c(\mathbb {R})$, $\varphi _0\in C^r_c(\mathbb {R}^3)$, we get

$$\begin{aligned} \int _{-\infty }^\infty \varphi _1'\int _\Omega p\varphi _0+ \int _{-\infty }^\infty B \varphi _1\left( \int _\Omega \textbf{v}\cdot \nabla \varphi _0+\int _{\Gamma _1} v_t\varphi _0\right) =0. \end{aligned}$$

(5.27)

By density (5.27) holds true for all $\varphi _0\in H^1(\Omega )$. Since $\int _\Omega p\varphi _0, \int _\Omega \textbf{v}\cdot \nabla \varphi _0,\int _{\Gamma _1}v_t\varphi _0\in C(\mathbb {R})$, by (5.27) we then get that $\int _\Omega p\varphi _0\in C^1(\mathbb {R})$ and $\frac{d}{dt} \int _\Omega p\varphi _0=B\int _\Omega \textbf{v}\cdot \nabla \varphi _0+ B \int _{\Gamma _1}v_t\varphi _0$ for all $\varphi _0\in H^1(\Omega )$. Consequently, setting $\textbf{r}\in C^1(\mathbb {R}; H^0_{{{\,\textrm{curl}\,}}0}(\Omega ))$ (since $\textbf{v}\in C(\mathbb {R};H^0_{{{\,\textrm{curl}\,}}0}(\Omega ))$ by $\textbf{r}(t)=\int _0^t \textbf{v}(\tau )\,d\tau$ for all $t\in \mathbb {R}$, since $p(0)=p_0=0$, $v(0)=v_0=0$ and $\textbf{r}(0)=0$, we have

$$\begin{aligned} \int _\Omega p(t)\varphi _0=B\int _\Omega \textbf{r}(t)\cdot \nabla \varphi _0+ B \int _{\Gamma _1}v(t)\varphi _0 \qquad \text {for all }t\in \mathbb {R}. \end{aligned}$$

(5.28)

Taking in (5.28) test functions $\varphi _0\in \mathcal {D}(\Omega )$, we get that $-B{{\,\textrm{div}\,}}\textbf{r}(t)=p(t)$ in $\mathcal {D}'(\Omega )$, for all $t\in \mathbb {R}$. Since $p(t)\in L^2(\Omega )$ for all $t\in \mathbb {R}$, we consequently get that $\textbf{r}(t)\in H({{\,\textrm{div}\,}},\Omega )$ for all $t\in \mathbb {R}$. Thus, by (5.4), for all $t\in \mathbb {R}$ and $\varphi _0\in H^1(\Omega )$ we have

$$\begin{aligned} \int _\Omega p(t)\varphi _0=B\int _\Omega \textbf{r}(t)\cdot \nabla \varphi _0- B\langle \textbf{r}(t)\cdot \varvec{\nu },{\varphi _0}_{|\Gamma } \rangle _{H^{1/2}(\Gamma )}. \end{aligned}$$

(5.29)

By comparing (5.28) and (5.29), since the trace operator $H^1(\Omega )\rightarrow H^{1/2}(\Gamma )$ is surjective, recalling the splitting (2.1), we then get that $\textbf{r}(t)\cdot \varvec{\nu }=-v(t)$ in $H^{-1/2}(\Gamma )$ for all $t\in \mathbb {R}$. Since $v(t)\in H^1(\Gamma _1)$ we have $\textbf{r}(t)\cdot \varvec{\nu }\in H^1(\Gamma )\hookrightarrow H^{1/2}(\Gamma )$. Summarizing we have ${{\,\textrm{curl}\,}}\textbf{r}(t)=0$, ${{\,\textrm{div}\,}}\textbf{r}(t)\in L^2(\Omega )$ and $\textbf{r}(t)\cdot \varvec{\nu }\in H^{1/2}(\Gamma )$; hence, we can apply [12, Chapter 9, Corollary 1, p. 212] to conclude that $\textbf{r}(t)\in H^1_{{{\,\textrm{curl}\,}}0}(\Omega )$ for all $t\in \mathbb {R}$. Hence, $\textbf{r}(t)$ solves problem (4.8) with $w=p(t)/\rho _0$ and, by Lemma 4.2, we get $\textbf{r}\in C(\mathbb {R}; H^1_{{{\,\textrm{curl}\,}}0}(\Omega ))$ and $\textbf{r}(t)\cdot \varvec{\nu }=-v(t)$ on $\Gamma _1$, $\textbf{r}(t)\cdot \varvec{\nu }=0$ on $\Gamma _0$ in the trace sense. Consequently, $(\textbf{r},v)\in X^1_{\mathcal {L}}$, so $(\textbf{r},v)\in X^1_{{\mathcal {L}}}$ and $\textbf{r}_t=\textbf{v}$. Consequently, (5.22) translates into (4.35) and $(\textbf{r},v)$ is a weak solution of $({\mathcal {L}}_0)$ with $\textbf{r}_0=\textbf{r}_1=\textbf{v}_0=0$ and $v_0=v_1=0$. By Theorem 4.16 we then get that $\textbf{r}\equiv 0$ and $v\equiv 0$ in $\mathbb {R}$, so also $\textbf{v}=\textbf{r}_t\equiv 0$ in $\mathbb {R}$, concluding the proof. $\square$

