1 Introduction

Motivated by the well-posedness results for the 2D Euler equations in non-smooth domains in [2, 3] and the questions about the Boussinesq system over non-smooth domains raised in [1, Sect. 4], we aim in this article to address the global well-posedness of the 2D Euler–Boussinesq equations in a non-smooth domain \(\Omega \subset \mathbb R^2\) of polygonal type. The 2D Euler–Boussinesq equations describing the evolution of mass and heat flow of an inviscid incompressible fluid read in the non-dimensional form:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\varvec{u} + \varvec{u}\cdot \nabla \varvec{u} + \nabla \pi = T \varvec{e}_2,\quad \varvec{e}_2=(0,1),\\ {\mathrm {div}\,}\varvec{u}=0,\\ \partial _t T - \kappa \Delta T + \varvec{u}\cdot \nabla T =0, \end{array}\right. } \end{aligned}$$
(1.1)

where \((x,y)\in \Omega \), \(t\in (0, t_1)\), \(\varvec{u}=(u_1,u_2)\) and \(T\) denote the velocity field and the temperature of the fluid, respectively, \(\pi \) stands for the pressure, and \(\kappa > 0\) is the thermal diffusivity. We associate with (1.1) the following initial and boundary conditions:

$$\begin{aligned} {\left\{ \begin{array}{ll} \varvec{u}(0,x,y)=\varvec{u}_0(x,y),\quad \quad T(0,x,y)=T_0(x,y),\\ \varvec{u}(t,x,y)\cdot \varvec{n}=0,\qquad (x,y)\in \partial \Omega ,\\ T=\eta ,\qquad (x,y)\in \partial \Omega , \end{array}\right. } \end{aligned}$$
(1.2)

where \(\varvec{n}\) is the outward unit normal vector to \(\partial \Omega \) and \(\varvec{u}_0, T_0\) and \(\eta \) are the given initial and boundary data. We also denote by \(\varvec{\tau }\) the unit tangent vector to \(\partial \Omega \).

The general 2D Boussinesq system with full viscosity \(\nu \) and diffusivity \(\kappa \) reads

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\varvec{u} -\nu \Delta \varvec{u}+ \varvec{u}\cdot \nabla \varvec{u} + \nabla \pi = T \varvec{e}_2,\quad \varvec{e}_2=(0,1),\\ {\mathrm {div}\,}\varvec{u}=0,\\ \partial _t T - \kappa \Delta T + \varvec{u}\cdot \nabla T=0. \end{array}\right. } \end{aligned}$$

From the mathematical point of view, the global well-posedness and global regularity of the Boussinesq system as well as the existence of the global attractor in presence of viscosity have been widely studied, see for example [4–15]. Recently, there are many works devoted to the study of the 2D Boussinesq system with partial viscosity, see for example [16–21] in the whole space \(\mathbb R^2\) and [1, 22, 23] in bounded smooth domains. There are also many works which considered the case when only the horizontal viscosity or vertical viscosity is present, see for example [24–27]. However, the global regularity for the 2D Boussinesq system when \(\nu =\kappa =0\) is still an outstanding open problem, and to the best of our knowledge, the well-posedness issue regarding the 2D Euler–Boussinesq system (1.1) in non-smooth domains has not yet been addressed in the literature, which is the goal of this article. In some realistic applications, the variation of the fluid viscosity and thermal diffusivity with the temperature may not be disregarded (see for example [28] and references therein) and there are many works on this direction too, see for example [28–32] where the existence of weak solutions, global regularity, and existence of global attractor have been studied.

The 2D Boussinesq system also has close connection to the fundamental fluid models, for example, the Euler equations. It is well known that the standard 2D Euler equations are globally well posed if the initial data satisfy the Yudovich’s type condition, see [33–35]. Roughly speaking, if the initial vorticity is bounded or unbounded but with small growth rate of the \(L^p\)-norm, then the 2D Euler equations possess a global unique solution and recently this result has been extended to non-smooth domains in [2, 3]. Note that the global well-posedness for the 2D Euler–Boussinesq system has been studied in [17] with Yudovich’s type data for the whole space \(\mathbb R^2\) and also studied in [23] with \(H^3\)-regular data for bounded smooth domains. Here, we would like to establish the global well-posedness result for the 2D Euler–Boussinesq system in non-smooth domains with Yudovich’s type data, which generalizes the results in [17, 23] and gives a definite answer to part of the questions asked in [1]. We also remark that the author in [23] only studied the case when the boundary data are constant, while here we will consider arbitrary boundary data for the 2D Euler–Boussinesq system.

