New Thought on Matsumura–Nishida Theory in the $L_p$–$L_q$ Maximal Regularity Framework

Abstract

This paper is devoted to proving the global well-posedness of initial-boundary value problem for Navier–Stokes equations describing the motion of viscous, compressible, barotropic fluid flows in a three dimensional exterior domain with non-slip boundary conditions. This was first proved by an excellent paper due to Matsumura and Nishida (Commun Math Phys 89:445–464, 1983). In [10], they used energy method and their requirement was that space derivatives of the mass density up to third order and space derivatives of the velocity fields up to fourth order belong to $L_2$ in space-time, detailed statement of Matsumura and Nishida theorem is given in Theorem 1 of Sect. 1 of context. This requirement is essentially used to estimate the $L_\infty$ norm of necessary order of derivatives in order to enclose the iteration scheme with the help of Sobolev inequalities and also to treat the material derivatives of the mass density. On the other hand, this paper gives the global wellposedness of the same problem as in [10] in $L_p$ ($1 <p \le 2$) in time and $L_2\cap L_6$ in space maximal regularity class, which is an improvement of the Matsumura and Nishida theory in [10] from the point of view of the minimal requirement of the regularity of solutions. In fact, after changing the material derivatives to time derivatives by Lagrange transformation, enough estimates obtained by combination of the maximal $L_p$ ($1 <p \le 2$) in time and $L_2\cap L_6$ in space regularity and $L_p$–$L_q$ decay estimate of the Stokes equations with non-slip conditions in the compressible viscous fluid flow case enable us to use the standard Banach’s fixed point argument. Moreover, one of the purposes of this paper is to present a framework to prove the $L_p$–$L_q$ maximal regularity for parabolic-hyperbolic type equations with non-homogeneous boundary conditions and how to combine the maximal $L_p$–$L_q$ regularity and $L_p$–$L_q$ decay estimates of linearized equations to prove the global well-posedness of quasilinear problems in unbounded domains, which gives a new thought of proving the global well-posedness of initial-boundary value problems for systems of parabolic or parabolic-hyperbolic equations appearing in mathematical physics.

1 Introduction

Matsumura and Nishida [10] proved the existence of unique solutions of equations governing the flow of viscous, compressible, and heat conduction fluids in an exterior domain of 3 dimensional Euclidean space $\mathbb {R}^3$ for all times, provided the initial data are sufficiently small. Although Matsumura and Nishida [10] considered the viscous, barotropic, and heat conductive fluid, in this paper we only consider the viscous, compressible, barotropic fluid for simplicity and reprove the Matsumura and Nishida theory in view of the $L_p$ in time ($1 < p \le 2$) and $L_2 \cap L_6$ in space maximal regularity theorem.

To describe in more detail, we start with description of equations considered in this paper. Let $\Omega$ be a three dimensional exterior domain, that is the complement, $\Omega ^c$, of $\Omega$ is a bounded domain in the three dimensional Euclidean space $\mathbb {R}^3$. Let $\Gamma$ be the boundary of $\Omega$, which is a compact $C^2$ hypersurface. Let $\rho =\rho (x, t)$ and $\mathbf{v} = (v_1(x, t), v_2(x, t), v_3(x, t))^\top$ be respective the mass density and the velocity field, where $M^\top$ denotes the transposed M, t is a time variable and $x=(x_1, x_2, x_3) \in \Omega$. Let ${\mathfrak {p}}= {\mathfrak {p}}(\rho )$ be the fluid pressure, which is a smooth function defined on $(0, \infty )$ such that ${\mathfrak {p}}'(\rho ) > 0$ for $\rho >0$. We consider the following equations:

$$\begin{aligned} \begin{aligned} \partial _t\rho + \mathrm{div}\,(\rho \mathbf{v} ) = 0&\quad&\hbox { in}\ \Omega \times (0, T), \\ \rho (\partial _t\mathbf{v} + \mathbf{v} \cdot \nabla \mathbf{v} ) - \mathrm{Div}\,(\mu \mathbf{D} (\mathbf{v} ) + \nu \mathrm{div}\,\mathbf{v} \mathbf{I} -{\mathfrak {p}}(\rho )\mathbf{I} ) = 0&\quad&\hbox { in}\ \Omega \times (0, T), \\ \mathbf{v} |_{\Gamma } = 0, \quad (\rho , \mathbf{v} )|_{t=0} = (\rho _* + \theta _0, \mathbf{v} _0)&\quad&\hbox { in}\ \Omega . \end{aligned}\end{aligned}$$

(1)

Here, $\partial _t = \partial /\partial t$, $\mathbf{D} (\mathbf{v} ) = \nabla \mathbf{v} + (\nabla \mathbf{v} )^\top$ is the deformation tensor, $\mathrm{div}\,\mathbf{v} = \sum _{j=1}^3 \partial v_j/\partial x_j$, for a $3\times 3$ matrix K with (i, j) th component $K_{ij}$, $\mathrm{Div}\,K =(\sum _{j=1}^3 \partial K_{1j}/\partial x_j, \sum _{j=1}^3 \partial K_{2j}/\partial x_j, \sum _{j=1}^3 \partial K_{3j}/\partial x_j)^\top$, $\mu$ and $\nu$ are two viscous constants such that $\mu > 0$ and $\mu + \nu > 0$, and $\rho _*$ is a positive constant describing the mass density of a reference body.

According to Matsumura and Nishida [10], we have the global well-posedness of Eq. (1) in the $L_2$ framework stated as follows:

Theorem 1

([10]). Let $\Omega$ be a three dimensional exterior domain, the boundary of which is a smooth 2 dimensional compact hypersurface. Then, there exsits a small number $\epsilon > 0$ such that for any initial data $(\theta _0, \mathbf{v} _0) \in H^3(\Omega )^4$ satisfying smallness condition: $\Vert (\theta _0, \mathbf{v} _0)\Vert _{H^3(\Omega )} \le \epsilon$ and compatibility conditions of order 1, that is $\mathbf{v} _0$ and $\partial _t\mathbf{v} |_{t=0}$ vanish at $\Gamma$, Problem (1) admits unique solutions $\rho = \rho _*+\theta$ and $\mathbf{v}$ with

$$\begin{aligned}&\theta \in C^0((0, \infty ), H^3(\Omega )) \cap C^1((0, \infty ), H^2(\Omega )), \quad \nabla \rho \in L_2((0, \infty ), H^2(\Omega )^3), \\&\mathbf{v} \in C^0((0, \infty ), H^3(\Omega )^3) \cap C^1((0, \infty ), H^1(\Omega )^3), \quad \nabla \mathbf{v} \in L_2((0, \infty ),, H^3(\Omega )^9). \end{aligned}$$

Matsumura and Nishida [10] proved Theorem 1 essentially by energy method. One of key issues in [10] is to estimate $\sup _{t \in (0, \infty )} \Vert \mathbf{v} (\cdot , t)\Vert _{H^1_\infty (\Omega )}$ by Sobolev’s inequality, namely

$$\begin{aligned} \sup _{t \in ((0, \infty )} \Vert \mathbf{v} (\cdot , t)\Vert _{H^1_\infty (\Omega )} \le C\sup _{t \in (0, \infty )} \Vert \mathbf{v} (\cdot , t))\Vert _{H^3(\Omega )}. \end{aligned}$$

(2)

Recently, Enomoto and Shibata [8] proved the global wellposedness of Eq. (1) for $(\theta _0, \mathbf{v} _0) \in H^2(\Omega )^4$ with small norms. Namely, they proved the following theorem.

Theorem 2

([8]). Let $\Omega$ be a three dimensional exterior domain, the boundary of which is a smooth 2 dimensional compact hypersurface. Then, there exsits a small number $\epsilon > 0$ such that for any initial data $(\theta _0, \mathbf{v} _0) \in H^2(\Omega )^4$ satisfying $\Vert (\theta _0, \mathbf{v} _0)\Vert _{H^2(\Omega )} \le \epsilon$ and compatibility condition: $\mathbf{v} _0|_{\Gamma }=0$, problem (1) admits unique solutions $\rho = \rho _*+\theta$ and $\mathbf{v}$ with

$$\begin{aligned}&\theta \in C^0((0, \infty ), H^2(\Omega )) \cap C^1((0, \infty ), H^1(\Omega )), \quad \nabla \rho \in L_2((0, \infty ), H^1(\Omega )^3), \\&\mathbf{v} \in C^0((0, \infty ), H^2(\Omega )^3) \cap C^1((0, \infty ), L_2(\Omega )^3), \quad \nabla \mathbf{v} \in L_2((0, \infty ), H^2(\Omega )^9). \end{aligned}$$

The method used in the proof of Enomoto and Shibata [8] is essentially the same as that in Matsumura and Nishida [10]. Only the difference is that (2) is replaced by $\int ^\infty _0\Vert \nabla \mathbf{v} \Vert _{L_\infty (\Omega )}^2\,dt \le C\int ^\infty _0\Vert \nabla \mathbf{v} \Vert _{H^2(\Omega )}^2\,dt$ in [8]. As a conclusion, in the $L_2$ framework the least regularity we need is that $\nabla \rho \in L_2((0, \infty ), H^1(\Omega )^3)$ and $\nabla \mathbf{v} \in L_2((0, \infty ), H^2(\Omega )^9)$. In this paper, we improve this point by solving the Eq. (1) in the $L_p$-$L_q$ maximal regularity class, that is the following theorem is a main result of this paper.

Theorem 3

Let $\Omega$ be an exterior domain in $\mathbb {R}^3$, whose boundary $\Gamma$ is a compact $C^2$ hypersurface and $T \in (0, \infty )$. Let p be an exponent with $1 < p \le 2$ and set $p' = p/(p-1)$. Let $\sigma \in (0, 1)$ and set $\ell =(5+\sigma )/(4+2\sigma )$ and $r= 2(2+\sigma )/(4+\sigma )=(1/2+1/(2+\sigma ))^{-1}$. Let b be a positive constant satisfying the condition

$$\begin{aligned} \frac{1}{p'}< b <\ell -\frac{1}{p}. \end{aligned}$$

(3)

Set

$$\begin{aligned}&{{\mathcal {I}}}= \left\{ (\theta _0, \mathbf{v} _0) \mid \theta _0 \in \left( \bigcap _{q=2, 6}H^1_q(\Omega )\right) \cap L_r(\Omega ), \quad \mathbf{v} _0 \in \left( \bigcap _{q=2, 6} B^{2(1-1/p)}_{q,p}(\Omega )^3\right) \cap L_r(\Omega )^3\right\} , \\&\quad \Vert (\theta _0, \mathbf{v} _0)\Vert _{{{\mathcal {I}}}} = \sum _{q=2, 6} \Vert \theta _0\Vert _{H^1_q(\Omega )} + \sum _{q=2, 6}\Vert \mathbf{v} _0\Vert _{B^{2(1-1/p)}_{q,p}(\Omega )} + \Vert (\theta _0, \mathbf{v} _0)\Vert _{L_r(\Omega )}. \end{aligned}$$

Then, there exists a small constant $\epsilon \in (0, 1)$ independent of T such that if initial data $(\theta _0, \mathbf{v} _0) \in {{\mathcal {I}}}$ satisfy the compatibility condition: $\mathbf{v} _0|_\Gamma =0$ and the smallness condition : $\Vert (\theta _0, \mathbf{v} _0)\Vert _{{{\mathcal {I}}}} \le \epsilon ^2$, then problem (1) admits unique solutions $\rho =\rho _*+\theta$ and $\mathbf{v}$ with

$$\begin{aligned} \begin{aligned} \theta&\in H^1_p((0, T), L_2(\Omega ) \cap L_6(\Omega )) \cap L_p((0, T), H^1_2(\Omega ) \cap H^1_6(\Omega )), \\ \mathbf{v}&\in H^1_p((0, T), L_2(\Omega )^3 \cap L_6(\Omega )^3) \cap L_p((0, T), H^2_2(\Omega )^3 \cap H^2_6(\Omega )^3). \end{aligned}\end{aligned}$$

(4)

Moreover, writing $\Vert (\theta , \mathbf{v} )\Vert _{H^{\ell , m}_q(\Omega )} = \Vert \theta \Vert _{H^\ell _q(\Omega )} + \Vert \mathbf{v} \Vert _{H^m_q(\Omega )}$ and setting

$$\begin{aligned} {{\mathcal {E}}}_T(\theta , \mathbf{v} )&= \Vert<t>^b(\theta , \mathbf{v} )\Vert _{L_\infty ((0, T), L_2(\Omega ) \cap L_6(\Omega ))}+ \Vert<t>^b\nabla (\theta , \mathbf{v} )\Vert _{L_p((0, T), H^{0,1}_2(\Omega ))} \\&\quad + \Vert<t>^b(\theta , \mathbf{v} )\Vert _{L_p((0, T), H^{1,2}_6(\Omega ))} + \Vert <t>^b\partial _t(\theta , \mathbf{v} )\Vert _{L_p((0, T), L_2(\Omega ) \cap L_6(\Omega ))}, \end{aligned}$$

we have ${{\mathcal {E}}}_T(\theta , \mathbf{v} ) \le \epsilon$.

Remark 4

(1) $T>0$ is taken arbitrarily and $\epsilon >0$ is chosen independently of T, and so Theorem 3 tells us the global wellposedness of Eq. (1) for $(0, \infty )$ time inverval.

(2) In the $p=2$ case, Theorem 3 gives an extension of Matsumura and Nishida theorem [10]. Roughly speaking, if we assume that $(\theta _0, \mathbf{v} _0) \in H^3_2(\Omega )^4$, then $(\theta _0, \mathbf{v} _0) \in (H^1_2(\Omega ) \cap H^1_6(\Omega ))\times (H^1_2(\Omega ) \cap B^1_{6,2}(\Omega ))$, and so the global wellposedness holds in the class as

$$\begin{aligned} \theta \in H^1_2((0, T), H^1_2(\Omega ) \cap H^1_6(\Omega )), \quad \mathbf{v} \in H^1_2((0, T), L_2(\Omega )^3 \cap L_6(\Omega )^3) \cap L_2((0, T), H^2_2(\Omega )^3\cap H^2_6(\Omega )^3) \end{aligned}$$

under the additional condition: $(\theta _0, \mathbf{v} _0) \in L_r(\Omega )^4$.

(3)  Since we assume that $1 < p \le 2$, it automatically follows that

$$\begin{aligned} b < \ell -\frac{1}{2}= \frac{3}{2(2+\sigma )}. \end{aligned}$$

(5)

(4)  Following the argument in [12, Theorem 3.8.1], we can also consider the case where $2< p < \infty$.

As related topics, we consider the Cauchy problem, that is $\Omega = \mathbb {R}^3$ without boundary condition. Matsumura and Nishida [9] proved the global wellposedness theorem, the statement of which is essentially the same as in Theorem 1 and the proof is based on energy method. Danchin [4] proved the global wellposedness in the critical space by using the Littlewood–Paley decomposition.

Theorem 5

([4]). Let $\Omega = \mathbb {R}^N$ $(N\ge 2)$. Assume that $\mu > 0$ and $\mu +\nu > 0$. Let $B^s = \dot{B}^s_{2,1}(\mathbb {R}^N)$ and

$$\begin{aligned} F^s = (L_2((0, \infty ), B^s) \cap C((0, \infty ), B^s \cap B^{s-1})) \times (L_1((0, \infty ), B^{s+1}) \cap C((0, \infty ), B^{s-1}))^N. \end{aligned}$$

Then, there exists an $\epsilon > 0$ such that if initial data $\theta _0 \in B^{N/2}(\mathbb {R}^N) \cap B^{N/2-1}(\mathbb {R}^N)$ and $\mathbf{v} _0 \in B^{N/2-1}(\mathbb {R}^N)^N$ satisfy the condition:

$$\begin{aligned} \Vert \theta _0\Vert _{B^{N/2}(\mathbb {R}^N) \cap B^{N/2-1}(\mathbb {R}^N)} + \Vert \mathbf{v} _0\Vert _{B^{N/2-1}(\mathbb {R}^N)} \le \epsilon , \end{aligned}$$

then problem (1) with $\Omega =\mathbb {R}^N$ and $T=\infty$ admits a unique solution $\rho =\rho _*+\theta$ and $\mathbf{v}$ with $(\theta , \mathbf{v} ) \in F^{N/2}$.

In the case where $\Omega = \mathbb {R}^3$ or $\mathbb {R}^N$, there are a lot of works concerning (1), but we do not mention them any more, because we are interested only in the global wellposedness in exterior domains. For more information on references, refer to Enomoto and Shibata [7].

Concerning the $L_1$ in time maximal regularity in exterior domains, the incompressible viscous fluid flows has been treated by Danchin and Mucha [5]. To obtain $L_1$ maximal regularity in time, we have to use $\dot{B}^s_{q,1}$ in space, which is slightly regular space than $H^s_q$, and the decay estimates for semigroup on $\dot{B}^s_{q,1}$ must be needed to controle terms arising from the cut-off procedure near the boundary. Detailed arguments related with these facts can be found in [5]. To treat (1) in an exterior domain in the $L_1$ in time maximal regularity framework, we have to prepare not only $L_1$ maximal regularity for model problems in the whole space and the half space but also decay properties of semigroup in $\dot{B}^s_{q,1}$, and so this will be a future work. From Theorem 3, we may say that problem (1) can be solved in $L_p$ in time and $L_2\cap L_6$ in space maximal regularity class for any exponet $p \in (1, 2]$.

The paper is organized as follows. In Sect. 2, Eq. (1) are rewriten in Lagrange coordinates to eliminate $\mathbf{v} \cdot \nabla \rho$ and a main result for equations with Lagrangian description is stated. In Sect. 3, we give an $L_p$–$L_q$ maximal regularity theorem in some abstract setting. In Sect. 4, we give estimates of nonlinear terms. In Sect. 5, we prove main results stated in Sect. 2. In Sect. 6, Theorem 3 is proved by using a main result in Sect. 2. In Sect. 7, we discuss the N dimensonal case.