5.5 Main results for problems $({\mathcal {E}})$, $({\mathcal {E}}_0)$, $({\mathcal {E}}^c)$ and $({\mathcal {E}}^c_0)$

We can finally prove Theorems 1.4 and 1.7 and deal with optimal regularity issues.

Proof of Theorem 1.4

By combining Theorem 5.2 and Lemma 5.6, for all $U_{0{\mathcal {E}}}\in \mathcal {H}^1_{\mathcal {E}}$ problem $({\mathcal {E}}_0)$ has a unique generalized solution $(p,\textbf{v},v)\in X^1_{\mathcal {E}}$, continuously depending on $U_{0{\mathcal {E}}}$. Moreover, $(p,\textbf{v},v)$ is the unique generalized solution of problem $({\mathcal {E}}^c_0)$ if and only if $U_{0{\mathcal {E}}}\in \mathcal {H}^1_{\mathcal {E}^c}$. By Lemma 5.8 the triple $(p,\textbf{v},v)$ is also a weak solution of $({\mathcal {E}}_0)$ (or of $({\mathcal {E}}^c_0)$, when $U_{0{\mathcal {E}}}\in \mathcal {H}^1_{\mathcal {E}^c}$) which, by Theorem 5.9, is unique among them. Moreover, again by Theorem 5.2 and Lemma 5.6, the solution is strong if and only if $U_{0{\mathcal {E}}}\in \mathcal {H}^2_{\mathcal {E}}$ and $(\textbf{v}_0,v_1)\in \mathbb {H}^1_{\mathcal {L}}$, i.e., $U_{0{\mathcal {E}}}\in D(A_{\mathcal {E}})$, which is dense in $\mathcal {H}^1_{\mathcal {E}}$. The same conclusions trivially hold for problem $({\mathcal {E}}^c_0)$. The continuous dependence in this case follows by Theorem 5.4. Finally, the energy identity (1.13) can be obtained as in the proof of Theorem 1.3 directly from (1.5), or re-deriving it for strong solutions and then using a density argument. $\square$

The uniqueness of weak solutions of $({\mathcal {E}}_0)$ in Theorem 5.9, together with the existence of a generalized and hence weak solution of it, shows that solutions of $({\mathcal {E}})$ are equivalently weak or generalized. Combining this remark with Lemma 5.8, we thus obtain that, also for Eulerian problems, all types of solutions in Definition 5.5coincide, strong solutions being defined only in the class $X^2_{\mathcal {E}}$. In the sequel we consequently shall only deal with weak solutions of $({\mathcal {E}})$ and $({\mathcal {E}^c})$, i.e., with elements of the spaces $\mathcal {S}^n_{\mathcal {E}}$ and $\mathcal {S}^n_{\mathcal {E}^c}$ defined in (1.14). Recalling the spaces $\mathcal {T}^n_{\mathcal {E}}$ and $\mathcal {T}^n_{\mathcal {E}^c}$ defined in (5.19), it is useful to point out the following result.