In this article, we are interested in the polygonal-like (non-smooth) domains with maximum aperture \(\max \alpha _j\le \pi /2\) because the elliptic regularity results are only available for such domains (see (2.7) below). Here, a domain \(\Omega \subset \mathbb R^2\) is said to be a polygonal-like domain if it is a bounded simply connected open set and the boundary \(\partial \Omega \) is enclosed by piecewise \(\mathcal C^{1,1}\) planar curves, with finitely many points \(\{O_j\}_{j=1}^N\) of discontinuity for the tangent vector, and such that, in some neighborhood of each point \(O_j\), \(\Omega \) coincides with the cone of vertex \(O_j\) and aperture \(\alpha _j\in (0, 2\pi )\).

In order to deal with the non-homogeneous boundary conditions on \(\partial \Omega \) for the temperature \(T\), we need to use classical lifting results (see for example [36, Theorem 1.5.2.3,Theorem 1.5.2.8]). But in order to avoid the technical conditions for \(\eta \) at the corner points of the domain \(\Omega \), we assume that the boundary data \(\eta \) are inferred from a function \(S\) defined on \(\Omega \), that is

$$\begin{aligned} S=\eta ,\quad \text { on }\partial \Omega . \end{aligned}$$

For the sake of simplicity, we also assume that the boundary data \(\eta \) and hence the function \(S\) are independent of time \(t\).

Following a traditional approach (see for example [14, 37]), we recast the 2D Euler–Boussinesq system in terms of the perturbative variable (perturbation away from the stationary state \((0, S)\)); namely we set

$$\begin{aligned} (\varvec{u},\theta )=(\varvec{u}, T-S). \end{aligned}$$

In the perturbative variables, the 2D Euler–Boussinesq system (1.1)–(1.2) reads

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\varvec{u} + \varvec{u}\cdot \nabla \varvec{u} + \nabla \pi = \theta \varvec{e}_2 + S\varvec{e}_2,\quad \varvec{e}_2=(0,1),\\ {\mathrm {div}\,}\varvec{u}=0,\\ \partial _t \theta - \Delta \theta + \varvec{u}\cdot \nabla \theta +\varvec{u}\cdot \nabla S = \Delta S, \\ \end{array}\right. } \end{aligned}$$
(1.3)

with the initial and boundary conditions

$$\begin{aligned} {\left\{ \begin{array}{ll} \varvec{u}(0,x,y)=\varvec{u}_0(x,y),\quad \quad \theta (0,x,y)=\theta _0(x,y):=T_0(x,y)-S(x,y),\\ \varvec{u}(t,x,y)\cdot \varvec{n}=0,\quad \quad \theta (t,x,y)=0,\qquad (x,y)\in \partial \Omega . \end{array}\right. } \end{aligned}$$
(1.4)

Note that we have set the diffusivity \(\kappa =1\) in (1.3) for simplicity.

The rest of the article is organized as follows. At the end of this introduction, we introduce the notion of quasi-strong solution for the 2D Euler–Boussinesq system (1.3)–(1.4) and state our main result. We prove the existence of the quasi-strong solution by the vanishing viscosity method, which was used by Bardos in [38] to study the 2D Euler equations. In Sect. 2, we collect the necessary tools for the analysis of the Boussinesq system in the polygonal-like domains. Section 3 is devoted to prove the uniform estimates for the approximated solutions constructed by the vanishing viscosity method. Finally in Sect. 4, we prove the main result, Theorem 1.1 below, that is the existence of the quasi-strong solution and also the regularity and uniqueness of the solutions for the 2D Euler–Boussinesq system (1.3)–(1.4). The proof of the uniqueness follows Yudovich’s energy method and relies on the endpoint \(L^\infty (\Omega )\rightarrow L^{\gamma _\mathrm {exp}}(\Omega )\) regularity result for the solution to the Dirichlet problem in the polygonal-like domains. In Appendix 1, we recast the standard \(L^p\)-estimate for the 2D Euler equations using the vorticity formulation.