The main point of our proof is to obtain maximal regularity estimates with decay properties of solutions to linearized equations, the Stokes equations with non-slip conditions. To explain the idea, we write linearized equations as $\partial _t u - Au = f$ and $u|_{t=0}=u_0$ symbolically, where f is a function corresponding to nonlinear terms and A is a closed linear operator with domain D(A). We write $u=u_1+u_2$, where $u_1$ is a solution to time shifted equations: $\partial _t u_1 + \lambda _1u_1- Au_1 = f$ with some large positive number $\lambda _1$ and $u_2$ is a solution to compensating equations: $\partial _t u_2 -Au_2 = \lambda _1u_1$ and $u_2|_{t=0} = u_0-u_1|_{t=0}$. Since the fundamental solutions to time shifted equations have exponential decay properties, $u_1$ has the same decay properties as these of nonlinear terms f. Moreover $u_1$ belongs to the domain of A for all positive time. By Duhamel principle $u_2$ is given by $u_2= T(t)(u_0-u_1|_{t=0})+ \lambda _1\int ^t_0 T(t-s)u_1(s)\,ds$, where $\{T(t)\}_{t\ge 0}$ is a continuous analytic semigroup associated with A. By using $L_p$-$L_q$ decay properties of $\{T(t)\}_{t\ge 0}$ in the interval $0< s < t-1$ and standard estimates of continuous analytic semigroup: $\Vert T(t-s)u_0\Vert _{D(A)} \le C\Vert u_0\Vert _{D(A)}$ for $t-1< s < t$, where $\Vert \cdot \Vert _{D(A)}$ denotes a domain norm, we obtain maximal $L_p$-$L_q$ regularity of $u_2$ with decay properties. This method seems to be a new thought to prove the global wellposedness and to be applicable to many quasilinear problems of parabolic type or parabolic-hyperbolic mixture type appearing in mathematical physics.

To end this section, symbols of functional spaces used in this paper are given. Let $L_p(\Omega )$, $H^m_p(\Omega )$ and $B^s_{q,p}(\Omega )$ denote the standard Lebesgue spaces, Sobolev spaces and Besov spaces, while their norms are written as $\Vert \cdot \Vert _{L_p(\Omega )}$, $\Vert \cdot \Vert _{H^m_p(\Omega )}$ and $\Vert \cdot \Vert _{B^s_{q,p}(\Omega )}$. We write $H^m(\Omega ) = H^m_2(\Omega )$, $H^0_q(\Omega )=L_q(\Omega )$ and $W^s_q(\Omega )= B^s_{q,q}(\Omega )$. For any Banach space X with norm $\Vert \cdot \Vert _X$, $L_p((a, b), X)$ and $H^m_p((a, b), X)$ denote respective the standard X-valued Lebesgue spaces and Sobolev spaces, while their time weighted norms are defined by

$$\begin{aligned} \Vert<t>^b f\Vert _{L_p((a, b), X)} = {\left\{ \begin{array}{ll} \Bigl (\int ^b_a(<t>^b\Vert f(t)\Vert _X)^p\,dt\Bigr )^{1/p} \quad &{} (1 \le p< \infty ), \\ \mathrm{esssup}_{t \in (a, b)} <t>^b\Vert f(t)\Vert _X\quad &{}(p=\infty ), \end{array}\right. } \end{aligned}$$

where $<t> = (1 + t^2)^{1/2}$. Let $X^n = \{ \mathbf{v} =(u_1, \ldots , u_n)) \mid u_i \in X (i=1, \ldots , n)\}$, but we write $\Vert \cdot \Vert _{X^n} = \Vert \cdot \Vert _X$ for simplicity. Let $H^{\ell , m}_q(\Omega ) = \{(\rho , \mathbf{v} ) \mid \rho \in H^\ell _q(\Omega ), \mathbf{v} \in H^m_q(\Omega )^3\}$ and $\Vert (\rho , \mathbf{v} )\Vert _{H^{\ell , m}_q(\Omega )} = \Vert \rho \Vert _{H^\ell _q(\Omega )} + \Vert \mathbf{v} \Vert _{H^m_q(\Omega )}$. The letter C denotes generic constants and $C_{a, b, \cdots }$ denotes that constants depend on quantities a, b, $\ldots$. C and $C_{a,b, \cdots }$ may change from line to line.

2 Equations in Lagrange Coordinates and Statment of Main Results

To prove Theorem 3, we write Eq. (1) in Lagrange coordinates $\{y\}$. Let $\zeta =\zeta (y, t)$ and $\mathbf{u} =\mathbf{u} (y, t)$ be the mass density and the velocity field in Lagrange coordinates $\{y\}$, and for a while we assume that

$$\begin{aligned} \mathbf{u} \in H^1_p((0, T), L_6(\Omega )^3) \cap L_p((0, T), H^2_6(\Omega )^3), \end{aligned}$$

(6)

and the quantity: $\Vert <t>^b\nabla \mathbf{u} \Vert _{L_p((0, T), H^1_6(\Omega )}$ is small enough for some $b > 0$ with $bp' > 1$, where $1/p + 1/p' = 1$. We consider the Lagrange transformation:

$$\begin{aligned} x = y + \int ^t_0 \mathbf{u} (y, s)\,ds \end{aligned}$$

(7)

and assume that

$$\begin{aligned} \int ^T_0\Vert \nabla \mathbf{u} (\cdot , t)\Vert _{L_\infty (\Omega )}\,dt <\delta \end{aligned}$$

(8)

with some small number $\delta > 0$. If $0< \delta < 1$, then for $x_i = y_i + \int ^t_0\mathbf{u} (y_i, s)\,ds$ we have

$$\begin{aligned} |x_1-x_2| \ge (1-\int ^T_0\Vert \nabla \mathbf{u} (\cdot , t)\Vert _{L_\infty (\Omega )}\,dt) |y_1-y_2|, \end{aligned}$$

and so the correspondence (7) is one to one. Moreover, applying a method due to Ströhmer [13], we see that the correspondence (7) is a $C^{1+\omega }$ ($\omega \in (0, 1/2)$) diffeomorphism from $\overline{\Omega }$ onto itself for any $t \in (0, T)$. In fact, let $J = \mathbf{I} + \int ^t_0\nabla \mathbf{u} (y, s)\,ds$, which is the Jacobian of the map defined by (7), and then by Sobolev’s imbedding theorem and Hölder’s inequality for $\omega \in (0, 1/2)$ we have

$$\begin{aligned} \sup _{t \in (0, T)} \Vert \int ^t_0\nabla \mathbf{u} (\cdot , s)\,ds\Vert _{C^{\omega }(\overline{\Omega })} \le C_\omega \Bigl (\int ^T_0<s>^{-bp'}\,ds\Bigr )^{1/p'} \Bigl (\int ^T_0\Vert<s>^b\nabla \mathbf{u} (\cdot , s)\Vert _{H^1_6(\Omega )}^p\,ds\Bigr )^{1/p}<\infty \nonumber \\ \end{aligned}$$

(9)

and we may assume that the right hand side of (9) is small enough and (8) holds in the process of constructing a solution. By (7), we have

$$\begin{aligned} \frac{\partial x}{\partial y} = \mathbf{I} + \int ^t_0\frac{\partial \mathbf{u} }{\partial y}(y, s)\,ds, \end{aligned}$$

and so choosing $\delta > 0$ small enough, we may assume that there exists a $3\times 3$ matrix $\mathbf{V} _0(\mathbf{k} )$ of $C^\infty$ functions of variables $\mathbf{k}$ for $|\mathbf{k} | < \delta$, where $\mathbf{k}$ is a corresponding variable to $\int ^t_0\nabla \mathbf{u} \,ds$, such that $\frac{\partial y}{\partial x} = \mathbf{I} + \mathbf{V} _0(\mathbf{k} )$ and $\mathbf{V} _0(0) = 0$. Let $V_{0ij}(\mathbf{k} )$ be the (i, j) th component of $3\times 3$ matrix $V_0(\mathbf{k} )$, and then we have

$$\begin{aligned} \frac{\partial }{\partial x_j} = \frac{\partial }{\partial y_j} + \sum _{j=1}^3V_{0ij}(\mathbf{k} )\frac{\partial }{\partial y_j}. \end{aligned}$$

(10)

Let $X_t(x) = y$ be the inverse map of Lagrange transform (7) and set $\rho (x, t) = \zeta (X_t(x), t)$ and $\mathbf{v} (x, t) = \mathbf{u} (X_t(x), t)$. Setting

$$\begin{aligned} {{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} )\nabla \mathbf{u} = \sum _{i, j=1}^3V_{0ij}(\mathbf{k} )\frac{\partial u_i}{\partial y_j}, \end{aligned}$$

we have $\mathrm{div}\,\mathbf{v} = \mathrm{div}\,\mathbf{u} + {{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} )\mathbf{u}$. Let $\zeta = \rho _* + \eta$, and then

$$\begin{aligned} \frac{\partial }{\partial t} \rho + \mathrm{div}\,(\rho \mathbf{u} )= \frac{\partial \eta }{\partial t} + (\rho _* + \eta )(\mathrm{div}\,\mathbf{u} + {{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} )\nabla \mathbf{u} ). \end{aligned}$$

Setting

$$\begin{aligned} {{\mathcal {D}}}_\mathbf{D} (\mathbf{k} )\nabla \mathbf{u} = \mathbf{V} _0(\mathbf{k} )\nabla \mathbf{u} + (\mathbf{V} _0(\mathbf{k} )\nabla \mathbf{u} )^\top , \end{aligned}$$

(11)

we have $\mathbf{D} (\mathbf{v} ) = \nabla \mathbf{v} + (\nabla \mathbf{v} )^\top = (\mathbf{I} + \mathbf{V} _0(\mathbf{k} ))\nabla \mathbf{u} + ((\mathbf{I} + \mathbf{V} _0(\mathbf{k} ))\nabla \mathbf{u} )^\top = \mathbf{D} (\mathbf{u} ) + {{\mathcal {D}}}_\mathbf{D} (\mathbf{k} )\nabla \mathbf{u}$. Moreover,

$$\begin{aligned}\mathrm{Div}\,(\mu \mathbf{D} (\mathbf{v} ) + \nu \mathrm{div}\,\mathbf{v} \mathbf{I} )&= (\mathbf{I} +\mathbf{V} _0(\mathbf{k} ))\nabla (\mu (\mathbf{D} (\mathbf{u} ) + {{\mathcal {D}}}_\mathbf{D} (\mathbf{k} )\nabla \mathbf{u} ) + \nu (\mathrm{div}\,\mathbf{u} + {{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} )\nabla \mathbf{u} ) \\&= \mathrm{Div}\,(\mu \mathbf{D} (\mathbf{u} ) + \nu \mathrm{div}\,\mathbf{u} \mathbf{I} ) + \mathbf{V} _1(\mathbf{k} )\nabla ^2\mathbf{u} + (\mathbf{V} _2(\mathbf{k} )\int ^t_0\nabla ^2\mathbf{u} \,ds)\nabla \mathbf{u} \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} \mathbf{V} _1(\mathbf{k} )\nabla ^2\mathbf{u}&= \mu {{\mathcal {D}}}_\mathbf{D} (\mathbf{k} )\nabla ^2\mathbf{u} + \nu {{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} )\nabla ^2\mathbf{u} \mathbf{I} \\&\quad + \mathbf{V} _0(\mathbf{k} )(\mu \nabla \mathbf{D} (\mathbf{u} ) + \nu \nabla \mathrm{div}\,\mathbf{u} \mathbf{I} + \mu {{\mathcal {D}}}_\mathbf{D} (\mathbf{k} )\nabla ^2\mathbf{u} + \nu {{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} )\nabla ^2\mathbf{u} \mathbf{I} ), \\ (\mathbf{V} _2(\mathbf{k} )\int ^t_0\nabla \mathbf{u} \,ds)\nabla \mathbf{u}&= (\mathbf{I} +\mathbf{V} _0(\mathbf{k} ))(\mu (d_\mathbf{k} {{\mathcal {D}}}_\mathbf{D} (\mathbf{k} )\int ^t_0\nabla ^2\mathbf{u} \,ds)\nabla \mathbf{u} +\nu (d_\mathbf{k} {{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} ) \int ^t_0\nabla ^2\mathbf{u} \,ds \nabla \mathbf{u} )\mathbf{I} . \end{aligned}\end{aligned}$$

(12)

Here, $d_\mathbf{k} F(\mathbf{k} )$ denotes the derivative of F with respect to $\mathbf{k}$. Note that $\mathbf{V} _1(0) = 0$. Moreover, we write

$$\begin{aligned} \nabla {\mathfrak {p}}(\rho ) = {\mathfrak {p}}'(\rho _*)\nabla \eta +({\mathfrak {p}}'(\rho _*+\eta ) - {\mathfrak {p}}'(\rho _*))\nabla \eta + {\mathfrak {p}}'(\rho _*+\eta )\mathbf{V} _0(\mathbf{k} )\nabla \theta . \end{aligned}$$

(13)

The material derivative $\partial _t\mathbf{v} + \mathbf{v} \cdot \nabla \mathbf{v}$ is changed to $\partial _t\mathbf{u}$.

Summing up, we have obtained

$$\begin{aligned} \begin{aligned} \partial _t\eta + \rho _*\mathrm{div}\,\mathbf{u} = F(\eta , \mathbf{u} )&\quad&\hbox { in}\ \Omega \times (0, T), \\ \rho _*\partial _t\mathbf{u} - \mathrm{Div}\,(\mu \mathbf{D} (\mathbf{u} ) + \nu \mathrm{div}\,\mathbf{u} \mathbf{I} - {\mathfrak {p}}'(\rho _*)\eta ) = \mathbf{G} (\eta , \mathbf{u} )&\quad&\hbox { in}\ \Omega \times (0, T), \\ \mathbf{u} |_\Gamma =0, \quad (\eta , \mathbf{u} )|_{t=0} = (\theta _0, \mathbf{v} _0)&\quad&\hbox { in}\ \Omega . \end{aligned}\end{aligned}$$

(14)

Here, we have set

$$\begin{aligned} \begin{aligned}&\mathbf{k} = \int ^t_0\nabla \mathbf{u} (\cdot , s)\,ds, \\&F(\eta , \mathbf{u} )= \rho _* {{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} )\nabla \mathbf{u} + \eta (\mathrm{div}\,\mathbf{u} + {{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} )\nabla \mathbf{u} ),\\&\mathbf{G} (\eta , \mathbf{u} ) = \eta \partial _t\mathbf{u} + \mathbf{V} _1(\mathbf{k} )\nabla ^2\mathbf{u} + (\mathbf{V} _2(\mathbf{k} )\int ^t_0\nabla ^2\mathbf{u} \,ds)\nabla \mathbf{u} \\&\qquad - ({\mathfrak {p}}'(\rho _*+\eta )-{\mathfrak {p}}'(\rho _*))\nabla \eta - {\mathfrak {p}}'(\rho _*+\eta )\mathbf{V} _0(\mathbf{k} )\nabla \eta \end{aligned}\end{aligned}$$

(15)

and ${{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} )\nabla \mathbf{u}$, $\mathbf{V} _1(\mathbf{k} )$ and $\mathbf{V} _2(\mathbf{k} )$ have been defined in (11), (12) and (13). Note that ${{\mathcal {D}}}_\mathrm{div}\,(0)=0$, $\mathbf{V} _0(0)=0$, and $\mathbf{V} _1(0)=0$. The following theorem is a main result in this paper.

Theorem 6

Let $\Omega$ be an exterior domain in $\mathbb {R}^3$, whose boundary $\Gamma$ is a compact $C^2$ hypersurface and $T \in (0, \infty )$. Let p be an exponent with $1 < p \le 2$ and set $p' = p/(p-1)$. Let $\sigma \in (0, 1)$ and set $\ell =(5+\sigma )/(4+2\sigma )$ and $r= 2(2+\sigma )/(4+\sigma )=(1/2+1/(2+\sigma ))^{-1}$. Let b be a positive constant satisfying the condition

$$\begin{aligned} \frac{1}{p'}< b <\ell -\frac{1}{p}. \end{aligned}$$

(16)

Set

$$\begin{aligned} {{\mathcal {I}}}&= \left\{ (\theta _0, \mathbf{v} _0) \mid \theta _0 \in \left( \bigcap _{q=2, 6} H^1_q(\Omega )\right) \cap L_r(\Omega ), \quad \mathbf{v} _0 \in \left( \bigcap _{q=2, 6} B^{2(1-1/p)}_{q,p}(\Omega )^3\right) \cap L_r(\Omega )^3\right\} , \\&\quad \Vert (\theta _0, \mathbf{v} _0)\Vert _{{{\mathcal {I}}}} = \sum _{q=2, 6} \Vert \theta _0\Vert _{H^1_q(\Omega )} + \sum _{q=2, 6}\Vert \mathbf{v} _0\Vert _{B^{2(1-1/p)}_{q,p}(\Omega )} + \Vert (\theta _0, \mathbf{v} _0)\Vert _{L_r(\Omega )}. \end{aligned}$$

Then, there exists a small constant $\epsilon \in (0, 1)$ independent of T such that if initial data $(\theta _0, \mathbf{v} _0) \in X$ satisfy the compatibility condition: $\mathbf{v} _0|_\Gamma =0$ and the smallness condition : $\Vert (\theta _0, \mathbf{v} _0)\Vert _{{{\mathcal {I}}}} \le \epsilon ^2$, then Problem (14) admits unique solutions $\zeta =\rho _*+\eta$ and $\mathbf{u}$ with

$$\begin{aligned} \begin{aligned} \eta&\in H^1_p((0, T), H^1_2(\Omega )) \cap H^1_6(\Omega )), \\ \mathbf{u}&\in H^1_p((0, T), L_2(\Omega )^3 \cap L_6(\Omega )^3) \cap L_p((0, T), H^2_2(\Omega )^3 \cap H^2_6(\Omega )^3) \end{aligned}\end{aligned}$$

(17)

possessing the estimate $E_T(\eta , \mathbf{u} ) \le \epsilon$. Here, we have set

$$\begin{aligned} E_T(\eta , \mathbf{u} ) = {{\mathcal {E}}}_T(\eta , \mathbf{u} ) + \Vert <t>^b\partial _t\nabla \eta \Vert _{L_p((0, T), L_2(\Omega )\cap L_6(\Omega ))} \end{aligned}$$

and ${{\mathcal {E}}}_T(\eta , \mathbf{u} )$ is the quantity defined in Theorem 3.

Remark 7

(1)  The choice of $\epsilon$ is independent of $T>0$, and so solutions of Eq. (14) exist for any time $t \in (0, \infty )$.