Lemma 5.10

The operator $\mathcal {I}''\in \mathcal {L}(X^1_{\mathcal {E}};Y^1_{\mathcal {E}})$, defined by $\mathcal {I}''(p,\textbf{v},v)=(p,\textbf{v},v,v_t)$, restricts to a bijective isomorphism $\mathcal {I}^n_{\mathcal {E}}\in \mathcal {L}(\mathcal {S}^n_{\mathcal {E}};\mathcal {T}^n_{\mathcal {E}})$ for each $n\in {\widetilde{\mathbb {N}}}$, $n\le r$. Its inverse $\left( \mathcal {I}^n_{\mathcal {E}}\right) ^{-1}\in \mathcal {L}(\mathcal {T}^n_{\mathcal {E}};\mathcal {S}^n_{\mathcal {E}})$ is simply given by $\left( \mathcal {I}^n_{\mathcal {E}}\right) ^{-1}(p,\textbf{v},v,z)=(p,\textbf{v},v)$.

Moreover, $\mathcal {I}^n_{\mathcal {E}}$ and $\left( \mathcal {I}^n_{\mathcal {E}}\right) ^{-1}$ further restrict to

$$\begin{aligned} \mathcal {I}^n_{\mathcal {E}^c}\in \mathcal {L}(\mathcal {S}^n_{\mathcal {E}^c};\mathcal {T}^n_{\mathcal {E}^c})\qquad \text {and}\quad \left( \mathcal {I}^n_{\mathcal {E}^c}\right) ^{-1}\in \mathcal {L}(\mathcal {T}^n_{\mathcal {E}^c};\mathcal {S}^n_{\mathcal {E}^c}). \end{aligned}$$

Proof

The first assertion follows by (5.2), Lemma 5.6 and the equivalence remarked just before the statement. The second one follows by the first one since $X^n_{\mathcal {E}^c}=\left\{ (p,\textbf{v},v)\in X^n_{\mathcal {E}}: \int _\Omega \frac{\partial ^ip}{\partial t^i}=B\int _{\Gamma _1}\frac{\partial ^iv}{\partial t^i}\quad \hbox { for}\ i=0,\ldots ,n-1\right\}$. $\square$

We can now give the

Proof of Theorem 1.7

By Proposition 3.12, Lemma 5.1, Lemma 5.10, (5.5) and (5.6), for any $(u,v)\in \mathcal {S}^1_{\mathcal {P}}$ we have $\left( \mathcal {I}^1_{\mathcal {E}}\right) ^{-1}\cdot \Phi _{_{{\mathcal {P}}{\mathcal {E}}}}\cdot \mathcal {J}^1_{\mathcal {P}}(u,v)=(\rho _0 u_t,-\nabla u,v)$, so (1.19) defines the operators

$$\begin{aligned} \Psi _{_{{\mathcal {P}}{\mathcal {E}}}}=\left( \mathcal {I}^1_{\mathcal {E}}\right) ^{-1}\!\!\!\!\cdot \Phi _{_{{\mathcal {P}}{\mathcal {E}}}}\!\!\cdot \mathcal {J}^1_{\mathcal {P}}\in \mathcal {L}(\mathcal {S}^1_{\mathcal {P}};\mathcal {S}^1_{\mathcal {E}}), \, \Psi _{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}=\left( \mathcal {I}^1_{\mathcal {E}^c}\right) ^{-1}\!\!\!\!\cdot \Phi _{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}\!\! \cdot \mathcal {J}^1_{\mathcal {P}^c}\in \mathcal {L}(\mathcal {S}^1_{\mathcal {P}^c};\mathcal {S}^1_{\mathcal {E}^c}) \end{aligned}$$