1.1 Definition of the Quasi-Strong Solution and the Main Result

In order to set up the framework of how to study the 2D Euler–Boussinesq system (1.3)–(1.4), we recall the classical space

$$\begin{aligned} V=\left\{ \varvec{u}\in H^1(\Omega )\,:\,{\mathrm {div}\,}\varvec{u}=0,\quad \varvec{u}\cdot \varvec{n}=0\quad \text { on }\partial \Omega \right\} , \end{aligned}$$

and we say that a couple \((\varvec{u}, \theta )\) satisfying

$$\begin{aligned}&\varvec{u}\in L^\infty (0, t_1; V),\qquad \partial _t\varvec{u} \in L^2(0, t_1; L^{3/2}(\Omega ));\nonumber \\&\theta \in \mathcal C([0, t_1]; H^1_0(\Omega ))\cap L^2(0,t_1; H^2(\Omega )),\qquad \partial _t\theta \in L^2(0, t_1; L^2(\Omega )), \end{aligned}$$
(1.5)

is a quasi-strong solution of the problem (1.3)–(1.4) if

$$\begin{aligned}&-\int _0^{t_1}\langle \varvec{u}(t), \tilde{\varvec{u}} \rangle _{L^2}\psi '(t){\mathrm {d}t} +\int _0^{t_1}\langle \varvec{u}(t)\cdot \nabla \varvec{u}(t), \tilde{\varvec{u}} \rangle _{L^2}\psi (t){\mathrm {d}t}\\&\quad =\langle \varvec{u}_0, \tilde{\varvec{u}} \rangle \psi (0)+\int _0^{t_1}\langle \theta \varvec{e}_2 + S\varvec{e}_2, \tilde{\varvec{u}} \rangle _{L^2}\psi (t){\mathrm {d}t}, \end{aligned}$$

for all \(\tilde{\varvec{u}}\in L^3_{\varvec{\tau }}(\Omega )\) and \(\psi \in \mathcal C^1([0, t_1])\) with \(\psi (t_1)=0\), and

$$\begin{aligned}&-\int _0^{t_1}\langle \theta , \tilde{\theta } \rangle _{L^2}\varphi '(t){\mathrm {d}t} - \int _0^{t_1}\langle \Delta \theta , \tilde{\theta } \rangle _{L^2}\varphi (t){\mathrm {d}t} + \int _0^{t_1}\langle \varvec{u}\cdot \nabla (\theta +S), \tilde{\theta } \rangle _{L^2}\varphi (t){\mathrm {d}t}\\&\quad =\langle \theta _0, \tilde{\theta } \rangle \varphi (0) + \int _0^{t_1}\langle \Delta S, \tilde{\theta } \rangle _{L^2}\varphi (t){\mathrm {d}t}, \end{aligned}$$

for all \(\tilde{\theta }\in L^2(\Omega )\) and \(\varphi \in \mathcal C^1([0, t_1])\) with \(\varphi (t_1)=0\). For the meaning of the notation \(L^p_{\varvec{\tau }}(\Omega )\) (\(1< p< \infty \)), see Sect. 2.

The existence of a global weak solution when the boundary data \(\eta \) and hence \(S\) are constants is obtained via the fixed point theory in [23]. It seems that the fixed point arguments could not be adapted to the case of arbitrary boundary data. Here, we are going to utilize the vanishing viscosity method to prove the existence of a global quasi-strong solution and furthermore prove the global well-posedness of the 2D Euler–Boussinesq (1.3)–(1.4) with Yudovich’s type data. We now state the main result of this article, with the proof presented in the Sects. 3–4.

Theorem 1.1

Let \(\Omega \) be a polygonal-like domain (piecewise \(\mathcal C^{1,1}\)-boundary) with maximum aperture \(\alpha _j\le \pi /2\) and let there be given \(S\in H^2(\Omega )\), \(\varvec{u}_0\in V\), \(\theta _0\in H_0^1(\Omega )\), and \(t_1>0\). Then there exists a global quasi-strong solution \((\varvec{u}, \theta )\in \mathcal C([0, t_1]; L^2_{\varvec{\tau }}(\Omega ))\times \mathcal C([0, t_1]; H_0^1(\Omega ))\) of the 2D Euler–Boussinesq system (1.3)–(1.4) such that the following estimates hold:

$$\begin{aligned} {\left\{ \begin{array}{ll} \Vert \varvec{u} \Vert _{L^\infty (0, t_1; V)} + \Vert \theta \Vert _{L^\infty (0, t_1; H^1_0(\Omega ))}+ \Vert \theta \Vert _{L^2(0, t_1; H^2(\Omega ))} \le \mathcal Q_2,\\ \Vert \varvec{u}_t \Vert _{L^2(0, t_1; L^{3/2}(\Omega ))} + \Vert \theta _t \Vert _{L^2(0, t_1; L^2(\Omega ))}\le \mathcal Q_2, \end{array}\right. } \end{aligned}$$
(1.6)

where \(\mathcal Q_2\) is a positive function defined by

$$\begin{aligned} \mathcal Q_2:=\mathcal Q_2(t_1, \Vert \varvec{u}_0 \Vert _{H^1}, \Vert \theta _0 \Vert _{H^1}, \Vert S \Vert _{H^2}), \end{aligned}$$

which is increasing in all its arguments.

Furthermore, if we additionally assume \(\omega _0={\mathrm {curl}\,}\varvec{u}_0\in L^{\infty }(\Omega )\), \(\theta _0\in H^2(\Omega )\), and \(S\in H^3(\Omega )\), then there exists a unique solution \((\varvec{u}, \theta )\) of the 2D Euler–Boussinesq system (1.3)–(1.4) satisfying

$$\begin{aligned} \omega= & {} {\mathrm {curl}\,}\varvec{u}\in L^\infty (0, t_1; L^{\infty }(\Omega )),\quad \theta \in \mathcal C([0, t_1]; H^2(\Omega ))\cap L^2(0, t_1; H^3(\Omega )),\\&\theta _t\in L^\infty (0,t_1; L^2(\Omega ))\cap L^2(0, t_1; H^1(\Omega )), \end{aligned}$$

and the estimates

$$\begin{aligned} \Vert \theta \Vert _{L^\infty (0, t_1; H^2(\Omega )) } + \Vert \theta \Vert _{L^2(0, t_1; H^3(\Omega )) }\le & {} \mathcal Q_3,\nonumber \\ \Vert \theta _t \Vert _{L^\infty (0, t_1; L^2(\Omega ))} + \Vert \theta _t \Vert _{L^2(0, t_1; H^1(\Omega )) }\le & {} \mathcal Q_3,\nonumber \\ \Vert \omega \Vert _{ L^\infty (\Omega \times (0, t_1))}\le & {} \mathcal Q_4, \end{aligned}$$
(1.7)

where \(\mathcal Q_3\) and \(\mathcal Q_4\) are positive functions defined by

$$\begin{aligned} \mathcal Q_3\!:=\!\mathcal Q_3(t_1, \Vert \omega _0 \Vert _{L^{4}}, \Vert \theta _0 \Vert _{H^2}, \Vert S \Vert _{H^3}),\quad \mathcal Q_4\!:=\!\mathcal Q_4(t_1, \Vert \omega _0 \Vert _{L^{\infty }(\Omega )}, \Vert \theta _0 \Vert _{H^2}, \Vert S \Vert _{H^3}), \end{aligned}$$

which are increasing in all their arguments.

Remark 1.1

We first note that the regularity of \(\theta \) in Theorem 1.1 only depends on the \(L^4\)-norm of the initial vorticity \(\omega _0\) and hence, as in [35], the estimate (2.3) can be used to show the uniqueness part of Theorem 1.1 under an assumption weaker than \(\omega _0\in L^\infty (\Omega )\), including unbounded initial vorticity with controlled growth rate of the \(L^p\)-norm of \(\omega _0\) as \(p\rightarrow \infty \). For instance, one can take

$$\begin{aligned} \omega _0\in \bigcap _{1<p<\infty }L^p(\Omega ),\qquad \text { with } \qquad \sup _{p>e^e} \frac{ \Vert \omega _0 \Vert _{L^p} }{ \log \log p} < \infty ; \end{aligned}$$

see [35, Sect. 5] for a precise definition of the class of allowed data.