(2)  For any natural number m, $B^m_{q, 2}(\Omega ) \subset H^m_q(\Omega )$ for $2< q < \infty$ and $B^m_{2,2} = H^m$.

(3)  Letting $\sigma >0$ be taken a small number such that $H^2_6 \subset C^{1+\sigma }$, we see that Theorem 6 implies

$$\begin{aligned} \int ^T_0\Vert \mathbf{u} (\cdot , s)\Vert _{C^{1+\sigma }(\Omega )}\,ds < \delta \end{aligned}$$

with some small number $\delta > 0$, which guarantees that Lagrange transform given in (7) is a $C^{1+\sigma }$ diffeomorphism on $\Omega$. Moreover, Theorem 3 follows from Theorem 6, the proof of which will be given in Sect. 6 below.

3 ${{\mathcal {R}}}$-Bounded Solution Operators

This section gives a general framework of proving the maximal $L_p$ regularity ($1< p < \infty$), and so problem is formulated in an abstract setting. Let X, Y, and Z be three UMD Banach spaces such that $X \subset Z \subset Y$ and X is dense in Y, where the inclusions are continuous. Let A be a closed linear operator from X into Y and let B be a linear operator from X into Z and also from Z into Y. Moreover, we assume that

$$\begin{aligned} \Vert Ax\Vert _Y \le C\Vert x\Vert _X, \quad \Vert Bx\Vert _Z \le C\Vert x\Vert _X, \quad \Vert Bz\Vert _Y \le C\Vert z\Vert _Z \end{aligned}$$

with some constant C for any $x \in X$ and $z \in Z$. Let $\omega \in (0, \pi /2)$ be a fixed number and set

$$\begin{aligned} \Sigma _\omega&= \{\lambda \in \mathbb {C}\setminus \{0\} \mid |\arg \lambda | < \pi -\omega \}, \quad \Sigma _{\omega , \lambda _0} = \{\lambda \in \Sigma _\omega \mid |\lambda | \ge \lambda _0\}. \end{aligned}$$

We consider an abstract boundary value problem with parameter $\lambda \in \Sigma _{\omega , \lambda _0}$:

$$\begin{aligned} \lambda u - A u = f, \quad Bu = g. \end{aligned}$$

(18)

Here, $Bu = g$ represents boundary conditions, restrictions like divergence condition for Stokes equations in the incompressible viscous fluid flows case, or both of them. The simplest example is the following:

$$\begin{aligned} \lambda u - \Delta u = f \hbox { in}\ \Omega , \quad \frac{\partial u}{\partial \nu } = g \hbox { on}\ \Gamma , \\ \end{aligned}$$

where $\Omega$ is a uniform $C^2$ domain in $\mathbb {R}^N$, $\Gamma$ its boundary, $\nu$ the unit outer normal to $\Gamma$, and $\partial /\partial \nu = \nu \cdot \nabla$ with $\nabla = (\partial /\partial x_1, \ldots , \partial /\partial x_N)$ for $x=(x_1, \ldots , x_N) \in \mathbb {R}^N$. In this case, it is standard to choose $X = H^2_q(\Omega )$, $Y = L_q(\Omega )$, $Z = H^1_q(\Omega )$ with $1< q < \infty$, $A = \Delta$, and $B = \partial /\partial \nu$.

Problem formulated in (18) is corresponding to parameter elliptic problems which have been studied by Agmon [1], Agmon et al. [2], Agranovich and Visik [3], Denk and Volevich [6] and references there in, and their arrival point is to prove the unique existence of solutions possessing the estimate:

$$\begin{aligned} |\lambda |\Vert u\Vert _Y + \Vert u\Vert _X \le C(\Vert f\Vert _Y + |\lambda |^\alpha \Vert g\Vert _Y + \Vert g\Vert _Z) \end{aligned}$$

for some $\alpha \in \mathbb {R}$. From this estimate, we can derive the generation of a continuous analytic semigroup associated with A when $Bu=0$. But to prove the maximal $L_p$ regularity with $1< p < \infty$ for the corresponding nonstationary problem:

$$\begin{aligned}&\partial _t v- A v = f, \quad Bv= g \quad \hbox { for}\ t > 0, \quad v|_{t=0} = v_0, \end{aligned}$$

(19)

especially in the cases where $Bv=g \not =0$, further consideration is needed. Below, we introduce a framework based on the Weis operator valued Fourier multiplier theorem. To state this theorem, we make a preparation.

Definition 8

Let E and F be two Banach spaces and let ${{\mathcal {L}}}(E, F)$ be the set of all bounded linear operators from E into F. We say that an operator family ${{\mathcal {T}}}\subset {{\mathcal {L}}}(E, F)$ is ${{\mathcal {R}}}$ bounded if there exist a constant C and an exponent $q \in [1, \infty )$ such that for any integer n, $\{T_j\}_{j=1}^n \subset {{\mathcal {T}}}$ and $\{f_j\}_{j=1}^n \subset E$, the inequality:

$$\begin{aligned} \int ^1_0\left\| \sum _{j=1}^n r_j(u)T_jf_j\right\| _F^q\,d u \le C\int ^1_0\left\| \sum _{j=1}^n r_j(u)f_j\right\| _E^q\,du \end{aligned}$$

is valid, where the Rademacher functions $r_k$, $k \in \mathbb {N}$, are given by $r_k: [0, 1] \rightarrow \{-1, 1\}$; $t \mapsto \mathrm{sign}(\sin 2^k\pi t)$. The smallest such C is called ${{\mathcal {R}}}$ bound of ${{\mathcal {T}}}$ on ${{\mathcal {L}}}(E, F)$, which is denoted by ${{\mathcal {R}}}_{{{\mathcal {L}}}(E, F)}{{\mathcal {T}}}$.

For $m(\xi ) \in L_\infty (\mathbb {R}\setminus \{0\}, {{\mathcal {L}}}(E, F))$, we set

$$\begin{aligned} T_mf = {{\mathcal {F}}}^{-1}_\xi [m(\xi ){{\mathcal {F}}}[f](\xi )] \quad f \in {{\mathcal {S}}}(\mathbb {R}, E), \end{aligned}$$

where ${{\mathcal {F}}}$ and ${{\mathcal {F}}}_\xi ^{-1}$ denote respective Fourier transformation and inverse Fourier transformation.

Theorem 9

(Weis’s operator valued Fourier multiplier theorem). Let E and F be two UMD Banach spaces. Let $m(\xi ) \in C^1(\mathbb {R}\setminus \{0\}, {{\mathcal {L}}}(E, F))$ and assume that

$$\begin{aligned}&{{\mathcal {R}}}_{{{\mathcal {L}}}(E, F)}(\{m(\xi ) \mid \xi \in \mathbb {R}\setminus \{0\}\}) \le r_b \\&{{\mathcal {R}}}_{{{\mathcal {L}}}(E, F)}(\{\xi m'(\xi ) \mid \xi \in \mathbb {R}\setminus \{0\}\}) \le r_b \end{aligned}$$

with some constant $r_b > 0$. Then, for any $p \in (1, \infty )$, $T_m \in {{\mathcal {L}}}(L_p(\mathbb {R}, E), L_p(\mathbb {R}, F))$ and

$$\begin{aligned} \Vert T_mf\Vert _{L_p(\mathbb {R}, F)} \le C_pr_b\Vert f\Vert _{L_p(\mathbb {R}, E)} \end{aligned}$$

with some constant $C_p$ depending solely on p.

Remark 10

For a proof, refer to Weis [14].

We introduce the following assumption. Recall that $\omega$ is a fixed number such that $0< \omega < \pi /2$.

Assumption 11

Let X, Y and Z be UMD Banach spaces. There exist a constant $\lambda _0$, $\alpha \in \mathbb {R}$, and an operator family ${{\mathcal {S}}}(\lambda )$ with

$$\begin{aligned} {{\mathcal {S}}}(\lambda )\in \mathrm{Hol}\,(\Sigma _{\omega ,\lambda _0}, {{\mathcal {L}}}(Y\times Y \times Z, X)) \end{aligned}$$

such that for any $f \in Y$ and $g \in Z$, $u={{\mathcal {S}}}(\lambda )(f, \lambda ^\alpha g, g)$ is a solution of Eq. (18), and the estimates:

$$\begin{aligned} {{\mathcal {R}}}_{{{\mathcal {L}}}(Y\times Y \times Z, X)}(\{(\tau \partial _\tau )^\ell {{\mathcal {S}}}(\lambda ) \mid \lambda \in \Sigma _{\omega ,\lambda _0}\})&\le r_b \\ {{\mathcal {R}}}_{{{\mathcal {L}}}(Y\times Y \times Z, Y)}(\{(\tau \partial _\tau )^\ell (\lambda {{\mathcal {S}}}(\lambda )) \mid \lambda \in \Sigma _{\omega ,\lambda _0}\})&\le r_b \end{aligned}$$

for $\ell =0,1$ are valid, where $\lambda = \gamma + i\tau \in \Sigma _{\omega , \lambda _0}$. ${{\mathcal {S}}}(\lambda )$ is called an ${{\mathcal {R}}}$-bounded solution operator or an ${{\mathcal {R}}}$ solver of Eq. (18).

We now consider an initial-boundary value problem:

$$\begin{aligned} \partial _t u - Au = f \quad Bu = g \quad (t>0), \quad u|_{t=0} = u_0. \end{aligned}$$

(20)

This problem is divided into the following two equations:

$$\begin{aligned} \partial _t u - Au&= f&\quad Bu&= g&\quad&(t \in \mathbb {R}); \end{aligned}$$

(21)

$$\begin{aligned} \partial _t u - Au&= 0&\quad Bu&= 0&\quad&(t>0), \quad u|_{t=0} = u_0. \end{aligned}$$

(22)

From the definition of ${{\mathcal {R}}}$-boundedness with $n=1$ we see that $u={{\mathcal {S}}}(\lambda )(f, 0, 0)$ satisifes equations:

$$\begin{aligned} \lambda u -Au = f, \quad Bu = 0, \end{aligned}$$

and the estimate:

$$\begin{aligned} |\lambda |\Vert u\Vert _Y + \Vert u\Vert _X \le C\Vert f\Vert _Y. \end{aligned}$$

Let ${{\mathcal {D}}}(A)$ be the domain of the operator A defined by

$$\begin{aligned} {{\mathcal {D}}}(A) = \{u_0 \in X \mid Bu_0=0\}. \end{aligned}$$

Then, the operator A generates continuous analytic semigroup $\{T_A(t)\}_{t\ge 0}$ such that $u = T_A(t)u_0$ solves Eq. (22) uniquely and the following estimates hold:

$$\begin{aligned} \Vert u(t)\Vert _Y \le r_be^{\lambda _0t}\Vert u_0\Vert _Y, \quad \Vert \partial _tu(t)\Vert _Y \le r_b\,t^{-1}\,e^{\lambda _0 t}\Vert u_0\Vert _Y, \quad \Vert \partial _tu(t)\Vert _Y \le r_be^{\lambda _0 t}\Vert u_0\Vert _X. \end{aligned}$$

(23)

These estimates and trace method of real-interpolation theory yield the following theorem.

Theorem 12

(Maximal regularity for initial value problem). Let $1< p < \infty$ and set ${{\mathcal {D}}}= (Y, {{\mathcal {D}}}(A))_{1-1/p, p}$, where $(\cdot , \cdot )_{1-1/p, p}$ denotes a real interpolation functor. Then, for any $u_0 \in {{\mathcal {D}}}$, Problem (22) admits a unique solution u with

$$\begin{aligned} e^{-\lambda _0t}u \in L_p(\mathbb {R}_+, X) \cap H^1_p(\mathbb {R}_+, Y) \quad (\mathbb {R}_+=(0, \infty )) \end{aligned}$$

possessing the estimate:

$$\begin{aligned} \Vert e^{-\lambda _0t}\partial _tu\Vert _{L_p(\mathbb {R}_+, Y)} + \Vert e^{-\lambda _0t}u\Vert _{L_p(\mathbb {R}_+, X)} \le C\Vert u_0\Vert _{(Y, {{\mathcal {D}}}(A))_{1-1/p, p}}. \end{aligned}$$

The ${{\mathcal {R}}}$-bounded solution operator plays an essential role to prove the following theorem.

Theorem 13

(Maximal regularity for boundary value problem). Let $1< p < \infty$. Then for any f and g with $e^{-\gamma t}f \in L_p(\mathbb {R}, Y)$ and $e^{-\gamma t}g \in L_p(\mathbb {R}, Z) \cap H^\alpha _p(\mathbb {R}, Y)$ for any $\gamma \ge \lambda _0$, Problem (21) admits a unique solution u with $e^{-\gamma t} u \in L_p(\mathbb {R}, X) \cap H^1_p(\mathbb {R}, Y)$ for any $\gamma \ge \lambda _0$ possessing the estimate:

$$\begin{aligned}&\Vert e^{-\gamma t}\partial _tu\Vert _{L_p(\mathbb {R}_+, Y)} + \Vert e^{-\gamma t}u\Vert _{L_p(\mathbb {R}_+, X)} \le C( \Vert e^{-\gamma t}f\Vert _{L_p(\mathbb {R}, Y)} \\&\quad + (1+\gamma )^\alpha \Vert e^{-\gamma t}g\Vert _{H^\alpha _p(\mathbb {R}, Y)} + \Vert e^{-\gamma t}g\Vert _{L_p(\mathbb {R}, Z)}) \end{aligned}$$

for any $\gamma \ge \lambda _0$. Here, the constant C may depend on $\lambda _0$ but independent of $\gamma$ whenever $\gamma \ge \lambda _0$, and we have set

$$\begin{aligned} H^\alpha _p(\mathbb {R}, Y) = \{h \in {{\mathcal {S}}}'(\mathbb {R}, Y) \mid \Vert h\Vert _{H^\alpha _p(\mathbb {R}, Y)} : = \Vert {{\mathcal {F}}}^{-1}_\xi [(1+|\xi |^2)^{\alpha /2}{{\mathcal {F}}}[h](\xi )]\Vert _{L_p(\mathbb {R}, Y)} < \infty \}. \end{aligned}$$

Proof

Let ${{\mathcal {L}}}$ and ${{\mathcal {L}}}^{-1}$ denote respective Laplace transformation and inverse Laplace transformation defined by setting

$$\begin{aligned}&{{\mathcal {L}}}[f](\lambda ) = \int _\mathbb {R}e^{-\lambda t}f(t)\,d t = \int _\mathbb {R}e^{-i\tau t}(e^{-\gamma t}f(t))\,d t = {{\mathcal {F}}}[e^{-\gamma t}f(t)](\tau ) \quad (\lambda = \gamma + i\tau ),\\&{{\mathcal {L}}}^{-1}[f](t) = \frac{1}{2\pi }\int _\mathbb {R}e^{\lambda t}f(\tau )\,d \tau = \frac{e^{\gamma t}}{2\pi }\int _\mathbb {R}e^{i\tau t}f(\tau )\,d \tau = e^{\gamma t}{{\mathcal {F}}}^{-1}[f](\tau ). \end{aligned}$$

We consider equations:

$$\begin{aligned} \partial _tu-Au = f, \quad Bu = g \quad \hbox { for}\ t \in \mathbb {R}. \end{aligned}$$

Applying Laplace transformation yields that

$$\begin{aligned} \lambda {{\mathcal {L}}}[u](\lambda ) -A{{\mathcal {L}}}[u](\lambda ) ={{\mathcal {L}}}[ f](\lambda ), \quad B{{\mathcal {L}}}[u](\lambda ) = {{\mathcal {L}}}[g](\lambda ). \end{aligned}$$

Applying ${{\mathcal {R}}}$-bounded solution operator ${{\mathcal {S}}}(\lambda )$ yields that

$$\begin{aligned} {{\mathcal {L}}}[u](\lambda ) = {{\mathcal {S}}}(\lambda )({{\mathcal {L}}}[f](\lambda ), \lambda ^\alpha {{\mathcal {L}}}[g](\lambda ), {{\mathcal {L}}}[g](\lambda )), \end{aligned}$$

and so

$$\begin{aligned} u = {{\mathcal {L}}}^{-1}[{{\mathcal {S}}}(\lambda ){{\mathcal {L}}}[(f, \Lambda ^\alpha g, g)](\lambda )], \end{aligned}$$

where $\Lambda ^\alpha g = {{\mathcal {L}}}^{-1}[\lambda ^\alpha {{\mathcal {L}}}[g]]$. Moreover,

$$\begin{aligned} \partial _tu = {{\mathcal {L}}}^{-1}[\lambda {{\mathcal {S}}}(\lambda ){{\mathcal {L}}}[f, \Lambda ^\alpha g, g)](\lambda )]. \end{aligned}$$

Using Fourier transformation and inverse Fourier transformation, we rewrite

$$\begin{aligned}&u = e^{\gamma t}{{\mathcal {F}}}^{-1}[{{\mathcal {S}}}(\lambda ){{\mathcal {F}}}[e^{-\gamma t}(f, \Lambda ^\alpha g, g)] (\tau )](t), \\&\partial _tu = e^{\gamma t}{{\mathcal {F}}}^{-1}[\lambda {{\mathcal {S}}}(\lambda ) {{\mathcal {F}}}[e^{-\gamma t}(f, \Lambda ^\alpha g, g)](\tau )](t). \end{aligned}$$

Applying the assumption of ${{\mathcal {R}}}$-bounded solution operators and Weis’s operator valued Fourier multiplier theorem yields that

$$\begin{aligned}&\Vert e^{-\gamma t}\partial _tu\Vert _{L_p(\mathbb {R}, Y)} + \Vert e^{-\gamma t}u\Vert _{L_p(\mathbb {R}, X)}\\&\quad \le C_pr_b(\Vert e^{-\gamma t}f\Vert _{L_p(\mathbb {R}, Y)} + (1 + \gamma )^\alpha \Vert e^{-\gamma t}g\Vert _{H^\alpha _p(\mathbb {R}, Y)} + \Vert e^{-\gamma t}g\Vert _{L_p(\mathbb {R}, Z)}) \end{aligned}$$

for any $\gamma \ge \lambda _0$. The uniqueness follows from the generation of analytic semigroup and Duhamel’s principle.