(5.30)

given in the statements. They are surjective and satisfy $\text {Ker } \Psi _{_{{\mathcal {P}}{\mathcal {E}}}}= \text {Ker } \Psi _{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}=\mathbb {C}_{X_{\mathcal {P}}}$. One consequently gets the bijective isomorphisms

$$\begin{aligned} {{\dot{\Psi }}}_{_{{\mathcal {P}}{\mathcal {E}}}}=\left( \mathcal {I}^1_{\mathcal {E}}\right) ^{-1}\!\!\!\!\cdot \Phi _{_{{\mathcal {P}}{\mathcal {E}}}}\!\!\cdot \dot{\mathcal {J}}^1_{\mathcal {P}}\in \mathcal {L}(\dot{\mathcal {S}}^1_{\mathcal {P}};\mathcal {S}^1_{\mathcal {E}}), \,\, {{\dot{\Psi }}}_{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}\!\!=\left( \mathcal {I}^1_{\mathcal {E}^c}\right) ^{-1}\!\!\!\!\cdot \Phi _{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}\! \!\cdot \dot{\mathcal {J}}^1_{\mathcal {P}^c}\!\!\in \mathcal {L}(\mathcal {S}^1_{\mathcal {P}^c};\mathcal {S}^1_{\mathcal {E}^c}) \end{aligned}$$

(5.31)

given in (1.20). Their inverses are the operators

$$\begin{aligned} {{\dot{\Psi }}}_{_{{\mathcal {E}}{\mathcal {P}}}}=\left( \dot{\mathcal {J}}^1_{\mathcal {P}}\right) ^{-1}\!\!\!\!\!\!\!\cdot \Phi _{_{{\mathcal {E}}{\mathcal {P}}}}\cdot \mathcal {I}^1_{\mathcal {E}}\in \mathcal {L}(\mathcal {S}^1_{\mathcal {E}};\dot{\mathcal {S}}^1_{\mathcal {P}}), \,\, {{\dot{\Psi }}}_{_{{\mathcal {E}^c}\!{\mathcal {P}^c}}}=\left( \dot{\mathcal {J}}^1_{\mathcal {P}^c}\right) ^{-1}\!\!\!\!\!\!\!\cdot \Phi _{_{{\mathcal {E}^c}\!{\mathcal {P}^c}}}\!\!\cdot \mathcal {I}^1_{\mathcal {E}^c}\!\in \! \mathcal {L}(\mathcal {S}^1_{\mathcal {E}^c};\dot{\mathcal {S}}^1_{\mathcal {P}^c}), \end{aligned}$$

(5.32)

so formulas (1.20) trivially hold.

To check that ${{\dot{\Psi }}}_{_{{\mathcal {E}}{\mathcal {P}}}}$ is given by (1.21)–(1.22) we recall that, by (5.5), (5.6) and Lemma 5.10, for all $(p,\textbf{v},v)\in \mathcal {S}^1_{\mathcal {E}}$ and $t\in \mathbb {R}$ we have

$$\begin{aligned}{}[\Phi _{_{{\mathcal {E}}{\mathcal {P}}}}\mathcal {I}^1_{\mathcal {E}}(p,\textbf{v},v)](t)=F_{_{{\mathcal {E}}{\mathcal {P}}}}(p(t), \textbf{v}(t), v(t),v_t(t))=({\dot{u}}(t), v(t), p(t)/\rho _0, v_t(t)), \end{aligned}$$

where ${\dot{u}}(t)\in {\dot{H}}^1(\Omega )$ is the unique solution of $-\nabla {\dot{u}}(t)=\textbf{v}(t)$. Since $\mathcal {J}^1_{\mathcal {P}}$ is surjective there is $(u,v)\in \mathcal {S}^1_{\mathcal {P}}$ such that

$$\begin{aligned} ({\dot{u}}(t), v(t), p(t)/\rho _0, v_t(t))=(\pi _0 u(t),v(t),u_t(t),v_t(t))\qquad \text {for all }t\in \mathbb {R}, \end{aligned}$$

(5.33)

that is

$$\begin{aligned} \pi _0 u(t)={\dot{u}}(t),\quad \text {and}\quad p(t)=\rho _0u_t(t)\qquad \text {for all }t\in \mathbb {R}. \end{aligned}$$

(5.34)

Clearly, (5.34) implies (1.21)–(1.22). Since ${{\dot{\Psi }}}_{_{{\mathcal {E}}{\mathcal {P}}}}$ is bijective, by (1.19) one gets (1.23), while (1.24) is trivial. To prove (1.25) we remark that, as we just proved, $(p,\textbf{v},v)=\Psi _{_{{\mathcal {P}}{\mathcal {E}}}}(u,v)$ is equivalent to (5.33), that is (5.32). By uniqueness then it is equivalent to (5.32) at $t=0$, that is to $-\nabla u_0=\textbf{v}_0$ and $p_0=\rho _0u_1$. $\square$

We now give an optimal regularity result for problems $({\mathcal {E}}_0)$, $({\mathcal {E}}^c_0)$. We also show that the operators in Theorem 1.7 behave well with respect to regularity classes.