2 Notations and Preliminaries

Here and throughout this article, we will not distinguish the notations for vector and scalar function spaces whenever they are self-evident from the context. For \(s\in \mathbb R\) and \(1\le p\le \infty \), we denote by \(W^{s,p}(\Omega )\) (resp. \(H^s(\Omega )\)) the classical Sobolev space of order \(s\) on \(\Omega \) with norm \(\Vert \cdot \Vert _{W^{s,p}}\) (resp. \(\Vert \cdot \Vert _{H^s}\)), by \(W^{s,p}_0(\Omega )\) (resp. \(H_0^s(\Omega )\)) the closure of \(\mathcal D(\Omega )\) in the space \(W^{s,p}(\Omega )\) (resp. \(H^s(\Omega )\)) when \(s>0\), and by \(L^p(\Omega )\) the classical \(L^p\)-Lebesgue space with norm \(\Vert \cdot \Vert _{L^p}\). For simplicity, we reserve the notation \(\Vert \cdot \Vert \) for the \(L^2\)-norm.

In this article, we denote by \(\mathcal Q_i(\cdot )\) (\(i=1,2,\ldots \)) the positive increasing functions in all their arguments, which may vary from line to line. The symbol \(C\) denotes a generic positive constant, which may depend on the domain \(\Omega \), but is independent of the data \(\varvec{u}_0\), \(\theta _0\), and \(S\) and of the time \(t_1\).

2.1 \(L^p\)-Tangential Vector Fields and Helmholtz Decomposition

We also introduce the \(L^p\)-tangential vector fields space as in [2, Sect. 2.2.1]:

$$\begin{aligned} L^p_{\varvec{\tau }}(\Omega )=\left\{ \varvec{u}\in L^p(\Omega )\,:\,{\mathrm {div}\,}\varvec{u}=0,\quad \varvec{u}\cdot \varvec{n}=0\quad \text { on }\partial \Omega \right\} ,\qquad 1<p<\infty , \end{aligned}$$

and the space of smooth functions

$$\begin{aligned} \mathcal V=\left\{ \varvec{u}\in \mathcal D(\Omega )\,:\, {\mathrm {div}\,}\varvec{u}=0 \right\} . \end{aligned}$$

It is well known that for general Lipschitz domains (see for example [39, Theorem I.1.4]), the space \(\mathcal V\) is dense in \(L^2_{\varvec{\tau }}(\Omega )\) and

$$\begin{aligned} L^2_{\varvec{\tau }}(\Omega )^{\perp }=\{ \nabla \pi \, :\, \pi \in H^1(\Omega ) \}. \end{aligned}$$

Let us denote by \(\mathrm {P}_\Omega \,:\,L^2(\Omega )\rightarrow L^2_{\varvec{\tau }}(\Omega )\) the corresponding orthogonal projection operator. Recently, \(\mathrm {P}_\Omega \) has been shown to extend to a bounded linear operator on \(L^p(\Omega )\) for bounded convex domains (see [40, Theorem 1.3]) with \(1<p<\infty \) and for general Lipschitz domains with the range \(p\in (3/2-\epsilon , 3+\epsilon )\) (see [41]). Here, we collect those results as follows.

Proposition 2.1

Let \(\Omega \subset \mathbb R^2\) be a polygonal-like domain (piecewise \(\mathcal C^{1,1}\)-boundaryFootnote 1). Then there holds

  1. (i)

    For \(p\in (1,\, \infty )\), the space \(\mathcal V\) is dense in \(L^p_{\varvec{\tau }}(\Omega );\)

  2. (ii)

    For \(p\in [3/2,\, 3]\), the operator \(\mathrm {P}_\Omega \) is extended to be a bounded linear projection operator from \(L^p(\Omega )\) to \(L_{\varvec{\tau }}^p(\Omega )\) with the operator norm only depending on \(p\) and the domain \(\Omega ;\)

  3. (iii)

    For \(p\in [3/2,\, 3]\) and for each \(\varvec{v} \in L^p(\Omega )\), there exists \(\pi \) belonging to the space \(W^{1,p}(\Omega )\), unique up to an additive constant such that

    $$\begin{aligned} \mathrm {P}_\Omega ^\perp \varvec{v}:=(\mathrm {1}-\mathrm {P}_\Omega ) \varvec{v}= \nabla \pi , \end{aligned}$$
    (2.1)

    and with the estimate

    $$\begin{aligned} \max \big \{ \Vert \mathrm {P}_\Omega \varvec{v} \Vert _{L^p},\; \Vert \nabla \pi \Vert _{L^p} \big \} \le C_{p, \Omega }\Vert \varvec{v} \Vert _{L^p}, \end{aligned}$$

    where \(C_{p, \Omega }>0\) depends only on \(p\) and on the domain \(\Omega \).