$\square$

We now explain our strategy to solve initial-boundary value problem:

$$\begin{aligned} \partial _tu - Au = f, \quad Bu = g \quad \hbox { for}\ t \in (0, \infty ), \quad u|_{t=0} = u_0. \end{aligned}$$

(24)

The point is how to get enough decay estimates. As a first step, we consider the following time shifted equations without initial data

$$\begin{aligned} \partial _tw + \lambda _1 w - Aw = f, \quad Bw = g \quad \hbox { for}\ t \in \mathbb {R}. \end{aligned}$$

(25)

Then, we have the following theorem which guarantees the polynomial decay of solutions.

Theorem 14

Let $\lambda _0$ be a constant appearing in Assumption 11 and let $\lambda _1 > \lambda _0$. Let $1< p < \infty$ and $b \ge 0$. Then, for any f and g with $<t>^bf \in L_p(\mathbb {R}, Y)$ and $<t>^bg \in L_p(\mathbb {R}, Z) \cap H^\alpha _p(\mathbb {R}, X)$, Problem (25) admits a unique solution $w \in H^1_p(\mathbb {R}, Y) \cap L_p(\mathbb {R}, X)$ possessing the estimate:

$$\begin{aligned} \begin{aligned}&\Vert<t>^bw\Vert _{L_p(\mathbb {R}, X)} + \Vert<t>^b\partial _tw\Vert _{L_p(\mathbb {R}, Y)} \\&\quad \le C(\Vert<t>^bf\Vert _{L_p(\mathbb {R}, Y)} + \Vert<t>^bg\Vert _{H^\alpha _p(\mathbb {R}, Y)} + \Vert <t>^bg\Vert _{L_p(\mathbb {R}, Z)}). \end{aligned} \end{aligned}$$

(26)

Proof

Since $ik + \lambda _1 \in \Sigma _{\omega , \lambda _0}$, for $k \in \mathbb {R}$ we set $w = {{\mathcal {F}}}^{-1}[{{\mathcal {M}}}(ik + \lambda _1)({{\mathcal {F}}}[f], (ik)^\alpha {{\mathcal {F}}}[g], {{\mathcal {F}}}[g])]$, and then w satisfies equations:

$$\begin{aligned} \partial _tw + \lambda _1w - Aw=f, \quad Bw=g \quad \hbox { for}\ t \in \mathbb {R}, \end{aligned}$$

and the estimate:

$$\begin{aligned} \Vert \partial _t w\Vert _{L_p(\mathbb {R}, Y)} + \Vert w\Vert _{L_p(\mathbb {R}, X)} \le C(\Vert f\Vert _{L_p(\mathbb {R}, Y)} + \Vert g\Vert _{H^\alpha _p(\mathbb {R}, Y)} + \Vert g\Vert _{L_p(\mathbb {R}, Z)}). \end{aligned}$$

(27)

This prove the theorem in the case where $b=0$. When $0 < b \le 1$, we observe that

$$\begin{aligned} \partial _t(<t>^bw) +\lambda _1(<t>^bw) - A(<t>^bw) =<t>^bf +<t>^{b-2}tw, \quad B(<t>^bw) = <t>^bg, \end{aligned}$$

and so noting that $\Vert <t>^{b-2}t w\Vert _Y \le C\Vert w\Vert _Y \le C\Vert w\Vert _X$, we have

$$\begin{aligned}&\Vert<t>^bw\Vert _{L_p((0, \infty ), X)} + \Vert<t>^b\partial _tw\Vert _{L_p((0, \infty ), Y)} \\&\quad \le C(\Vert<t>^{b-2}tw\Vert _{L_p(\mathbb {R}, Y)} + \Vert<t>^bf\Vert _{L_p(\mathbb {R}, Y)} + \Vert<t>^bg\Vert _{H^\alpha _p(\mathbb {R}, Y)} + \Vert<t>^bg\Vert _{L_p(\mathbb {R}, Z)}) \\&\quad \le C( \Vert<t>^bf\Vert _{L_p(\mathbb {R}, Y)} + \Vert<t>^bg\Vert _{H^\alpha _p(\mathbb {R}, Y)} + \Vert <t>^bg\Vert _{L_p(\mathbb {R}, Z)}). \end{aligned}$$

If $b > 1$, then repeated use of this argument yields the theorem, which completes the proof of Theorem 14.

$\square$

To compensate solutions, let $v_1$ be a solution of time shifted equations:

$$\begin{aligned} \partial _tv_1 + \lambda _1 v_1 -Av_1 = \lambda _1w, \quad Bv_1=0 \quad \hbox { for}\ t \in \mathbb {R}. \end{aligned}$$

By Theorem 14,

$$\begin{aligned} \begin{aligned}&\Vert<t>^b\partial _t v_1\Vert _{L_p(\mathbb {R}, Y)} + \Vert<t>^bv_1\Vert _{L_p(\mathbb {R}, X)} \le C\Vert<t>w\Vert _{L_p(\mathbb {R}, Y)} \\&\quad \le C( \Vert<t>^bf\Vert _{L_p(\mathbb {R}, Y)} + \Vert<t>^bg\Vert _{H^\alpha _p(\mathbb {R}, Y)} + \Vert <t>^bg\Vert _{L_p(\mathbb {R}, Z)}). \end{aligned} \end{aligned}$$

(28)

Here, we used the assumption that X is continuously embedded into Y, that is $\Vert w\Vert _Y \le C\Vert w\Vert _X$ for some constant C. The role of $v_1$ is to controle the compatibility conditions, that is

$$\begin{aligned} v_1 \in {{\mathcal {D}}}(A) \quad \hbox { for all}\ t \in \mathbb {R}. \end{aligned}$$

(29)

Thus, if $g=0$ in (24) like Dirichlet zero condition case, then we need not this step.

To solve Eq. (24), we now consider a second compensation function $v_2$, which is a solution of the following initial problem with zero boundary condition:

$$\begin{aligned} \partial _tv_2 -Av_2 = \lambda _1v_1, \quad Bv_2=0 \text {for~}t \in (0, \infty ), \quad v_2|_{t=0} = u_0 - (w|_{t=0} + v|_{t=0}). \end{aligned}$$

(30)

To solve (30) with the help of semi-group $\{T_A(t)\}_{t\ge 0}$, we need the compatibility condition:

$$\begin{aligned} B(u_0 - (w|_{t=0}+ v_1|_{t=0})) = Bu_0 - g|_{t=0} =0. \end{aligned}$$

(31)

Since (29) holds, assuming the compatibility condition: $Bu_0 = g|_{t=0}$, by Duhamel’s principle, $v_2$ is represented as

$$\begin{aligned} v_2 = T_A(t)(u_0 - (w|_{t=0} + v_1|_{t=0}) + \int ^t_0T_A(t-s)(\lambda _1v_1(s))\,ds. \end{aligned}$$

(32)

And then, $u = w + v_1 + v_2$ is a required solution of Eq. (24). Concerning the estimate of $v_2$, for $t \in (0, 2)$ we use the estimate:

$$\begin{aligned} \Vert T_A(t)v_0\Vert _{D(A)} \le C\Vert v_0\Vert _{D(A)} \end{aligned}$$

where $\Vert \cdot \Vert _{D(A)}$ denotes the norm of domain D(A). And, for $t \in [2, \infty )$ we use so called $L_p$-$L_q$ decay estimate for the semigroup $\{T_A(t)\}_{t\ge 0}$. In this paper, we use the $L_p$-$L_q$ decay estimate for the Stokes equations for the compressible viscous fluid, which will be given in (68) in Sect. 5 below.

4 Estimates of Nonlinear Terms

In what follows, let $T > 0$ be any positive time and let b and p be positive numbers and an exponents given in Theorem 3 and Theorem 6. Let ${{\mathcal {U}}}^i_\epsilon$ ($i=1,2$) be underlying spaces for linearized equations of equations (14), which is defined by

(33)

Recall that our energy $E_T(\eta ,\mathbf{u} )$ has been defined by

$$\begin{aligned} E_T(\eta , \mathbf{u} )&= \Vert<t>^{b}\nabla (\eta , \mathbf{u} )\Vert _{L_p((0, T), H^{0,1}_2(\Omega ))} + \Vert<t>^{b}(\eta , \mathbf{u} )\Vert _{L_\infty ((0, T), L_2(\Omega )\cap L_6(\Omega ))} \\&\quad + \Vert<t>^{b}\partial _t(\eta , \mathbf{u} )\Vert _{L_p((0, T), H^{1,0}_2(\Omega ) \cap H^{1,0}_{6} (\Omega ))} +\Vert <t>^{b}(\eta , \mathbf{u} )\Vert _{L_p((0, T), H^2_6(\Omega ))}. \end{aligned}$$

To estimate $L_{2+\sigma }$ norm, we use standard interpolation inequality:

$$\begin{aligned} \Vert f\Vert _{L_{2+\sigma }(\Omega )} \le \Vert f\Vert _{L_2(\Omega )}^{\frac{4-\sigma }{2(2+\sigma )}} \Vert f\Vert _{L_6(\Omega )}^{\frac{3\sigma }{2(2+\sigma )}} \le \frac{4-\sigma }{2(2+\sigma )}\Vert f\Vert _{L_2(\Omega )} + \frac{3\sigma }{2(2+\sigma )}\Vert f\Vert _{L_6(\Omega )}. \end{aligned}$$

(34)

In estimating $L_r$ norm, we meet $L_{2+\sigma }$ norm in view of Hölder’s inequality, but this norm is estimate by $L_2$ and $L_6$ norm with the help of (34). In particular, for $(\theta , \mathbf{v} ) \in {{\mathcal {U}}}^1_T\times {{\mathcal {U}}}^2_T$, we know that

$$\begin{aligned} \begin{aligned} \Vert<t>^b(\theta , \mathbf{v} )\Vert _{L_\infty ((0, T), L_{2+\sigma }(\Omega )}&\le C_\sigma \sum _{q=2, 6}\Vert<t>^b(\theta , \mathbf{v} ) \Vert _{L_\infty ((0, T), L_q(\Omega ))}, \\ \Vert<t>^b\nabla (\theta , \mathbf{v} )\Vert _{L_p((0, T), H^{0,1}_{2+\sigma }(\Omega )}&\le C_\sigma \sum _{q=2, 6}\Vert<t>^b\nabla (\theta , \mathbf{v} ) \Vert _{L_p((0, T), H^{0,1}_q(\Omega ))}, \\ \Vert<t>^b\partial _t(\theta , \mathbf{v} )\Vert _{L_p((0, T), H^{1,0}_{2+\sigma }(\Omega )}&\le C_\sigma \sum _{q=2, 6}\Vert <t>^b\partial _t(\theta , \mathbf{v} ) \Vert _{L_p((0, T), H^{1,0}_q(\Omega ))}. \end{aligned} \end{aligned}$$

(35)

Notice that for any $\theta \in {{\mathcal {U}}}^1_T$ we see that

$$\begin{aligned} \rho _{*}/2 \le |\rho _{*}+\tau \theta (y, t)| \le 3\rho _{*}/2 \quad \text { for } (y,t) \in \Omega \times (0, T) \text { and } |\tau | \le 1. \end{aligned}$$

(36)

For $\mathbf{v} \in {{\mathcal {U}}}^2_T$ let $\mathbf{k} _\mathbf{v} = \int ^t_0\nabla \mathbf{v} (\cdot ,s)\,ds$, and then $|\mathbf{k} _\mathbf{v} (y, t)| \le \delta$ for any $(y, t) \in \Omega \times (0, T)$. Moreover, for $q=2, 2+\sigma$ and 6 by Hölder’s inequality

$$\begin{aligned} \sup _{t \in (0, T)} \Vert \mathbf{k} _\mathbf{v} \Vert _{H^1_q(\Omega )} \le \int ^T_0\Vert \nabla \mathbf{v} (\cdot , t)\Vert _{H^1_q(\Omega )}dt \le C\Bigl (\int ^\infty _0<t>^{-p'b}dt\Bigr )^{1/p'}\Vert <t>^b\nabla \mathbf{v} \Vert _{L_p((0, T), H^1_q(\Omega ))}, \end{aligned}$$

(37)

where $bp' > 1$.

In what follows, for notational simplicity we use the following abbreviation: $\Vert f\Vert _{H^1_q(\Omega )} = \Vert f\Vert _{H^1_q}$, $\Vert f\Vert _{L_q(\Omega )} = \Vert f\Vert _{L_q}$, $\Vert f\Vert _{L_\infty ((0, T), X)} = \Vert f\Vert _{L_\infty (X)}$, and $\Vert <t>^bf\Vert _{L_p((0, T), X)} = \Vert f\Vert _{L_{p,b}(X)}$. Let $(\theta , \mathbf{v} ) \in {{\mathcal {U}}}^1_T\times {{\mathcal {U}}}^2_T$ and $(\theta _i, \mathbf{v} _i) \in {{\mathcal {U}}}^1_T\times {{\mathcal {U}}}^2_T$ ($i=1,2$). The purpose of this section is to give necessary estimates of $(F(\theta , \mathbf{v} ), \mathbf{G} (\theta , \mathbf{v} ))$ and difference: $(F(\theta _1, \mathbf{v} _1) -F(\theta _2, \mathbf{v} _2), \mathbf{G} (\theta _1, \mathbf{v} _1) - \mathbf{G} (\theta _2, \mathbf{v} _2)))$ to prove the global wellposedness of Eq. (14). Recall that

$$\begin{aligned} \begin{aligned} F(\theta , \mathbf{v} )&= \rho _*{{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} )\nabla \mathbf{v} +\theta \mathrm{div}\,\mathbf{v} + \theta {{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} )\nabla \mathbf{v} , \\ \mathbf{G} (\theta , \mathbf{v} )&= \theta \partial _t\mathbf{v} + \mathbf{V} _1(\mathbf{k} )\nabla ^2\mathbf{v} + (\mathbf{V} _2(\mathbf{k} )\int ^t_0\nabla ^2\mathbf{v} \,ds)\nabla \mathbf{v} \\&\quad - ({\mathfrak {p}}'(\rho _*+\theta )-{\mathfrak {p}}'(\rho _*))\nabla \theta - {\mathfrak {p}}'(\rho _*+\theta )\mathbf{V} _0(\mathbf{k} )\nabla \theta . \end{aligned} \end{aligned}$$

(38)

We start with estimating $\Vert F(\theta , \mathbf{v} )\Vert _{L_{p,b}(H^1_r)}$. Recall that $r^{-1} = 2^{-1} + (2+\sigma )^{-1}$ and we use the estimates:

$$\begin{aligned} \begin{aligned} \Vert fg\Vert _{L_{p,b}(H^1_r)}&\le C\Vert f\Vert _{L_\infty (H^1_{2+\sigma })} \Vert g\Vert _{L_{p,b}(H^1_2)}, \\ \Vert fgh\Vert _{L_{p,b}(H^1_r)}&\le C(\Vert f\Vert _{L_\infty (H^1_6)} \Vert g\Vert _{L_\infty (H^1_{2+\sigma })} + \Vert f\Vert _{L_\infty (H^1_{2+\sigma })}\Vert g\Vert _{L_\infty (H^1_6)}) \Vert h\Vert _{L_{p,b}(H^1_2)}, \end{aligned} \end{aligned}$$

(39)

as follows from Hölder’s inequality and Sobolev’s inequality : $\Vert f\Vert _{L_\infty } \le C\Vert f\Vert _{H^1_6}$. Let $dG(\mathbf{k} )$ denote the derivative of $G(\mathbf{k} )$ with respect to $\mathbf{k}$ and $C_\mathrm{div}\,$ be a constan such that $\sup _{|\mathbf{k} |< \delta }|{{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} )| < C_\mathrm{div}\,$, $\sup _{|\mathbf{k} |< \delta }|d{{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} )| < C_\mathrm{div}\,$, and $\sup _{|\mathbf{k} |< \delta }|d(d{{\mathcal {D}}}_\mathrm{div}\,)(\mathbf{k} )| < C_\mathrm{div}\,$. Then, noting ${{\mathcal {D}}}_\mathrm{div}\,(0)=0$, by (37) we have

$$\begin{aligned} \begin{aligned} \Vert {{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} _\mathbf{v })\Vert _{H^1_q}&\le C_\mathrm{div}\,\Vert \mathbf{k} _\mathbf{v }\Vert _{H^1_q} \le C\Vert \nabla \mathbf{v} \Vert _{L_{p,b}(H^1_q)} \quad \text {for~} \mathbf{v} \in {{\mathcal {U}}}^2_T \text {~and~} q=2, 2+\sigma \text {~and~} 6. \end{aligned} \end{aligned}$$

(40)

Moreover, for $\mathbf{v} _1$, $\mathbf{v} _2 \in {{\mathcal {U}}}^2_T$ writing

$$\begin{aligned} {{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} _\mathbf{v _1}) - {{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} _\mathbf{v _2}) = \int ^t_0d{{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} _\mathbf{v _2}+ \tau (\mathbf{k} _\mathbf{v _1}-\mathbf{k} _\mathbf{v _2}))\,d\tau \,(\mathbf{k} _\mathbf{v _1}-\mathbf{k} _\mathbf{v _2}), \end{aligned}$$

and noting that $|\mathbf{k} _\mathbf{v _2}+ \tau (\mathbf{k} _\mathbf{v _1}-\mathbf{k} _\mathbf{v _2})| = |(1-\tau )\mathbf{k} _\mathbf{v _2} + \tau \mathbf{k} _\mathbf{v _1}| \le (1-\tau )\delta + \tau \delta = \delta$, we have

$$\begin{aligned} \begin{aligned}&\Vert {{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} _\mathbf{v _1})- {{\mathcal {D}}}_\mathrm{div}\,(\mathbf{k} _\mathbf{v _2})\Vert _{H^1_q}\\&\quad \le C_\mathrm{div}\,(\Vert \mathbf{k} _\mathbf{v _1}-\mathbf{k} _\mathbf{v _2}\Vert _{L_\infty (H^1_q)} + \sum _{i=1,2}\Vert \nabla \mathbf{k} _\mathbf{v _i}\Vert _{L_\infty (L_q)} \Vert \mathbf{k} _\mathbf{v _1}-\mathbf{k} _\mathbf{v _2}\Vert _{L_\infty (L_\infty )})\\&\quad \le C(\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_q)} + \sum _{i=1,2}\Vert \nabla \mathbf{v} _i\Vert _{L_{p,b}(H^1_q)} \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)}. \end{aligned} \end{aligned}$$