Theorem 5.11

(Regularity for $({\mathcal {E}}_0)$, $({\mathcal {E}}^c_0)$ and $({\mathcal {E}}) \rightleftarrows ({\mathcal {P}})$, $({\mathcal {E}^c}) \rightleftarrows ({\mathcal {P}^c})$) For all data $U_{0{\mathcal {E}}}=(p_0,\textbf{v}_0, v_0,v_1)\in \mathcal {H}^1_{\mathcal {E}}$ and $n\in {\widetilde{\mathbb {N}}}$, $2\le n\le r$, the weak solution $(p,\textbf{v},v)$ given in Theorem 1.4 belongs to $X_{\mathcal {E}}^n$ if and only if $U_{0{\mathcal {E}}}\in D^{n-1}_{\mathcal {E}}$. In this case $(p,\textbf{v},v)$ continuously depends on the data $U_{0{\mathcal {E}}}$ in the topologies of the respective spaces. In previous assertions, we can replace problem $({\mathcal {E}}_0)$ with problem $({\mathcal {E}}^c_0)$, provided we, respectively, replace $H^1_{\mathcal {E}}$, $D^{n-1}_{\mathcal {E}}$ and $X^n_{\mathcal {E}}$ with $H^1_{\mathcal {E}^c}$, $D^{n-1}_{\mathcal {E}^c}$ and $X^n_{\mathcal {E}^c}$.

Moreover, the operators $\Psi _{_{{\mathcal {P}}{\mathcal {E}}}}$ and $\Psi _{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}$ in Theorem 1.7, respectively, restrict to surjective operators $\Psi ^n_{_{{\mathcal {P}}{\mathcal {E}}}}\in \mathcal {L}(\mathcal {S}^n_{\mathcal {P}};\mathcal {S}^n_{\mathcal {E}})$ and $\Psi ^n_{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}\in \mathcal {L}(\mathcal {S}^n_{\mathcal {P}^c};\mathcal {S}^n_{\mathcal {E}^c})$, and we have $\ker \Psi ^n_{_{{\mathcal {P}}{\mathcal {E}}}}=\ker \Psi ^n_{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}=\mathbb {C}_{X_{\mathcal {P}}}$. Consequently, the bijective isomorphisms ${\dot{\Psi }}_{_{{\mathcal {P}}{\mathcal {E}}}}$, ${\dot{\Psi }}_{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}$, ${\dot{\Psi }}_{_{{\mathcal {E}}{\mathcal {P}}}}$ and ${\dot{\Psi }}_{_{{\mathcal {E}^c}\!{\mathcal {P}^c}}}$ in the same result, respectively, restrict to bijective isomorphisms ${\dot{\Psi }}_{_{{\mathcal {P}}{\mathcal {E}}}}^n\in \mathcal {L}(\dot{\mathcal {S}}^n_{\mathcal {P}};\mathcal {S}^n_{\mathcal {E}})$, ${\dot{\Psi }}_{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}^n\in \mathcal {L}(\dot{\mathcal {S}}^n_{\mathcal {P}^c};\mathcal {S}^n_{\mathcal {E}^c})$, ${\dot{\Psi }}_{_{{\mathcal {E}}{\mathcal {P}}}}^n\in \mathcal {L}(\mathcal {S}^n_{\mathcal {E}};\dot{\mathcal {S}}^n_{\mathcal {P}})$, and ${\dot{\Psi }}_{_{{\mathcal {E}^c}\!{\mathcal {P}^c}}}^n\in \mathcal {L}(\mathcal {S}^n_{\mathcal {E}^c};\dot{\mathcal {S}}^n_{\mathcal {P}^c})$.