In Proposition 2.1, item (i) is contained in [40, Lemma 6.1], and items (ii) and (iii) are proved in [41].

2.2 The Dirichlet Problem and the Biot–Savart Law

Let \(F=\mathrm {G}_\Omega f\) be the solution of the Dirichlet problem

$$\begin{aligned} -\Delta F = f,\qquad F|_{\partial \Omega }=0. \end{aligned}$$
(2.2)

The Lax–Milgram lemma tells us that, if \(f\in H^{-1}(\Omega )\), then there exists a unique \(F\in H_0^1(\Omega )\) denoted by \(\mathrm {G}_\Omega f\) satisfying (2.2) in the distributional sense. If we further assume \(f\in L^p(\Omega )\) with \(p\ge 2\), then the elliptic regularity result in [36], which is improved in [3, Theorem 1], for the polygonal-like domains with maximum aperture \(\max \alpha _j\le \pi /2\) guarantees that \(\mathrm {G}_\Omega f\) still has two derivatives in \(L^p(\Omega )\) and the following estimate holds:

$$\begin{aligned} \mathrm {G}_\Omega f\in W^{2,p}(\Omega )\cap W_0^{1,p}(\Omega ),\qquad \Vert \mathrm {G}_\Omega f \Vert _{ W^{2,p} }\le C_{\Omega } p \Vert f \Vert _{L^p},\quad 2\le p<\infty , \end{aligned}$$
(2.3)

where \(C_{\Omega }\) only depends on \(\Omega \). We also infer from [36, Theorem 5.1.1.4] that

$$\begin{aligned} \Vert \mathrm {G}_\Omega f \Vert _{H^3} \le C_{\Omega }\Vert f \Vert _{H^1},\qquad \forall f\in H_0^1(\Omega ). \end{aligned}$$
(2.4)

We now set

$$\begin{aligned} \varvec{\mathrm {K}}_\Omega :=\nabla ^{\perp }(\mathrm {G}_\Omega f),\qquad f\in H^{-1}(\Omega ), \end{aligned}$$

where \(\nabla ^{\perp }=(\partial _y, -\partial _x)\). Then the Biot–Savart law reads that for all \(2\le p<\infty \), there holds

$$\begin{aligned} {\left\{ \begin{array}{ll} \varvec{\mathrm {K}}_\Omega \in \fancyscript{L}( H^{-1}(\Omega ),\; L^2_{\varvec{\tau }}(\Omega )), \\ \varvec{\mathrm {K}}_\Omega \in \fancyscript{L}( L^p(\Omega ),\; W^{1,p}(\Omega )\cap L^2_{\varvec{\tau }}(\Omega )),\\ \varvec{\mathrm {K}}_\Omega \in \fancyscript{L}( H_0^1(\Omega ),\; H^2(\Omega )\cap L^2_{\varvec{\tau }}(\Omega )). \end{array}\right. } \end{aligned}$$
(2.5)

To prove (2.5), due to the regularity estimates (2.3)–(2.4), we only need to verify that

$$\begin{aligned} {\mathrm {div}\,} \varvec{\mathrm {K}}_\Omega f = 0,\qquad \varvec{\mathrm {K}}_\Omega f\cdot \varvec{n}=0,\qquad \forall f\in H^{-1}(\Omega ), \end{aligned}$$

which follows from the fact that \(\mathcal V\) is dense in \(L^2_{\varvec{\tau }}(\Omega )\) and the following identity:

$$\begin{aligned} \langle \varvec{\mathrm {K}}_\Omega f, \nabla \varphi \rangle _\Omega= & {} \langle \nabla ( \mathrm {G}_\Omega f), \nabla ^\perp \varphi \rangle _\Omega \\= & {} -\langle \mathrm {G}_\Omega f, {\mathrm {div}\,} \nabla ^\perp \varphi \rangle _\Omega + \int _{\partial \Omega } (\mathrm {G}_\Omega f )\nabla ^\perp \varphi \cdot \mathbf {n}=0, \quad \forall \varphi \in \mathcal D(\overline{\Omega }). \end{aligned}$$

2.3 Elliptic Regularity at \(p\rightarrow \infty \)

In order to extend the elliptic regularity (2.3) to the end point when \(p\rightarrow \infty \), one needs to work with the Orlicz spaces, where the elliptic regularity result in these spaces was recently proved in [3] for the polygonal-like domains.