(41)

Since $\theta = \theta |_{t=0} + \int ^t_0\partial _s\theta \,ds$, for $X \in \{L_q, H^1_q\}$ with $q=2$, $2+\sigma$ and 6

$$\begin{aligned} \begin{aligned} \Vert \theta (\cdot , t)\Vert _{X}&\le \Vert \theta _0\Vert _{X} + \int ^T_0\Vert (\partial _s\theta )(\cdot , s)\Vert _{X}\,ds \\&\le \Vert \theta _0\Vert _{X} + \Bigl (\int ^\infty _0<t>^{-p'b}\,dt\Bigr )^{1/p'} \Vert \partial _s\theta \Vert _{L_{p,b}(X)}. \end{aligned} \end{aligned}$$

(42)

In particular, by Sobolev’s inequality

$$\begin{aligned} \Vert \theta (\cdot , t)\Vert _{L_\infty } \le C(\Vert \theta _0\Vert _{H^1_6} + \Vert \partial _t\theta \Vert _{L_{p,b}(H^1_{6})}). \end{aligned}$$

(43)

For $\theta \in {{\mathcal {U}}}^1_T$ and $\mathbf{v} \in {{\mathcal {U}}}^2_T$, combining (39), (40), (41), (42), and (43) yields that

$$\begin{aligned} \begin{aligned}&\Vert F(\theta , \mathbf{v} )\Vert _{L_{p,b}(H^1_r)} \le C[\Vert \nabla \mathbf{v} \Vert _{L_{p,b}(H^1_{2+\sigma })} \Vert \nabla \mathbf{v} \Vert _{L_{p,b}(H^1_2)} + (\Vert \theta _0\Vert _{H^1_{2+\sigma }}+\Vert \partial _t\theta \Vert _{L_{p,b}(H^1_{2+\sigma })}) \Vert \nabla \mathbf{v} \Vert _{L_{p,b}(H^1_2)}\\&\quad +\{(\Vert \theta _0\Vert _{H^1_6}+\Vert \partial _t\theta \Vert _{L_{p,b}(H^1_6)}) \Vert \nabla \mathbf{v} \Vert _{L_{p,b}(H^1_{2+\sigma })} + (\Vert \theta _0\Vert _{H^1_{2+\sigma }}+\Vert \partial _t\theta \Vert _{L_{p,b}(H^1_{2+\sigma })}) \Vert \nabla \mathbf{v} \Vert _{L_{p,b}(H^1_6)}\}\\&\times \Vert \nabla \mathbf{v} \Vert _{L_{p,b}(H^1_2)}]. \end{aligned} \end{aligned}$$

(44)

Analogously, for $\theta _i \in {{\mathcal {U}}}^1_T$ and $\mathbf{v} _i \in {{\mathcal {U}}}^2_T$ ($i=1,2$),

$$\begin{aligned}&\Vert F(\theta _1, \mathbf{v} _1)-F(\theta _2, \mathbf{v} _2)\Vert _{L_{p,b}(L_r)} \nonumber \\&\quad \le C[(\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_{2+\sigma })} + \sum _{i=1,2}\Vert \nabla \mathbf{v} _i\Vert _{L_{p,b}(H^1_{2+\sigma })} \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)})\Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_2)} \nonumber \\&\qquad + \Vert \nabla \mathbf{v} _2\Vert _{L_{p,b}(H^1_{2+\sigma })}\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_2)} + \Vert \partial _t(\theta _1-\theta _2)\Vert _{L_{p,b}(H^1_{2+\sigma })} \Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_2)} \nonumber \\&\qquad + (\Vert \theta _0\Vert _{H^1_{2+\sigma }}+\Vert \partial _t\theta _2\Vert _{L_{p, b}(H^1_{2+\sigma })}) \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_2)} \nonumber \\&\qquad +(\Vert \partial _t(\theta _1-\theta _2)\Vert _{L_{p,b}(H^1_6)} \Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_{2+\sigma })} + \Vert \partial _t(\theta _1-\theta _2)\Vert _{L_{p,b}(H^1_{2+\sigma })} \Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_6)})\Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_2)} \nonumber \\&\qquad +\{(\Vert \theta _0\Vert _{H^1_6}+\Vert \partial _t\theta _2\Vert _{L_{p,b}(H^1_6)}) (\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_{2+\sigma })} + \sum _{i=1,2}\Vert \nabla \mathbf{v} _i\Vert _{L_{p,b}(H^1_{2+\sigma })} \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6))}) \nonumber \\&\qquad + (\Vert \theta _0\Vert _{H^1_{2+\sigma }}+\Vert \partial _t\theta \Vert _{L_{p,b}(H^1_{2+\sigma })}) (\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)} + \sum _{i=1,2}\Vert \nabla \mathbf{v} _i\Vert _{L_{p,b}(H^1_6)} \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)})\} \nonumber \\&\times \Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_2)} \nonumber \\&\qquad +\{(\Vert \theta _0\Vert _{H^1_6}+\Vert \partial _t\theta _2\Vert _{L_{p,b}(H^1_6)}) \Vert \nabla \mathbf{v} _2\Vert _{L_{p,b}(H^1_{2+\sigma })} + (\Vert \theta _0\Vert _{H^1_{2+\sigma }}+\Vert \partial _t\theta _2\Vert _{L_{p,b}(H^1_{2+\sigma })} \Vert \nabla \mathbf{v} _2\Vert _{L_{p,b}(H^1_6)}\} \nonumber \\&\times \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_2)}]. \end{aligned}$$

(45)

We now estimate $\Vert F(\theta , \mathbf{v} )\Vert _{L_{p,b}(H^1_q)}$ and $\Vert F(\theta _1, \mathbf{v} _1) - F(\theta _2, \mathbf{v} _2)\Vert _{L_{p, b}(H^1_q)}$ with $q=2$ and 6. For this purpose, we use the following estimates:

$$\begin{aligned} \Vert fg\Vert _{L_{p, b}(H^1_q)}&\le C\{\Vert f\Vert _{L_\infty (H^1_q)}\Vert g\Vert _{L_{p, b}(H^1_6)} + \Vert f\Vert _{L_\infty (H^1_q)}\Vert g\Vert _{L_{p,b}(H^1_6)}\}, \\ \Vert fgh\Vert _{L_{p, b}(H^1_q)}&\le C\{\Vert f\Vert _{L_\infty (H^1_q)} \Vert g\Vert _{L_\infty (H^1_6)}\Vert h\Vert _{L_{p, b}(H^1_6)} + \Vert f\Vert _{L_\infty (H^1_6)}\Vert g\Vert _{L_\infty (H^1_q)}\Vert h\Vert _{L_{p, b}(H^1_6)} \\&\quad + \Vert f\Vert _{L_\infty (H^1_6)}\Vert g\Vert _{L_\infty (H^1_6)}\Vert h\Vert _{L_{p, b}(H^1_q)}\}. \end{aligned}$$

And then, using (40), (41), (42), we have

$$\begin{aligned}&\Vert F(\theta ,\mathbf{v} )\Vert _{L_{p,b}(H^1_q)} \le C\{\Vert \nabla \mathbf{v} \Vert _{L_{p,b}(H^1_q)}\Vert \nabla \mathbf{v} \Vert _{L_{p,b}(H^1_6)} + (\Vert \theta _0\Vert _{H^1_q}+\Vert \partial _t\theta \Vert _{L_{p,b}(H^1_q)})\Vert \nabla \mathbf{v} \Vert _{L_{p,b}(H^1_6)} \nonumber \\&\qquad + (\Vert \theta _0\Vert _{H^1_6}+\Vert \partial _t\theta \Vert _{L_{p,b}(H^1_6)}) \Vert \nabla \mathbf{v} \Vert _{L_{p,b}(H^1_q)} + (\Vert \theta _0\Vert _{H^1_q}+\Vert \partial _t\theta \Vert _{L_{p,b}(H^1_q)})\Vert \nabla \mathbf{v} \Vert _{L_{p,b}(H^1_6)}^2 \nonumber \\&\qquad + (\Vert \theta _0\Vert _{H^1_6}+\Vert \partial _t\theta \Vert _{L_{p,b}(H^1_6)})\Vert \nabla \mathbf{v} \Vert _{L_{p,b}(H^1_q)} \Vert \nabla \mathbf{v} \Vert _{L_{p,b}(H^1_6)}\}; \end{aligned}$$

(46)

$$\begin{aligned}&\Vert F(\theta _1, \mathbf{v} _1) - F(\theta _2, \mathbf{v} _2)\Vert _{L_{p, b}(H^1_q)} \nonumber \\&\quad \le C \{(\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_q)} + \sum _{i=1,2}\Vert \nabla \mathbf{v} _i\Vert _{L_{p,b}(H^1_q)}\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)}) \Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_6)} \nonumber \\&\qquad + (\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)} + \sum _{i=1,2}\Vert \nabla \mathbf{v} _i\Vert _{L_{p,b}(H^1_6)}\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)}) \Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_q)}\nonumber \\&\qquad +\Vert \nabla \mathbf{v} _2\Vert _{L_{p,b}(H^1_q)}\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)} + \Vert \nabla \mathbf{v} _2\Vert _{L_{p,b}(H^1_6)}\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_q)} \nonumber \\&\qquad + \Vert \partial _t(\theta _1-\theta _2)\Vert _{L_{p,b}(H^1_q)}\Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_6)} + \Vert \partial _t(\theta _1-\theta _2)\Vert _{L_{p,b}(H^1_6)} \Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_q)} \nonumber \\&\qquad + (\Vert \theta _0\Vert _{H^1_q} + \Vert \partial _t\theta _2\Vert _{L_{p,b}(H^1_q)}) \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)} + (\Vert \theta _0\Vert _{H^1_6} + \Vert \partial _t\theta _2\Vert _{L_{p,b}(H^1_6)}) \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_q)} \nonumber \\&\qquad +\Vert \partial _t(\theta _1-\theta _2)\Vert _{L_{p,b}(H^1_q)}\Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_6)}^2 +\Vert \partial _t(\theta _1-\theta _2)\Vert _{L_{p,b}(H^1_6)}\Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_q)} \Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_6)} \nonumber \\&\qquad + (\Vert \theta _0\Vert _{H^1_q}+\Vert \partial _t\theta _2\Vert _{L_{p,b}(H^1_q)})( \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)} + \sum _{i=1,2}\Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_6)}\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)}) \nonumber \\&\times \Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_6)}\nonumber \\&\qquad + (\Vert \theta _0\Vert _{H^1_6}+\Vert \partial _t\theta _2\Vert _{L_{p,b}(H^1_6)})( \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_q)} + \sum _{i=1,2}\Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_q)}\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)}) \nonumber \\&\times \Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_6)}\nonumber \\&\qquad + (\Vert \theta _0\Vert _{H^1_6}+\Vert \partial _t\theta _2\Vert _{L_{p,b}(H^1_6)})( \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)} + \sum _{i=1,2}\Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_6)}\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)}) \nonumber \\&\times \Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_q)}\nonumber \\&\qquad +(\Vert \theta _0\Vert _{H^1_q}+\Vert \partial _t\theta _2\Vert _{L_{p,b}(H^1_q)}) \Vert \nabla \mathbf{v} _2\Vert _{L_{p,b}(H^1_6)} \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)} \nonumber \\&\qquad +(\Vert \theta _0\Vert _{H^1_6}+\Vert \partial _t\theta _2\Vert _{L_{p,b}(H^1_6)}) \Vert \nabla \mathbf{v} _2\Vert _{L_{p,b}(H^1_q)} \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)} \nonumber \\&\qquad +(\Vert \theta _0\Vert _{H^1_6}+\Vert \partial _t\theta _2\Vert _{L_{p,b}(H^1_6)}) \Vert \nabla \mathbf{v} _2\Vert _{L_{p,b}(H^1_6)} \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_q)}\}. \end{aligned}$$

(47)

We next estimate $\Vert \mathbf{G} (\theta , \mathbf{v} )\Vert _{L_{p,b}(L_r)}$ and $\Vert \mathbf{G} (\theta _1, \mathbf{v} _1)-\mathbf{G} (\theta _2, \mathbf{v} _2)\Vert _{L_{p,b}(L_r)}$. For this purpose, we use the estimates:

$$\begin{aligned} \begin{aligned} \Vert fg\Vert _{L_{p,b}(L_r)}&\le \Vert f\Vert _{L_\infty (L_{2+\sigma })}\Vert g\Vert _{L_{p,b}(L_2)}, \\ \Vert fgh\Vert _{L_{p,b}(L_r)}&\le \Vert f\Vert _{L_\infty (L_\infty )} \Vert g\Vert _{L_\infty (L_{2+\sigma )}}\Vert h\Vert _{L_{p,b}(L_2)}. \end{aligned} \end{aligned}$$

(48)

Employing the same argument as in (40) and (41) and using $\mathbf{V} _i(0)=0$ ($i=0,1$), for $i=0,1$ we have

$$\begin{aligned} \begin{aligned}&\Vert \mathbf{V} _i(\mathbf{k} )\Vert _{L_\infty (L_q)} \le \sup _{|\mathbf{k} | < \delta }|d\mathbf{V} _i(\mathbf{k} )| \int ^T_0\Vert \nabla \mathbf{v} (\cdot , s)\Vert _{L_q} \le C\Vert \nabla \mathbf{v} \Vert _{L_{p,b}(L_q)}; \\&\Vert \mathbf{V} _i(\mathbf{k} _\mathbf{v _1}) - \mathbf{V} _i(\mathbf{k} _\mathbf{v _2})\Vert _{L_\infty (L_q)} \le C\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(L_q)}, \end{aligned} \end{aligned}$$

(49)

where $q=2, 2+\sigma$ and 6. Moreover, $\Vert \mathbf{V} _2(\mathbf{k} )\Vert _{L_\infty (L_\infty )} = \sup _{|\mathbf{k} | < \delta }|\mathbf{V} _1(\mathbf{k} )|$,

$$\begin{aligned}&\Vert \mathbf{V} _i(\mathbf{k} )\Vert _{L_\infty (L_\infty )} \le \sup _{|\mathbf{k} | < \delta }|d\mathbf{V} _i(\mathbf{k} )| \int ^T_0\Vert \nabla \mathbf{v} (\cdot , s)\Vert _{H^1_6} \le C\Vert \nabla \mathbf{v} \Vert _{L_{p,b}(H^1_6)}; \quad (i=0,1), \\&\Vert \mathbf{V} _i(\mathbf{k} _\mathbf{v _1}) - \mathbf{V} _i(\mathbf{k} _\mathbf{v _2})\Vert _{L_\infty (L_\infty )} \le C\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)} \quad (i=0,1,2) \end{aligned}$$

as follows from $|\mathbf{V} _2(\mathbf{k} _\mathbf{v _1}) - \mathbf{V} _2(\mathbf{k} _\mathbf{v _2})| \le \sup _{|\mathbf{k} | \le \delta }|(d\mathbf{V} _i)(\mathbf{k} )||\mathbf{k} _\mathbf{v _1}-\mathbf{k} _\mathbf{v _2}|$. Writing

$$\begin{aligned} {\mathfrak {p}}'(\rho _*+\theta )-{\mathfrak {p}}'(\rho _*)&= \int ^1_0{\mathfrak {p}}''(\rho _* +\tau \theta )\,d\tau \,\theta , \\ {\mathfrak {p}}'(\rho _*+\theta _1)-{\mathfrak {p}}'(\rho _*+\theta _2)&= \int ^1_0{\mathfrak {p}}''(\rho _* + \theta _2 +\tau (\theta _1-\theta _2))\,d\tau \,(\theta _1-\theta _2), \end{aligned}$$

by (36) and (42) we have

$$\begin{aligned} \begin{aligned}&\Vert ({\mathfrak {p}}'(\rho _*+\theta )-{\mathfrak {p}}'(\rho _*))\nabla \theta \Vert _{L_{p,b}(L_r)} \le C(\Vert \theta _0\Vert _{L_{2+\sigma }}+\Vert \partial _t\theta \Vert _{L_{p,b}(L_{2+\sigma })}) \Vert \nabla \theta \Vert _{L_{p,b}(L_2)}, \\&\Vert ({\mathfrak {p}}'(\rho _*+\theta _1)-{\mathfrak {p}}'(\rho _*))\nabla \theta _1 -({\mathfrak {p}}'(\rho _*+\theta _2)-{\mathfrak {p}}'(\rho _*))\nabla \theta _2\Vert _{L_{p,b}(L_r)}) \\&\quad \le C\{\Vert \partial _t(\theta _1-\theta _2)\Vert _{L_{p,b}(L_{2+\sigma })} \Vert \nabla \theta \Vert _{L_{p,b}(L_2)} + (\Vert \theta _0\Vert _{L_{2+\sigma }}+\Vert \partial _t\theta _2\Vert _{L_{p,b}(L_{2+\sigma })} \Vert \nabla (\theta _1-\theta _2)\Vert _{L_{p, b}(L_2)},\} \\&\Vert ({\mathfrak {p}}'(\rho _*+\theta )-{\mathfrak {p}}'(\rho _*))\nabla \theta \Vert _{L_{p,b}(L_q)} \le C(\Vert \theta _0\Vert _{H^1_6})+\Vert \partial _t\theta \Vert _{L_{p,b}(H^1_6)} \Vert \nabla \theta \Vert _{L_{p,b}(L_q)}), \\&\Vert ({\mathfrak {p}}'(\rho _*+\theta _1)-{\mathfrak {p}}'(\rho _*))\nabla \theta _1 -({\mathfrak {p}}'(\rho _*+\theta _2)-{\mathfrak {p}}'(\rho _*))\nabla \theta _2\Vert _{L_{p,b}(L_q)} \\&\quad \le C\{\Vert \partial _t(\theta _1-\theta _2)\Vert _{L_{p,b}(H^1_6)} \Vert \nabla \theta _1\Vert _{L_{p,b}(L_q)} + (\Vert \theta _0\Vert _{H^1_6})+\Vert \partial _t\theta _2\Vert _{L_{p,b}(H^1_6)} \Vert \nabla (\theta _1-\theta _2)\Vert _{L_{p, b}(L_q)}\}, \end{aligned} \end{aligned}$$