Proof

The optimal regularity for problem $({\mathcal {E}}_0)$, and hence also for problem $({\mathcal {E}}^c_0)$, follows by combining Theorem 5.4 and Lemma 5.10. Thank to formulas (5.30)–(5.32), the restriction properties of the operators ${\dot{\Psi }}_{_{{\mathcal {P}}{\mathcal {E}}}}$, ${\dot{\Psi }}_{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}$, ${\dot{\Psi }}_{_{{\mathcal {E}}{\mathcal {P}}}}$ and ${\dot{\Psi }}_{_{{\mathcal {E}^c}\!{\mathcal {P}^c}}}$ follow by combining Proposition 3.12, Lemmas 5.1 and 5.10. $\square$

5.6 The relation between $({\mathcal {E}^c})$ and $({\mathcal {L}})$ or $({\mathcal {L}}')$

We start by completing the proofs of the main results stated in Sect. 1.

Proof of Corollary 1.9

The result is obtained by simply combining Theorems 1.5 and 1.7. Indeed, we define the bijective isomorphism $\Psi _{{\mathcal {E}^c}{\mathcal {L}}}$ by (1.29), so its inverse $\Psi _{{\mathcal {L}}{\mathcal {E}^c}}$ is given by $\Psi _{_{{\mathcal {L}}{\mathcal {E}^c}}}={\dot{\Psi }}_{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}\cdot {\dot{\Psi }}_{_{{\mathcal {L}}{\mathcal {P}^c}}}$ as stated. To evaluate the action of these two isomorphisms, we use Theorems 1.5 and 1.7. In particular, for all $(p,\textbf{v},v)\in \mathcal {S}^1_{\mathcal {E}^c}$, by Theorem 1.7 we have ${\dot{\Psi }}_{{\mathcal {E}^c}{\mathcal {P}^c}}(p,\textbf{v},v)=(u,v)+\mathbb {C}_{X_{\mathcal {P}}}$, where u is given by (1.22), and (1.23) holds true. Hence, by Theorem 1.5, $\Psi _{{\mathcal {E}^c}{\mathcal {L}}}(p,\textbf{v},v)={{\dot{\Psi }}}_{{\mathcal {P}^c}{\mathcal {L}}}[(u,v)+\mathbb {C}_{X_{\mathcal {P}}}]=(\textbf{r},v)$ where $\textbf{r}$ is given by (1.26), and (1.27) holds, together with (1.28). Conversely, for any $(\textbf{r},v)\in \mathcal {S}^1_{\mathcal {L}}$, by Theorem 1.5 we have ${{\dot{\Psi }}}_{{\mathcal {L}}{\mathcal {P}^c}}(\textbf{r},v)=(u,v)+\mathbb {C}_{X_{\mathcal {P}}}$, where u is given by (1.17) and (1.16) holds true. Hence, by Theorem 1.7, we get (1.30). Finally, (1.31) follows by (1.18) and (1.25). $\square$

Although, as we have just seen, the relation between solutions of $({\mathcal {E}^c})$ and $({\mathcal {L}})$ follows by Theorems 1.5 and 1.7, this relation can be understood at a more abstract level by considering the groups $\{{\mathcal {E}}^1_c(t),t\in \mathbb {R}\}$ and $\{{\mathcal {L}}^1(t),t\in \mathbb {R}\}$. Indeed, we combine the main isomorphisms between phase spaces in Sects. 4.2 and 5.2 by setting

$$\begin{aligned} F_{_{{\mathcal {E}^c}\!{\mathcal {L}}}}=F_{_{{\mathcal {P}^c}\!{\mathcal {L}}}}\cdot F_{_{{\mathcal {E}^c}\!{\mathcal {P}^c}}}\in \mathcal {L}(\mathcal {H}^1_{\mathcal {E}^c};\mathcal {H}^1_{\mathcal {L}}),\quad F_{_{{\mathcal {L}}{\mathcal {E}^c}}}=F_{_{{\mathcal {P}^c}\!{\mathcal {E}^c}}}\cdot F_{_{{\mathcal {L}}{\mathcal {P}^c}}}\in \mathcal {L}(\mathcal {H}^1_{\mathcal {L}};\mathcal {H}^1_{\mathcal {E}^c}), \end{aligned}$$