2.3.1 The Orlicz Spaces

Here, we briefly recall some preliminaries on the Orlicz spaces; see [42, 43] for more details. A function \(\gamma \,:\,[0, \infty ]\mapsto [0, \infty ]\) is said to be a Young function if

  1. (1)

    \(\gamma \) is increasing and \(\gamma (0)=0\), \(\lim _{s\rightarrow \infty }\gamma (s)=\infty \);

  2. (2)

    \(\gamma \) is a convex lower-semicontinuous \([0, \infty ]\)-valued function on \(\mathbb R\);

  3. (3)

    \(\gamma \) is non-trivial, that is there exists a number \(0<s_0<\infty \) such that \(0<\gamma (s_0)<\infty \).

The convex conjugate \(\gamma ^*\) of a \(\gamma \) is defined by

$$\begin{aligned} \gamma ^*(t):=\sup \{ st - \gamma (s),\qquad s\ge 0\}, \end{aligned}$$

and one can show that \(\gamma \) is a Young function if and only if \(\gamma ^*\) is a Young function. The convex conjugacy allows us to obtain the Orlicz space version of Hölder’s inequality. The typical examples of Young functions are

$$\begin{aligned} \gamma _p(s) = s^p,\quad p> 1,\qquad \qquad \gamma _{\mathrm {exp}}(s) = e^s - 1 \end{aligned}$$

and their corresponding convex conjugates:

$$\begin{aligned} \gamma _p(t) = t^{p'},\quad p'=\frac{p}{p-1},\qquad \qquad \gamma _{\mathrm {exp}}^*(t) = {\left\{ \begin{array}{ll} t\ln t - t + 1, &{}\forall t\ge 1,\\ 0, &{}0\le t\le 1. \end{array}\right. } \end{aligned}$$

We now define the Orlicz spaces \(L^\gamma (\Omega )\) to be the set of all measurable functions such that the Luxemburg norm is finite, that is

$$\begin{aligned} L^\gamma (\Omega )=\{ f\text { is a measurable function on }\Omega \,:\, \Vert f \Vert _{L^{\gamma }} <\infty \}, \end{aligned}$$

where the Luxemburg norm \(\Vert f \Vert _{L^{\gamma }}\) is defined by

$$\begin{aligned} \Vert f \Vert _{L^{\gamma }}:=\inf \{ \lambda >0\,:\, \int _\Omega \gamma (|f |/\lambda ){\mathrm {d}x}{\mathrm {d}y} \le 1 \}. \end{aligned}$$

One can easily verify that \(L^{\gamma _p}(\Omega )=L^p(\Omega )\) for all \(p> 1\) and we also have the following Hölder’s inequality for the Orlicz spaces.

Proposition 2.2

Let \(\gamma =\gamma (s)\) be a Young function. Then the space \(L^\gamma (\Omega )\) with the norm \(\Vert \cdot \Vert _{\gamma }\) is a Banach space and for all \(f\in L^{\gamma }(\Omega )\) and \(f\in L^{\gamma ^*}(\Omega )\), there holds

$$\begin{aligned} \int _\Omega |f ||g |{\mathrm {d}x}{\mathrm {d}y}\le 2\Vert f \Vert _{L^{\gamma }}\Vert g \Vert _{L^{\gamma ^*}}. \end{aligned}$$

In this article, we are interested in the Young function \(\gamma _{\mathrm {exp}}\) and its convex conjugate \(\gamma _{\mathrm {exp}}^*\). Direct calculation shows that

$$\begin{aligned} \gamma _{\mathrm {exp}}^*(t) \le \frac{ t^{1+\epsilon } }{ \epsilon },\qquad \forall \,t\ge 0,\quad \forall \,0<\epsilon \le 1, \end{aligned}$$

which permits us to conclude the following:

$$\begin{aligned} \Vert f \Vert _{L^{\gamma _{\mathrm {exp}}^*}} \le \epsilon ^{-1/(1+\epsilon )}\Vert f \Vert _{L^{1+\epsilon }},\qquad \, \forall \, 0<\epsilon \le 1. \end{aligned}$$
(2.6)