(50)

for $q=2, 2+\sigma$ and 6. Combining these estimates above, we have

$$\begin{aligned}&\Vert \mathbf{G} (\theta , \mathbf{v} )\Vert _{L_{p, b}(L_r)} \le C\{(\Vert \theta _0\Vert _{L_{2+\sigma }}+\Vert \partial _t\theta \Vert _{L_{p,b}(L_{2+\sigma })}) (\Vert \partial _t\mathbf{v} \Vert _{L_{p,b}(L_2)} + \Vert \nabla \theta \Vert _{L_{p,b}(L_2)}) \nonumber \\&\quad + \Vert \nabla \mathbf{v} \Vert _{L_{p,b}(L_{2+\sigma })}(\Vert \nabla ^2\mathbf{v} \Vert _{L_{p,b}(L_2)} + \Vert \nabla \theta \Vert _{L_{p,b}(L_2)})\}; \end{aligned}$$

(51)

$$\begin{aligned}&\Vert \mathbf{G} (\theta _1, \mathbf{v} _1) - \mathbf{G} (\theta _2, \mathbf{v} _2)\Vert _{L_{p,b}(L_r)} \le C\{\Vert \partial _t(\theta _1-\theta _2)\Vert _{L_{p,b}(L_{2+\sigma })}\Vert \partial _t\mathbf{v} _1\Vert _{L_{p,b}(L_2)} \nonumber \\&\quad + (\Vert \theta _0\Vert _{L_{2+\sigma }} + \Vert \partial _t\theta _2\Vert _{L_{p,b}(L_{2+\sigma })}) \Vert \partial _t(\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(L_2)} + \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(L_2)}\Vert \nabla ^2\mathbf{v} _1\Vert _{L_{p, b}(L_{2+\sigma })} \nonumber \\&\quad + \Vert \nabla \mathbf{v} _2\Vert _{L_{p,b}(L_{2+\sigma })}\Vert \nabla ^2(\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(L_2)} + \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(L_2)}\Vert \nabla ^2\mathbf{v} _1\Vert _{L_{p,b}(L_{2+\sigma })} \Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_6)} \nonumber \\&\quad + \Vert \nabla ^2(\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(L_2)}\Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(L_{2+\sigma })} + \Vert \nabla ^2\mathbf{v} _2\Vert _{L_{p,b}(L_{2+\sigma })}\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(L_2)} \nonumber \\&\quad +\Vert \partial _t(\theta _1-\theta _2)\Vert _{L_{p,b}(L_2)} \Vert \nabla \theta _1\Vert _{L_{p,b}(L_{2+\sigma })} + \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(L_2)}\Vert \nabla \theta _1\Vert _{L_{p,b}(L_{2+\sigma })} \nonumber \\&\quad + \Vert \nabla \mathbf{v} _2\Vert _{L_{p,b}(L_{2+\sigma })}\Vert \nabla (\theta _1-\theta _2)\Vert _{L_{p,b}(L_2)} +(\Vert \theta _0\Vert _{L_{2+\sigma }} + \Vert \partial _t\theta _2\Vert _{L_{p,b}(L_{2+\sigma })} \Vert \nabla (\theta _1-\theta _2)\Vert _{L_{p,b}(L_2)}. \end{aligned}$$

(52)

Finally, we estimate $\Vert G(\theta , \mathbf{v} )\Vert _{L_{p,b}(L_q)}$ and $\Vert G(\theta _1, \mathbf{v} _1) - \mathbf{G} (\theta _2, \mathbf{v} _2)\Vert _{L_{p,b}(L_q)}$ with $q=2$ and 6. For this purpose, we use the following estimates:

$$\begin{aligned} \Vert fg\Vert _{L_{p, b}(L_q)}&\le C\Vert f\Vert _{L_\infty (H^1_q)}\Vert g\Vert _{L_{p, b}(L_q)}, \\ \Vert fgh\Vert _{L_{p, b}(L_q)}&\le C\Vert f\Vert _{L_\infty (L_\infty )} \Vert g\Vert _{L_\infty (H^1_6)}\Vert h\Vert _{L_{p, b}(L_q)}. \end{aligned}$$

And then, using (49), (50), (42) and (43), for $q=2$ and 6 we have

$$\begin{aligned}&\Vert \mathbf{G} (\theta , \mathbf{v} )\Vert _{L_{p,b}(L_q)} \le C\{(\Vert \theta _0\Vert _{H^1_6} + \Vert \partial _t\theta \Vert _{L_{p,b}(H^1_6)}) (\Vert \partial _t\mathbf{v} \Vert _{L_{p,b}(L_q)} + \Vert \nabla \theta \Vert _{L_{p,b}(L_q)}) \nonumber \\&\quad +\Vert \nabla \mathbf{v} \Vert _{L_{p,b}(H^1_6)} (\Vert \nabla ^2\mathbf{v} \Vert _{L_{p,b}(L_q)} + \Vert \nabla \theta \Vert _{L_{p,b}(L_q)}\}; \end{aligned}$$

(53)

$$\begin{aligned}&\Vert \mathbf{G} (\theta _1, \mathbf{v} _1) - \mathbf{G} (\theta _2, \mathbf{v} _2)\Vert _{L_{p,b}(L_q)} \le C(\Vert \partial _t(\theta _1-\theta _2)\Vert _{L_{p,b}(H^1_6)}\Vert \partial _t\mathbf{v} _1\Vert _{L_{p,b}(L_q)} \nonumber \\&\quad +(\Vert \theta _0\Vert _{H^1_6} + \Vert \partial _t\theta _2\Vert _{L_{p,b}(H^1_6)}) \Vert \partial _t(\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(L_q)} + \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)}\Vert \nabla ^2\mathbf{v} _1\Vert _{L_{p,b}(L_q)} \nonumber \\&\quad + \Vert \nabla \mathbf{v} _2\Vert _{L_{p,b}(H^1_6)}\Vert \nabla ^2(\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(L_q)} +\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)}\Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_6)} \Vert \nabla ^2\mathbf{v} _1\Vert _{L_{p,b}(L_q)} \nonumber \\&\quad + \Vert \nabla ^2(\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(L_q)}\Vert \nabla \mathbf{v} _1\Vert _{L_{p,b}(H^1_6)} + \Vert \nabla ^2\mathbf{v} _2\Vert _{L_{p,b}(L_q)}\Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)} \nonumber \\&\quad +\Vert \partial _t(\theta _1-\theta _2)\Vert _{L_{p,b}(H^1_6)} \Vert \nabla \theta _1\Vert _{L_{p,b}(L_q)} + \Vert \nabla (\mathbf{v} _1-\mathbf{v} _2)\Vert _{L_{p,b}(H^1_6)}\Vert \nabla \theta _1\Vert _{L_{p,b}(L_q)} \nonumber \\&\quad + \Vert \nabla \mathbf{v} _2\Vert _{L_{p,b}(H^1_6)}\Vert \nabla (\theta _1-\theta _2)\Vert _{L_{p,b}(L_q)} +(\Vert \theta _0\Vert _{H^1_6} + \Vert \partial _t\theta _2\Vert _{L_{p,b}(H^1_6)} \Vert \nabla (\theta _1-\theta _2)\Vert _{L_{p,b}(L_q)}). \end{aligned}$$

(54)

5 A Priori Estimates for Solutions of Linearized Equations

Let ${{\mathcal {V}}}_{T, \epsilon } = \{(\theta , \mathbf{v} ) \in {{\mathcal {U}}}^1_T\times {{\mathcal {U}}}^2_T \mid E_T(\theta , \mathbf{v} ) \le \epsilon \}$. For $(\theta , \mathbf{v} ) \in {{\mathcal {V}}}_{T, \epsilon }$, we consider linearized equations:

$$\begin{aligned} \begin{aligned} \partial _t\eta + \rho _*\mathrm{div}\,\mathbf{u} = F(\theta , \mathbf{v} )&\quad&\hbox { in}\ \Omega \times (0, T), \\ \rho _*\partial _t\mathbf{u} - \mathrm{Div}\,(\mu \mathbf{D} (\mathbf{u} ) + \nu \mathrm{div}\,\mathbf{u} \mathbf{I} - {\mathfrak {p}}'(\rho _*)\eta ) = \mathbf{G} (\theta , \mathbf{v} )&\quad&\hbox { in}\ \Omega \times (0, T), \\ \mathbf{u} |_\Gamma =0, \quad (\eta , \mathbf{u} )|_{t=0} = (\theta _0, \mathbf{v} _0)&\quad&\hbox { in}\ \Omega . \end{aligned}\end{aligned}$$

(55)

We first show that Eq. (55) admit unique solutions $\eta$ and $\mathbf{u}$ with

$$\begin{aligned} \begin{aligned} \eta&\in H^1_p((0, T), H^1_2(\Omega ) \cap H^1_6(\Omega )), \\ \mathbf{u}&\in H^1_p((0, T), L_2(\Omega )^3 \cap L_6(\Omega )^3) \cap L_p((0, T), H^2_2(\Omega )^3\cap H^2_6(\Omega )^3) \end{aligned}\end{aligned}$$

(56)

possessing the estimate:

$$\begin{aligned} E_T(\eta , \mathbf{u} ) \le C(\epsilon ^2 + \epsilon ^3+\epsilon ^4) \end{aligned}$$

(57)

with some constant C independent of T and $\epsilon$.

To prove (57), we divide $\eta$ and $\mathbf{u}$ into two parts: $\eta =\eta _1+ \eta _2$ and $\mathbf{u} =\mathbf{u} _1+\mathbf{u} _2$, where $\eta _1$ and $\mathbf{u} _1$ are solutions of time shifted equations:

$$\begin{aligned} \begin{aligned} \partial _t\eta _1 +\lambda _1\eta _1+ \rho _*\mathrm{div}\,\mathbf{u} _1= F(\theta , \mathbf{v} )&\quad&\hbox { in}\ \Omega \times (0, T), \\ \rho _*(\partial _t\mathbf{u} _1+ \lambda \mathbf{u} _1) - \mathrm{Div}\,(\mu \mathbf{D} (\mathbf{u} _1) + \nu \mathrm{div}\,\mathbf{u} _1\mathbf{I} - {\mathfrak {p}}'(\rho _*)\eta _1) = \mathbf{G} (\theta , \mathbf{v} )&\quad&\hbox { in}\ \Omega \times (0, T), \\ \mathbf{u} _1|_\Gamma =0, \quad (\eta _1, \mathbf{u} _1)|_{t=0} = (0, 0)&\quad&\hbox { in}\ \Omega , \end{aligned}\end{aligned}$$

(58)

and $\eta _2$ and $\mathbf{u} _2$ are solutions to compensation equations:

$$\begin{aligned} \begin{aligned} \partial _t\eta _2 + \rho _*\mathrm{div}\,\mathbf{u} _2= \lambda _1\eta _1&\quad&\hbox { in}\ \Omega \times (0, T), \\ \rho _*\partial _t\mathbf{u} _2 - \mathrm{Div}\,(\mu \mathbf{D} (\mathbf{u} _2) + \nu \mathrm{div}\,\mathbf{u} _2\mathbf{I} - {\mathfrak {p}}'(\rho _*)\eta _2) = \rho _*\lambda _1\mathbf{u} _1&\quad&\hbox { in}\ \Omega \times (0, T), \\ \mathbf{u} _2|_\Gamma =0, \quad (\eta _2, \mathbf{u} _2)|_{t=0} = (\theta _0, \mathbf{v} _0)&\quad&\hbox { in}\ \Omega . \end{aligned}\end{aligned}$$

(59)

We first treat with Eq. (58). For this purpose, we use results stated in Sect. 3. We consider resolvent problem corresponding to Eq. (55) given as follows:

$$\begin{aligned} \begin{aligned} \lambda \zeta + \rho _*\mathrm{div}\,\mathbf{w} = f&\quad&\hbox { in}\ \Omega , \\ \rho _*\lambda \mathbf{w} - \mathrm{Div}\,(\mu \mathbf{D} (\mathbf{w} ) + \nu \mathrm{div}\,\mathbf{w} \mathbf{I} - {\mathfrak {p}}'(\rho _*)\zeta ) = \mathbf{g}&\quad&\hbox { in}\ \Omega , \\ \mathbf{w} |_\Gamma =0&\quad&\hbox { in}\ \Omega . \end{aligned}\end{aligned}$$

(60)

Enomoto and Shibata [7] proved the existence of ${{\mathcal {R}}}$ bounded solution operators associated with (60). Namely, we know the following theorem.

Theorem 15

Let $\Omega$ be a uniform $C^2$ domain in $\mathbb {R}^N$. Let $0< \omega < \pi /2$ and $1< q < \infty$. Set $H^{1,0}_q(\Omega ) = H^1_q(\Omega )\times L_q(\Omega )^3$ and $H^{1,2}_q(\Omega ) = H^1_q(\Omega )\times H^2_q(\Omega )^3$. Then, there exist a large number $\lambda _0 > 0$ and operator families ${{\mathcal {P}}}(\lambda )$ and ${{\mathcal {S}}}(\lambda )$ with

$$\begin{aligned} {{\mathcal {P}}}(\lambda ) \in \mathrm{Hol}\,(\Sigma _{\omega , \lambda _0}, {{\mathcal {L}}}(H^{1,0}_q(\Omega ), H^1_q(\Omega ))), \quad {{\mathcal {S}}}(\lambda ) \in \mathrm{Hol}\,(\Sigma _{\omega , \lambda _0}, {{\mathcal {L}}}(H^{1,0}_q(\Omega ), H^2_q(\Omega ))) \end{aligned}$$

such that for any $\lambda \in \Sigma _{\omega , \lambda _0}$ and $(f, \mathbf{g} ) \in H^{1,0}_q(\Omega )$, $\zeta = {{\mathcal {P}}}(\lambda )(f, \mathbf{g} )$ and $\mathbf{w} = {{\mathcal {S}}}(\lambda )(f, \mathbf{g} )$ are unique solutions of Stokes resolvent problem (60) and

$$\begin{aligned} {{\mathcal {R}}}_{{{\mathcal {L}}}(H^{1,0}_q(\Omega ), H^1_q(\Omega ))}(\{(\tau \partial _\tau )^\ell (\lambda ^k {{\mathcal {P}}}(\lambda )) \mid \lambda \in \Sigma _{\omega , \lambda _0}\}) \le r_b, \\ {{\mathcal {R}}}_{{{\mathcal {L}}}(H^{1,0}_q(\Omega ), H^{2-j}_q(\Omega )^3)}(\{(\tau \partial _\tau )^\ell (\lambda ^{j/2}{{\mathcal {S}}}(\lambda )) \mid \lambda \in \Sigma _{\omega , \lambda _0}\}) \le r_b \end{aligned}$$

for $\ell =0,1$, $k=0,1$ and $j = 0,1,2$.

From Theorem 14 we have the following theorem.

Theorem 16

Let $1<p, q < \infty$. Let $b \ge 0$. Then, there exists a large constant $\lambda _1 > 0$ such that for any $(f, \mathbf{g} )$ with $<t>^b(f, \mathbf{g} ) \in L_p((0, T), H^{1,0}_q(\Omega ))$, problem:

$$\begin{aligned} \begin{aligned} \partial _t\rho +\lambda _1\rho + \rho _*\mathrm{div}\,\mathbf{w} = f&\quad&\hbox { in}\ \Omega \times (0, T), \\ \rho _*(\partial _t\mathbf{w} + \lambda _1\mathbf{w} ) - \mathrm{Div}\,(\mu \mathbf{D} (\mathbf{w} ) + \nu \mathrm{div}\,\mathbf{w} \mathbf{I} - {\mathfrak {p}}'(\rho _*)\rho ) = \mathbf{g}&\quad&\hbox { in}\ \Omega \times (0, T), \\ \mathbf{w} |_\Gamma =0, \quad (\rho , \mathbf{w} )|_{t=0} = (0, 0)&\quad&\hbox { in}\ \Omega , \end{aligned}\end{aligned}$$

(61)

admits unique solutions $\rho \in H^1_p((0, T), H^1_q(\Omega ))$ and $\mathbf{w} \in H^1_p((0, T), L_q(\Omega )^3) \cap L_p((0, T), H^2_q(\Omega )^3)$ possessing the estimate:

$$\begin{aligned} \Vert<t>^b(\rho , \partial _t\rho )\Vert _{L_p((0, T), H^1_q(\Omega ))}&+ \Vert<t>^b\partial _t\mathbf{w} \Vert _{L_p((0, T), L_q(\Omega ))} + \Vert<t>^b\mathbf{w} \Vert _{L_p((0, T), H^2_q(\Omega ))} \\&\quad \le C\Vert <t>^b(f, \mathbf{g} )\Vert _{L_p((0, T), H^{1,0}_q(\Omega ))}. \end{aligned}$$

Here, C is a constant independent of $T>0$.