(5.35)

trivially being the inverse of each other. By (4.18)–(4.19), (4.20) and (5.5) they are given by

$$\begin{aligned} F_{_{{\mathcal {E}^c}\!{\mathcal {L}}}}(p,\textbf{v},v,z)=(\textbf{r},v,\textbf{v},z),\qquad F_{_{{\mathcal {L}}{\mathcal {E}^c}}}(\textbf{r},v,\textbf{s},z)=(-B{{\,\textrm{div}\,}}\textbf{r},\textbf{s},v,z), \end{aligned}$$

(5.36)

where, in the first identity, $\textbf{r}\in H^1_{{{\,\textrm{curl}\,}}0}(\Omega )$ is the unique solution of (4.8) with $w=p/\rho _0$. We then get the following result.

Theorem 5.12

The groups $\{{\mathcal {L}}^1(t),t\in \mathbb {R}\}$ and $\{{\mathcal {E}}^1_c(t),t\in \mathbb {R}\}$, given in Theorems 4.6 and 5.2, are similar since

$$\begin{aligned} {\mathcal {L}}^1(t)=F_{_{{\mathcal {E}^c}\!{\mathcal {L}}}}\cdot {\mathcal {E}}^1_c(t)\cdot F_{_{{\mathcal {L}}{\mathcal {E}^c}}}\qquad \text {for all }t\in \mathbb {R}. \end{aligned}$$

(5.37)

Moreover, for all $n\in \mathbb {N}$, $2\le n\le r$, the similarity (5.37) extends to the groups $\{{\mathcal {L}}^n(t),t\in \mathbb {R}\}$ and $\{{\mathcal {E}}^n_c(t),t\in \mathbb {R}\}$ given in Theorems 4.8 and 5.4, since

$$\begin{aligned} {\mathcal {L}}^n(t)=F_{_{{\mathcal {E}^c}\!{\mathcal {L}}}}\cdot {\mathcal {E}}^n_c(t)\cdot F_{_{{\mathcal {L}}{\mathcal {E}^c}}}\qquad \text {for all } t\in \mathbb {R}. \end{aligned}$$

(5.38)

Proof

Formula (5.37) follows by (4.31), (5.18) and (5.35). Moreover, by (4.23), (5.9) and (5.35) we have $F_{_{{\mathcal {E}^c}\!{\mathcal {L}}}}D^{n-1}_{\mathcal {E}^c}=D^{n-1}_{\mathcal {L}}$, so formula (5.38) follows by (4.34) and Theorem 5.4-II). $\square$

To complete the study of the relation between $({\mathcal {E}^c})$ and $({\mathcal {L}})$, we also point out that we can simply combine Corollary 1.9 with(4.53) and Theorem 5.11 as follows.

Corollary 5.13

(Regularity for $({\mathcal {E}^c}) \rightleftarrows ({\mathcal {L}})$) For all $n\in {\widetilde{\mathbb {N}}}$, $2\le n\le r$, the bijective isomorphisms $\Psi _{_{{\mathcal {L}}{\mathcal {E}^c}}}$ and $\Psi _{_{{\mathcal {E}^c}\!{\mathcal {L}}}}$ in Corollary 1.9, respectively, restrict to bijective isomorphisms

$$\begin{aligned} {\dot{\Psi }}_{_{{\mathcal {E}^c}\!{\mathcal {L}}}}^n\in \mathcal {L}(\mathcal {S}^n_{\mathcal {L}};\mathcal {S}^n_{\mathcal {E}^c}), \qquad {\dot{\Psi }}_{_{{\mathcal {E}^c}\!{\mathcal {L}}}}^n\in \mathcal {L}(\mathcal {S}^n_{\mathcal {E}^c};\mathcal {S}^n_{\mathcal {L}}). \end{aligned}$$

(5.39)

By combining Theorem 5.11 with Corollary 5.13, one gets that the commutative diagrams in  (1.32) extend to classes of more regular solutions, i.e., that in these diagrams one can replace 1 with $n\in {\widetilde{\mathbb {N}}}$, $2\le n\le r$.