Proof

Our situation is that $Bu = u$ and $g=0$ in Sect. 3. Let $f_0$ and $\mathbf{g} _0$ be the zero extensions of f and $\mathbf{g}$ outside of (0, T). Applying Theorem 14 yields the unique existence of solutions $\rho$ and $\mathbf{w}$ defined on the whole time interval $\mathbb {R}$ possessing the estimate (26). But, what $f_0$ and $\mathbf{g} _\mathbf{0}$ vanish for $t < 0$ implies that $\rho$ and $\mathbf{w}$ also vanish for $t < 0$, which can be proved by using the uniqueness argument due to Saito [11, Sect. 7]. Thus, these $\rho$ and $\mathbf{w}$ are required solutions to Eq. (61). This completes the proof of Theorem 16. $\square$

We now consider Eq. (59). The corresponding Cauchy problem is equations:

$$\begin{aligned} \begin{aligned} \partial _t\zeta + \rho _*\mathrm{div}\,\mathbf{z} = 0&\quad&\hbox { in}\ \Omega \times (0, T), \\ \rho _*\partial _t\mathbf{z} - \mathrm{Div}\,(\mu \mathbf{D} (\mathbf{z} ) + \nu \mathrm{div}\,\mathbf{z} \mathbf{I} - {\mathfrak {p}}'(\rho _*)\zeta ) =0&\quad&\hbox { in}\ \Omega \times (0, T), \\ \mathbf{z} |_\Gamma =0, \quad (\zeta , \mathbf{z} )|_{t=0} = (\theta _0, \mathbf{v} _0)&\quad&\hbox { in}\ \Omega . \end{aligned}\end{aligned}$$

(62)

As was seen in Sect. 3, Theorem 15 implies generation of continuous analytic semigroup $\{T(t)\}_{t\ge 0}$ associated with equations (62). Thus, by Duhamel’s principle we have

$$\begin{aligned} (\eta _2, \mathbf{u} _2) = T(t)(\theta _0, \mathbf{v} _0) + \int ^t_0 T(t-s)(\lambda _1\eta _1(\cdot , s), \rho _*\lambda _1\mathbf{u} _1(\cdot , s))\,ds. \end{aligned}$$

(63)

Now, we shall estimate $(\eta _1, \mathbf{u} _1)$ and $(\eta _2, \mathbf{u} _2)$. Applying Theorem 16 to Eq. (58) yields that

$$\begin{aligned} \begin{aligned} \Vert<t>^b\partial _t(\eta _1, \mathbf{u} _1)&\Vert _{L_p((0, T), H^{1,0}_q(\Omega ))} + \Vert<t>^b(\eta _1, \mathbf{u} _1)\Vert _{L_p((0, T), H^{1,2}_q(\Omega ))} \\&\le C\Vert <t>^b(F(\theta , \mathbf{v} ), \mathbf{G} (\theta , \mathbf{v} ))\Vert _{L_p((0, T), H^{1,0}_q(\Omega ))} \end{aligned}\end{aligned}$$

(64)

for $q=r$, 2 and 6. Recalling that $\Vert (\theta _0, \mathbf{v} _0)\Vert _{{{\mathcal {I}}}} \le \epsilon ^2$ and $E_T(\theta , \mathbf{v} ) \le \epsilon$, by (34), (44), (46), (51), (53), and (64), we have

$$\begin{aligned} \begin{aligned} \Vert<t>^b\partial _t(\eta _1, \mathbf{u} _1)\Vert _{L_p((0, T), H^{1,0}_q(\Omega ))}&+ \Vert <t>^b(\eta _1, \mathbf{u} _1)\Vert _{L_p((0, T), H^{1,2}_q(\Omega ))} ) \le C(\epsilon ^2 + \epsilon ^3 + \epsilon ^4). \end{aligned}\end{aligned}$$

(65)

for $q=r$, 2, and 6. Here, C is a constant independent of T and $\epsilon$. By the trace method of real interpolation theorem,

$$\begin{aligned} \Vert<t>^b\mathbf{u} _1\Vert _{L_\infty ((0, T), L_q(\Omega ))} \le C(\Vert<t>^b\partial _t\mathbf{u} _1\Vert _{L_p((0, T), L_q(\Omega ))} + \Vert <t>^b\mathbf{u} _1\Vert _{L_p((0, T), H^2_q(\Omega ))}), \end{aligned}$$

and so by (65),

$$\begin{aligned} \Vert <t>^b\mathbf{u} _1\Vert _{L_\infty ((0, T), L_q(\Omega ))} \le C(\epsilon ^2 + \epsilon ^3 + \epsilon ^4), \end{aligned}$$

(66)

for $q=2$ and 6, which, combined with (65), yields that

$$\begin{aligned} E_T(\eta _1, \mathbf{u} _1) \le C(\epsilon ^2 + \epsilon ^3 + \epsilon ^4) \end{aligned}$$

(67)

with some constant $C>0$ independent of $T \in (0, \infty )$.

To estimate $\eta _2$ and $\mathbf{u} _2$, we shall use the following $L_p$-$L_q$ decay estimates due to Enomoto and Shibata [8]. Setting $(\theta , \mathbf{v} ) = T(t)(f, \mathbf{g} )$, we have

$$\begin{aligned} \begin{aligned} \Vert (\theta , \mathbf{v} )(\cdot , t)\Vert _{L_p(\Omega )}&\le C_{p,q}t^{-\frac{3}{2}\left( \frac{1}{q}-\frac{1}{p}\right) } [(f, \mathbf{g} )]_{p,q}\quad (t>1);\\ \Vert \nabla (\theta , \mathbf{v} )(\cdot , t)\Vert _{L_p(\Omega )}&\le C_{p,q}t^{-\sigma (p,q)} [(f, \mathbf{g} )]_{p,q} \quad (t>1);\\ \Vert \nabla ^2\mathbf{v} (\cdot , t)\Vert _{L_p(\Omega )}&\le C_{p,q}t^{-\frac{3}{2q}}[(f, \mathbf{g} )]_{p,q} \quad (t> 1); \\ \Vert \partial _t (\theta , \mathbf{v} )(\cdot , t)\Vert _{L_p(\Omega )}&\le Ct^{-\frac{3}{2q}} [(f, \mathbf{g} )]_{p,q}\quad (t > 1). \end{aligned}\end{aligned}$$

(68)

Here, $1 \le q \le 2 \le p < \infty$, $[(f, \mathbf{g} )]_{p,q} = \Vert (f, \mathbf{g} )\Vert _{H^{1,0}_p(\Omega )} + \Vert (f, \mathbf{g} )\Vert _{L_q(\Omega )}$, and

$$\begin{aligned} \sigma (p,q) = \frac{3}{2}\left( \frac{1}{q}- \frac{1}{p}\right) + \frac{1}{2} \quad (2 \le p \le 3), \quad \text {and}\quad \frac{3}{2q}\quad (p \ge 3). \end{aligned}$$

Moreover, we use

$$\begin{aligned} \Vert (\theta , \mathbf{v} )(\cdot , t)\Vert _{H^{1,2}_q(\Omega )} \le M\Vert (f, \mathbf{g} )\Vert _{H^{1,2}_q(\Omega )} \quad (0< t < 2), \end{aligned}$$

(69)

for $q=2$, $2+\sigma$, and 6, which follows from standard estimates for continuous analytic semigroup. In (63), we set

$$\begin{aligned} (\eta ^1_2, \mathbf{u} ^1_2) = T(t)(\theta _0, \mathbf{v} _0), \quad (\eta ^2_2, \mathbf{u} ^2_2) = \int ^t_0T(t-s)(\lambda _1\eta _1(\cdot , s), \rho _*\lambda _1\mathbf{u} _1(\cdot , s))\,ds. \end{aligned}$$

Recall that

$$\begin{aligned} \ell = \frac{1}{2} + \frac{3}{2(2+\sigma )} = \frac{1}{2}+\frac{3}{2}\Bigl (\frac{1}{2}+\frac{1}{2+\sigma } - \frac{1}{2}\Bigr ) = \frac{3}{2}\Bigl (\frac{1}{2}+\frac{1}{2+\sigma } - \frac{1}{6}\Bigr ), \end{aligned}$$

and

$$\begin{aligned} \sigma (2, r) = \ell , \quad \sigma (2+\sigma ,r) = \frac{1}{2}+\frac{3}{4} > \ell , \quad \sigma (6, r) = \ell , \quad \ell \le 3/2r. \end{aligned}$$

In particular, we use (68) estimate with decay rate $\ell$, replacing q with r, except for the first inequality in (68).

We first consider the case where $T > 2$. Direct use of (68) with $q=r$ for $t \in (1, T)$ and (69) for $t \in (0, 2]$ yields immediately that

$$\begin{aligned} E_T(\eta ^1_2, \mathbf{u} ^1_2) \le C\Vert (\theta _0, \mathbf{v} _0)\Vert _{{{\mathcal {I}}}} \le C\epsilon ^2. \end{aligned}$$

(70)

Here, the estimate of $\sup _{1< t < T}\, t^b\Vert \eta ^1_2(\cdot , t), \mathbf{u} ^1_2(\cdot , t) \Vert _{L_q(\Omega )}$ is a little bit exceptional. In fact, since $b \le \ell -1/2 \le (3/2)(1/r-1/q)$ for $q = 2$ and 6 as follows from (5), we have

$$\begin{aligned} \sup _{1< t < T}\, t^b\Vert (\eta ^1_2(\cdot , t), \mathbf{u} ^1_2(\cdot , t)) \Vert _{L_q(\Omega )} \le C(\Vert (\theta _0, \mathbf{v} _0)\Vert _{L_r(\Omega )} + \Vert (\theta _0, \mathbf{v} _0)\Vert _{H^{1,0}_q(\Omega )}). \end{aligned}$$

To estimate $(\eta ^2_2, \mathbf{u} ^2_2)$, we set

$$\begin{aligned}{}[[(\eta _1, \mathbf{u} _1)(\cdot , s)]] = \Vert (\eta _1, \mathbf{u} _1)(\cdot , s)\Vert _{L_r(\Omega )} + \sum _{q=2, 6}( \Vert (\eta _1, \mathbf{u} _1)(\cdot , s)\Vert _{H^{1,2}_q(\Omega )} + \Vert \partial _t(\eta _1, \mathbf{u} _1)(\cdot , s)\Vert _{H^{1,0}_q(\Omega )}). \end{aligned}$$

We set

$$\begin{aligned} {\tilde{E}}_T(\eta _1, \mathbf{u} _1) : = \Bigl (\int ^T_0(<t>^b[[\eta _1, \mathbf{u} _1)(\cdot , t)]])^p\, dt\Bigr )^{1/p}, \end{aligned}$$

and then, by (65) we have

$$\begin{aligned} {\tilde{E}}_T(\eta _1, \mathbf{u} _1) \le C(\epsilon ^2 + \epsilon ^3 + \epsilon ^4). \end{aligned}$$

(71)

First we consider the case: $2 \le t \le T$. Let $(\eta _3, \mathbf{u} _3) = (\nabla \eta ^2_2, {\bar{\nabla }}^1 \nabla \mathbf{u} ^2_2)$ when $q=2$, and $(\eta _3, \mathbf{u} _3) = ({\bar{\nabla }}^1\eta ^2_2, {\bar{\nabla }}^2\mathbf{u} ^2_2)$ when $q=6$. Here, ${\bar{\nabla }}^m f = (\partial _x^\alpha f \mid |\alpha | \le m)$. And then,

$$\begin{aligned}&\Vert (\eta _3, \mathbf{u} _3)(\cdot , t)\Vert _{L_q(\Omega )} \\&\le C\Bigl \{\int ^{t/2}_0 + \int ^{t-1}_{t/2} + \int ^t_{t-1}\Bigr \} \Vert (\nabla , {\bar{\nabla }}^1\nabla )\text {or}({\bar{\nabla }}^1, {\bar{\nabla }}^2) T(t-s)(\lambda _1\eta _1, \rho _*\lambda _1\mathbf{u} _1)(\cdot , s)\Vert _{L_q(\Omega )}\,ds \\&= I_{q} + II_{q} + III_{q}. \end{aligned}$$

By (68) and $bp' > 1$ (cf. (3)), we have

$$\begin{aligned} I_{q}(t)&\le C\int ^{t/2}_0 (t-s)^{-\ell } [[(\eta _1, \mathbf{u} _1)]]\,ds \\&\le C(t/2)^{-\ell } \int ^{t/2}_0<s>^{-b}<s>^b [[(\eta _1, \mathbf{u} _1)(\cdot , s)]]\,ds \\&\le Ct^{-\ell } \Bigl (\int ^T_0<s>^{-bp'}\,ds\Bigr )^{1/p'}\Bigl (\int ^T_0 (<s>^b[[(\eta _1, \mathbf{u} _1)(\cdot , s)]])^p\,ds\Bigr )^{1/p} \\&\le Ct^{-\ell }{\tilde{E}}_T(\eta _1, \mathbf{u} _1). \end{aligned}$$

Recalling that $(\ell -b)p > 1$ (cf. (16)), we have

$$\begin{aligned} \int ^T_1(<t>^bI_{q}(t))^p\,dt \le C{\tilde{E}}_T(\eta _1, \mathbf{u} _1)^p. \end{aligned}$$

We next estimate $II_q(t)$. By (68) we have

$$\begin{aligned} II_{q}(t) \le C\int ^{t-1}_{t/2}(t-s)^{-\ell } [[(\eta _1, \mathbf{u} _1)(\cdot , s)]]\,ds. \end{aligned}$$

By Hölder’s inequality and $<t>^b \le C_b<s>^b$ for $s \in (t/2, t-1)$, we have

$$\begin{aligned}<t>^bII_{q}(t)&\le C\int ^{t-1}_{t/2}(t-s)^{-\ell /p'}(t-s)^{-\ell /p}<s>^b [[(\eta _1, \mathbf{u} _1)(\cdot , s)]]\, ds\\&\le C\Bigl (\int ^{t-1}_{t/2}(t-s)^{-\ell }\,ds\Bigr )^{1/p'} \Bigl (\int ^{t-1}_{t/2}(t-s)^{-\ell }(<s>^b[[(\eta _1, \mathbf{u} _1)(\cdot , s)]])^p\,ds\Bigr )^{1/p}. \end{aligned}$$

Setting $\int ^\infty _1 s^{-\ell }\,ds =L$, by Fubini’s theorem we have

$$\begin{aligned} \int ^T_2(<t>^b II_{q}(t))^p\,dt&\le CL^{p/p'}\int ^{T-1}_1(<s>^b[[(\eta _1, \mathbf{u} _1)(\cdot , s)]])^p \Bigl (\int ^{2s}_{s+1} (t-s)^{-\ell }\,dt\Bigr ) \,ds \\&\le CL^p{\tilde{E}}_T(\eta _1, \mathbf{u} _1)^p. \end{aligned}$$

Using a standard estimate (69) for continuous analytic semigroup, we have

$$\begin{aligned} III_{q}(t)&\le C\int ^t_{t-1} \Vert (\eta _1, \mathbf{u} _1)(\cdot , s)\Vert _{H^{1,2}_{q}(\Omega )}\,ds \le C\int ^t_{t-1} [[(\eta _1, \mathbf{u} _1)(\cdot , s)]]\,ds. \end{aligned}$$

Thus, employing the same argument as in estimating $II_{q}(t)$, we have

$$\begin{aligned} \int ^T_2(<t>^b III_{q}(t))^p\,dt \le C{\tilde{E}}_T(\eta _1, \mathbf{u} _1)^p. \end{aligned}$$

Combining these three estimates yields that

$$\begin{aligned} \int ^T_2(<t>^b\Vert (\eta _3, \mathbf{u} _3)(\cdot , t)\Vert _{L_q(\Omega )})^p\,dt \le C{\tilde{E}}_T(\eta _1, \mathbf{u} _1)^p, \end{aligned}$$

(72)

when $T > 2$.

For $0< t < \min (2, T)$, using (69) and employing the same argument as in estimating $III_q(t)$ above, we have

$$\begin{aligned} \int ^{\min (2, T)}_0(<t>^b\Vert (\eta _3, \mathbf{u} _3)(\cdot , t)\Vert _{L_q(\Omega )})^p\,dt \le C{\tilde{E}}_T(\eta _1, \mathbf{u} _1)^p, \end{aligned}$$

which, combined with (72), yields that

$$\begin{aligned} \int ^T_0(<t>^b\Vert (\eta _3, \mathbf{u} _3)(\cdot , t)\Vert _{L_q(\Omega )})^p\,dt \le C{\tilde{E}}_T(\eta _1, \mathbf{u} _1)^p \end{aligned}$$

(73)

for $q=2$ and 6.

Since

$$\begin{aligned} \partial _t(\eta ^2_2, \mathbf{u} ^2_2) = -\lambda _1(\eta _1, \rho _*\mathbf{u} _1)(\cdot , t) -\lambda _1\int ^t_0\partial _tT(t-s)(\eta _1, \rho _*\mathbf{u} _1)(\cdot ,s)\,ds, \end{aligned}$$

employing the same argument as in proving (73), we have

$$\begin{aligned} \int ^T_0(<t>^b\Vert \partial _t(\eta ^2_2, \mathbf{u} ^2_2)(\cdot , t)\Vert _{L_q(\Omega )})^p\,dt \le C{\tilde{E}}_T(\eta _1, \mathbf{u} _1)^p \end{aligned}$$

(74)

for $q=2$ and 6.

We now estimate $\sup _{2< t< T}<t>^b \Vert (\eta ^2_2, \mathbf{u} ^2_2)\Vert _{L_q(\Omega )}$ for $q=2$ and 6. Let $q=2$ and 6 in what follows. For $2< t < T$,

$$\begin{aligned} \Vert (\eta ^2_2, \mathbf{u} ^2_2)(\cdot , t)\Vert _{L_q(\Omega )}&\le C\Bigl \{\int ^{t/2}_0 + \int ^{t-1}_{t/2} + \int ^t_{t-1}\Bigr \} \Vert T(t-s)(\lambda _1\eta _1, \lambda _1\rho _*\mathbf{u} _1)(\cdot , s)\Vert _{L_q(\Omega )}\,ds \\&= I_{q, 0} + II_{q, 0} + III_{q, 0}. \end{aligned}$$

By (68), we have

$$\begin{aligned} I_{q,0}(t)&\le C\int ^{t/2}_0(t-s)^{-3/2(2+\sigma )}[[(\eta _1, \mathbf{u} _1)(\cdot , s)]]\,ds \\&\le C(t/2)^{-3/2(2+\sigma )}\int ^{t/2}_0<s>^{-b}<s>^b[[(\eta _1, \mathbf{u} _1)(\cdot , s)]]\,ds \\&\le Ct^{-3/2(2+\sigma )}\Bigl (\int ^\infty _0<s>^{-p'b}\,ds\Bigr )^{1/p'} {\tilde{E}}_T(\eta _1, \mathbf{u} _1). \end{aligned}$$

Noting that $(3/2(2+\sigma ))p'> bp' > 1$ and using (68), we have

$$\begin{aligned} II_{q,0}(t)&\le C\int ^{t-1}_{t/2}(t-s)^{-3/2(2+\sigma )}\Vert (\eta _1, \mathbf{u} _1)(\cdot , s)]]\,ds \\&\le C\Bigl (\int ^{t-1}_{t/2}((t-s)^{-3/2(2+\sigma )}<s>^{-b})^{p'}\,ds\Bigr )^{1/p'} \Bigl (\int ^{t-1}_{t/2}(<s>^b[[(\eta _1, \mathbf{u} _1)(\cdot , s)]])^p\,ds\Bigr )^{1/p}\\&\le C<t>^{-b}{\tilde{E}}_T(\eta _1, \mathbf{u} _1). \end{aligned}$$

By (69), we have

$$\begin{aligned} III_{q,0}(t)&\le C\int ^t_{t-1}[[(\eta _1, \mathbf{u} _1)(\cdot , s)]]\,ds\\&\le C<t>^{-b}\int ^t_{t-1}<s>^b[[(\eta _1, \mathbf{u} _1)(\cdot , s)]]\,ds \\&\le C<t>^{-b}\Bigl (\int ^t_{t-1}\,ds\Bigr )^{1/p'}{\tilde{E}}_T(\eta _1, \mathbf{u} _1). \end{aligned}$$

Since $b< 3/2(2+\sigma )$, combining these estimates yields that

$$\begin{aligned} \sup _{2< t< T}<t>^b\Vert (\eta _2^2, \mathbf{u} _2^2)(\cdot , t)\Vert _{L_q(\Omega )} \le C{\tilde{E}}_T(\eta _1, \mathbf{u} _1). \end{aligned}$$

(75)

For $0< t < \min (2, T)$, by standard estimate (69) of continuous analytic semigroup, we have

$$\begin{aligned} \sup _{0< t< \min (2, T)}<t>^b\Vert (\eta ^2_2, \mathbf{u} _2^2)(\cdot , t)\Vert _{L_q(\Omega )} \le C{\tilde{E}}_T(\eta _1, \mathbf{u} _1) \end{aligned}$$

which, combined with (75), yields that

$$\begin{aligned} \Vert <t>^b(\eta _2^2, \mathbf{u} _2^2)(\cdot , t)\Vert _{L_\infty ((0, T), L_q(\Omega )} \le C{\tilde{E}}_T(\eta _1, \mathbf{u} _1) \end{aligned}$$

(76)

for $q=2$ and 6.

Recalling that $\eta =\eta _1+\eta _2$ and $\mathbf{u} =\mathbf{u} _1+\mathbf{u} _2$, noting that $E_T(\eta _1, \mathbf{u} _1) \le C({\tilde{E}}_T(\eta _1, \mathbf{u} _1)+\Vert (\theta _0, \mathbf{v} _0)\Vert _{{\mathcal {I}}})$ as follows from (66), and combining (73), (74), (76), and (71) yield that

$$\begin{aligned} E_T(\eta , \mathbf{u} ) \le C(\epsilon ^2 + \epsilon ^3 + \epsilon ^4). \end{aligned}$$

(77)

If we choose $\epsilon > 0$ so small that $C(\epsilon + \epsilon ^2 + \epsilon ^3) < 1$ in (77), we have $E_T(\eta , \mathbf{u} ) \le \epsilon$. Moreover, by (43)

$$\begin{aligned} \sup _{t \in (0, T)} \Vert \eta (\cdot , t)\Vert _{L_\infty (\Omega )} \le C(\Vert \eta _0\Vert _{H^1_6} + \Vert \partial _t\eta \Vert _{L_p((0, T), H^1_6(\Omega ))}) \le C(\epsilon ^2 +\epsilon ^3 + \epsilon ^4). \end{aligned}$$

Thus, choosing $\epsilon > 0$ so small that $C(\epsilon ^2 +\epsilon ^3 + \epsilon ^4) \le \rho _*/2$, we see that $\sup _{t \in (0, T)} \Vert \eta (\cdot , t)\Vert _{L_\infty (\Omega )} \le \rho _*/2$. And also,

$$\begin{aligned} \int ^T_0\Vert \nabla \mathbf{u} (\cdot , s)\Vert _{L_\infty (\Omega )}\,ds \le \Bigl (\int ^\infty _0<s>^{-p'b}\,ds\Bigr )^{1/p'} \Vert <t>^b\nabla \mathbf{u} \Vert _{L_p((0, T), H^1_6(\Omega ))} \le C_{p', b}(\epsilon ^2 + \epsilon ^3 + \epsilon ^4). \end{aligned}$$

Thus, choosing $\epsilon > 0$ so small that $C_{p', b}(\epsilon ^2 + \epsilon ^3 + \epsilon ^4) \le \delta$, we see that $\int ^T_0\Vert \nabla \mathbf{u} (\cdot , s)\Vert _{L_\infty (\Omega )}\,ds \le \delta$. From consideration above, it follows that $(\eta , \mathbf{u} ) \in {{\mathcal {V}}}_{T, \epsilon }$. Let ${{\mathcal {S}}}$ be an operator defined by ${{\mathcal {S}}}(\theta , \mathbf{v} ) = (\eta , \mathbf{u} )$ for $(\theta , \mathbf{v} ) \in {{\mathcal {V}}}_{T, \epsilon }$, and then ${{\mathcal {S}}}$ maps ${{\mathcal {V}}}_{T, \epsilon }$ into itself.

We now show that ${{\mathcal {S}}}$ is a contraction map. Let $(\theta _i, \mathbf{v} _i) \in {{\mathcal {V}}}_{T, \epsilon }$ ($i=1,2$) and set $(\eta , \mathbf{u} ) = (\eta _1, \mathbf{u} _1) - (\eta _2, \mathbf{u} _2) = {{\mathcal {S}}}(\theta _1, \mathbf{v} _1)- {{\mathcal {S}}}(\theta _2, \mathbf{v} _2)$, and $F = F(\theta _1, \mathbf{v} _1)-F(\theta _2, \mathbf{v} _2)$ and $\mathbf{G} = \mathbf{G} (\theta _1, \mathbf{v} _1) - \mathbf{G} (\theta _2, \mathbf{v} _2)$. And then, from (55) it follows that

$$\begin{aligned} \begin{aligned} \partial _t\eta + \rho _*\mathrm{div}\,\mathbf{u} = F&\quad&\hbox { in}\ \Omega \times (0, T), \\ \rho _*\partial _t\mathbf{u} - \mathrm{Div}\,(\mu \mathbf{D} (\mathbf{u} ) + \nu \mathrm{div}\,\mathbf{u} \mathbf{I} - {\mathfrak {p}}'(\rho _*)\eta ) = \mathbf{G}&\quad&\hbox { in}\ \Omega \times (0, T), \\ \mathbf{u} |_\Gamma =0, \quad (\eta , \mathbf{u} )|_{t=0} = (0, 0)&\quad&\hbox { in}\ \Omega . \end{aligned} \end{aligned}$$

(78)

By (34), (45), (47), (52), and (54), we have

$$\begin{aligned} \Vert (F, \mathbf{G} )\Vert _{L_p((0, T), H^{1,0}_r(\Omega ))} + \sum _{q=2, 2+\sigma , 6}\Vert (F, \mathbf{G} )\Vert _{L_p((0, T), H^{1,0}_q(\Omega ))} \le C(\epsilon + \epsilon ^2+ \epsilon ^3)E_T((\theta _1, \mathbf{v} _1)-(\theta _2, \mathbf{v} _2)). \end{aligned}$$

Applying the same argument as in proving (77) to Eq. (78) and recalling $(\eta , \mathbf{u} ) = {{\mathcal {S}}}(\theta _1, \mathbf{v} _1)- S(\theta _2, \mathbf{v} _2)$, we have

$$\begin{aligned} E_T( {{\mathcal {S}}}(\theta _1, \mathbf{v} _1)- S(\theta _2, \mathbf{v} _2)) \le C(\epsilon + \epsilon ^2+ \epsilon ^3)E_T((\theta _1, \mathbf{v} _1)-(\theta _2, \mathbf{v} _2)), \end{aligned}$$

for some constant C independent of $\epsilon$ and T. Thus, choosing $\epsilon > 0$ so small that $C(\epsilon + \epsilon ^2+ \epsilon ^3) < 1$, we have that ${{\mathcal {S}}}$ is a contraction map on ${{\mathcal {V}}}_{T, \epsilon }$, which proves Theorem 6. Since the contraction mapping principle yields the uniqueness of solutions in ${{\mathcal {V}}}_{T, \epsilon }$, we have completed the proof of Theorem 6.

6 A Proof of Theorem 3

We shall prove Theorem 3 with the help of Theorem 6. In what follows, let b and p be the constants given in Theorem 6, and $q=2$ and 6. As was stated in Sect. 2, the Lagrange transform (7) gives a $C^{1+\omega }$ ($\omega \in (0, 1/2)$) diffeomorphism on $\Omega$ and $dx= \det (\mathbf{I} + \mathbf{k} )\,dy$, where $\{x\}$ and $\{y\}$ denote respective Euler coordinates and Lagrange coordinates on $\Omega$ and $\mathbf{k} = \int ^t_0\nabla \mathbf{u} (\cdot , s)\,ds$. By (8), $\Vert \mathbf{k} \Vert _{L_\infty (\Omega )} \le \delta < 1$. In particular, choosing $\delta >0$ smaller if necessary, we may assume that $C^{-1}\le \det (\mathbf{I} + \int ^t_0\nabla \mathbf{u} (\cdot , s) \,ds)\le C$ with some constant $C > 0$ for any $(x, t) \in \Omega \times (0, T)$. Let $y = X_t(x)$ be an inverse map of Lagrange transform (7), and set $\theta (x, t) = \eta (X_t(x), t)$ and $\mathbf{v} (x, t) = \mathbf{u} (X_t(x), t)$. We have

$$\begin{aligned} \Vert (\theta , \mathbf{v} )\Vert _{L_q(\Omega )} \le C\Vert (\eta , \mathbf{u} )\Vert _{L_q(\Omega )}. \end{aligned}$$

Noting that $(\eta , \mathbf{u} )(y, t) = (\theta , \mathbf{v} )(y+\int ^t_0\mathbf{u} (y, s)\,ds, t)$, the chain rule of composite functions yields that

$$\begin{aligned}&\Vert (\nabla (\theta , \mathbf{v} )\Vert _{L_q(\Omega )} \le C (1-\Vert \mathbf{k} \Vert _{L_\infty (\Omega )})^{-1}\Vert \nabla (\eta , \mathbf{u} )\Vert _{L_q(\Omega )}; \\&\quad \Vert \nabla ^2\mathbf{v} \Vert _{L_q(\Omega )} \le C((1-\Vert \mathbf{k} \Vert _{L_\infty (\Omega )})^{-2} \Vert \nabla ^2\mathbf{u} \Vert _{L_q(\Omega )} + (1-\Vert \mathbf{k} \Vert _{L_\infty (\Omega )})^{-1} \Vert \nabla \mathbf{k} \Vert _{L_q(\Omega )}\Vert \nabla \mathbf{u} \Vert _{L_\infty (\Omega )}). \end{aligned}$$

Thus, using $\Vert \nabla \mathbf{k} \Vert _{L_q(\Omega )} \le C\Vert <t>^b\nabla ^2\mathbf{u} \Vert _{L_p((0, T), L_q(\Omega ))}$ and $\Vert \nabla \mathbf{u} \Vert _{L_\infty (\Omega )}\le C\Vert \nabla \mathbf{u} \Vert _{H^1_6(\Omega )}$, we have

$$\begin{aligned}&\Vert<t>^b\nabla (\theta , \mathbf{v} )\Vert _{L_\infty ((0, T), L_2(\Omega ) \cap L_6(\Omega ))} \le C\Vert<t>^b\nabla (\theta , \mathbf{v} )\Vert _{L_\infty ((0, T), L_2(\Omega ) \cap L_6(\Omega ))}; \\&\Vert<t>^b(\theta , \mathbf{v} )\Vert _{L_p((0, T), L_6(\Omega ))} \le C\Vert<t>^b(\theta , \mathbf{v} )\Vert _{L_p((0, T), L_6(\Omega ))};\\&\Vert<t>^b(\theta , \mathbf{v} )\Vert _{L_\infty ((0, T), L_2(\Omega ) \cap L_6(\Omega ))} \le C\Vert<t>^b(\theta , \mathbf{v} )\Vert _{L_p((0, T), L_2(\Omega ) \cap L_6(\Omega ))};\\&\Vert<t>^b\nabla ^2\mathbf{v} \Vert _{L_p((0, T), L_2(\Omega ) \cap L_6(\Omega ))} \le C(\Vert<t>^b\nabla ^2\mathbf{u} \Vert _{L_p((0, T), L_2(\Omega ) \cap L_6(\Omega ))}\\&\quad + \Vert<t>^b\nabla ^2\mathbf{u} \Vert _{L_p((0, T), L_q(\Omega ))}\Vert <t>^b\nabla \mathbf{u} \Vert _{L_p((0, T), H^1_6(\Omega ))}). \end{aligned}$$

Since $\partial _t(\eta , \mathbf{u} )(y, t) = \partial _t[(\theta ,\mathbf{v} )(y+ \int ^t_0\mathbf{u} (y, s)\,ds, t)] = \partial _t(\theta , \mathbf{v} )(x, t) +\mathbf{u} \cdot \nabla (\theta , \mathbf{v} )(x, t)$, we have

$$\begin{aligned} \Vert \partial _t(\theta , \mathbf{v} )\Vert _{L_q(\Omega )} \le C(\Vert \partial _t(\eta , \mathbf{u} )\Vert _{L_q(\Omega )} + \Vert \mathbf{u} \Vert _{L_\infty (\Omega )} \Vert \nabla \eta \Vert _{L_q(\Omega )} + \Vert \mathbf{u} \Vert _{L_q(\Omega )}\Vert \nabla \mathbf{u} \Vert _{L_\infty (\Omega )}). \end{aligned}$$

Since $\Vert \nabla \eta \Vert _{L_\infty ((0, T), L_q(\Omega ))} \le \Vert \nabla \theta _0\Vert _{L_q(\Omega )} + C\Vert <t>^b\partial _t\eta \Vert _{L_p((0, T), H^1_q(\Omega ))}$, we have

$$\begin{aligned} \Vert<t>^b\partial _t(\theta , \mathbf{v} )\Vert _{L_p((0, T), L_q(\Omega ))}&\le C(\Vert<t>^b\partial _t(\eta , \mathbf{u} )\Vert _{L_p((0, T), L_q(\Omega ))} \\&\quad + (\Vert \nabla \theta _0\Vert _{L_q(\Omega )} + \Vert<t>^b\partial _t\eta \Vert _{L_p((0, T), H^1_q(\Omega ))})\Vert<t>^b \mathbf{u} \Vert _{L_p((0, T), H^1_6(\Omega ))}\\&\quad + \Vert<t>^b\mathbf{u} \Vert _{L_\infty ((0, T), L_q(\Omega ))} \Vert <t>^b\nabla \mathbf{u} \Vert _{L_p((0, T), H^1_6(\Omega ))}). \end{aligned}$$

By Theorem 6, we see that there exists a small constant $\epsilon > 0$ such that if initial data $(\theta _0,\mathbf{v} _0) \in {{\mathcal {I}}}$ satisifes the compatibility condition: $\mathbf{v} _0|_\Gamma =0$ and the smallness condition: $\Vert (\theta _0, \mathbf{v} _0)\Vert _{{{\mathcal {I}}}} \le \epsilon ^2$ then problem (1) admits unique solutions $\rho = \rho _*+\theta$ and $\mathbf{v}$ satisfying the regularity conditions (4) and ${{\mathcal {E}}}(\theta , \mathbf{v} ) \le \epsilon$. This completes the proof of Theorem 3.

7 Comment on the Proof

Let $N \ge 3$ and $\Omega$ be an exterior domain in $\mathbb {R}^N$. Assume that $L_p$-$L_q$ decay estmates for continuous analytic semigroup like (68) are valid. We choose $q_1=2$, $q_2 = 2+\sigma$, and $q_3$ in such a way that $q_3 > N$ and

$$\begin{aligned} \frac{1}{2} + \frac{N}{2(2+\sigma )} \le \frac{N}{2}\Bigl (\frac{1}{2}+\frac{1}{2+\sigma } -\frac{1}{q_3}\Bigr ). \end{aligned}$$

Namely, $q_3 =6$ ($N=3$) and $q_3 >N \ge 2N/(N-2)$ for $N \ge 4$. If $L_1$ in space estimates hold, then the global well-posedness is established with $q_1=q_2=2$. But, so far $L_1$ in space estimates does not hold, and so we have chosen $q_1=2$ and $q_2 = 2+\sigma$. Let p and b be chosen in such a way that

$$\begin{aligned} \Bigl (\frac{1}{2} + \frac{N}{2(2+\sigma )} - b\Bigr )p> 1, \quad bp' > 1. \end{aligned}$$

If we write equations as

$$\begin{aligned} \partial _tu - Au = f, \quad Bu = g \quad (t >0), \quad u|_{t=0}=u_0. \end{aligned}$$

Here, $Bu=g$ is corresponding to boundary conditions, and f and g are corresponding to nonlinear terms. The first reduction is that $u_1$ is a solution to equations:

$$\begin{aligned} \partial _tu_1 + \lambda _1 u_1 - Au_1 = f, \quad Bu_1 = g \quad (t\in \mathbb {R}). \end{aligned}$$

Then, $u_1$ has the same decay properties as nonlinear terms f and g have. If $u_1$ does not belong to the domain of the operator (A, B) (free boundary conditions or slip boundary conditions cases)), in addition we choose $u_2$ as a solution of equations:

$$\begin{aligned} \partial _tu_2 + \lambda _1 u_2 - Au_2 = \lambda _1u_1, \quad Bu_2 = 0 \quad (t \in \mathbb {R}) \end{aligned}$$

with very large constant $\lambda _1 > 0$. Since $u_2$ belongs to the domain of operator A for any $t > 0$, we choose $u_3$ as a solution of equations:

$$\begin{aligned} \partial _tu_3 - Au_3 = \lambda _1 u_2, \quad Bu_3 = 0 \quad (t > 0), \quad u_3|_{t=0} = u_0 -(u_1+u_2)|_{t=0}. \end{aligned}$$

And then, by the Duhamel principle, we have

$$\begin{aligned} u_3= T(t)(u_0 -(u_1+u_2)|_{t=0}) + \lambda _1\int ^t_0T(t-s)u_2(s)\,ds, \end{aligned}$$

and we use $L_p$-$L_q$ decay estimate like (68) for $0< s < t-1$ and a standard semigroup estimate for $t-1< s < t$, that is $\Vert T(t-s)u_2(s)\Vert _{D(A)} \le C\Vert u(s)\Vert _{D(A)}$ for $t-1<s<t$, where $\Vert \cdot \Vert _{D(A)}$ is a domain norm.

When $N=2$, the method above is fail, because

$$\begin{aligned} \frac{1}{2} + \frac{2}{2(2+\sigma )} < 1. \end{aligned}$$

And so, Matsumura–Nishida method seems to be only the way to prove the global wellposedness in two dimensional exterior domains.