2.3.2 Elliptic Regularity

The following result, which we borrow from [3, Theorem 1], gives an analog to the elliptic regularity (2.3) at the end point \(p\rightarrow \infty \). In our case, when \(\Omega \) is a polygonal-like domain (piecewise \(\mathcal C^{1,1}\)-boundary) with maximum aperture \(\max \alpha _j\le \pi /2\), there holds

$$\begin{aligned} \Vert D^2\mathrm {G}_\Omega f \Vert _{L^{\gamma _{\mathrm {exp}}}} \le C_\Omega \Vert f \Vert _{L^\infty }, \end{aligned}$$
(2.7)

where the constant \(C_\Omega >0\) depends only on the domain \(\Omega \).

Remark 2.1

In the case, when \(\Omega \) is a polygonal-like domain with the aperture \(\alpha _j\) of the form \(\frac{\pi }{k}\) for some integer \(k\ge 2\), the elliptic regularity results in [2, Proposition 3.1 and Remark 5.2] tell us that

$$\begin{aligned} \Vert D^2\mathrm {G}_\Omega f \Vert _{\mathrm {bmo}_r(\Omega )} \le C_\Omega \Vert f \Vert _{\mathrm {bmo}_z(\Omega ) }, \end{aligned}$$
(2.8)

which is a stronger inequality than (2.7). For a definition of the local \(\mathrm {bmo}_\star (\Omega )\) (\(\star =z, r\)) spaces, see [44] or [2, Sect. 3.1]. The extension of (2.8) to general polygonal-like domains is still an open problem, to the best of our knowledge (see also [3, Remark 1.1]).

3 Approximate Solutions

Inspired by [38] where the vanishing viscosity method is applied to the 2D Euler equations in a bounded smooth domain, we here utilize the same method to study the 2D Euler–Boussinesq system. Hence, we introduce the 2D Boussinesq system with full viscosity \(0<\nu \le 1\) and diffusivity \(\kappa =1\):

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\varvec{u}_\nu -\nu \Delta \varvec{u}_\nu + \varvec{u}_\nu \cdot \nabla \varvec{u}_\nu + \nabla \pi _\nu = \theta _\nu \varvec{e}_2 + S\varvec{e}_2,\quad \varvec{e}_2=(0,1),\\ {\mathrm {div}\,}\varvec{u}_\nu =0,\\ \partial _t \theta _\nu - \Delta \theta _\nu + \varvec{u}_\nu \cdot \nabla \theta _\nu +\varvec{u}_\nu \cdot \nabla S = \Delta S, \\ \end{array}\right. } \end{aligned}$$
(3.1)

with the initial and boundary conditions

$$\begin{aligned} \varvec{u}_\nu (0)= & {} \varvec{u}_0,\quad \quad \theta _\nu (0)=\theta _0,\qquad \text { in }\Omega ,\nonumber \\ \varvec{u}_\nu \cdot \varvec{n}= & {} 0,\quad \frac{\partial (\varvec{u}_\nu \cdot \varvec{\tau })}{\partial \varvec{n}}=0,\quad \quad \theta _\nu =0,\qquad \text { on }\partial \Omega . \end{aligned}$$
(3.2)

The existence and uniqueness of a global strong solution \((\varvec{u}_\nu , \theta _\nu )\) of the 2D Boussinesq system (3.1)–(3.2) in the polygonal-like domain \(\Omega \) are classically obtained using the Galerkin procedure, see for example [9, 37]. Here, we only need to prove some uniform estimates independent of \(\nu \).

Lemma 3.1

Assume that \(S\in H^2(\Omega ), \varvec{u}_0\in V\), and \(\theta _0\in H_0^1(\Omega )\). Then the solutions \((\varvec{u}_\nu , \theta _\nu )\) of (3.1)–(3.2) satisfy the following estimates:

$$\begin{aligned}&\sup _{t\in [0, t_1]}(\Vert \varvec{u}_\nu (t) \Vert _{H^1}^2 + \Vert \theta _\nu (t) \Vert _{H^1}^2) + \int _0^{t_1}\Vert \Delta \theta _\nu (t) \Vert ^2{\mathrm {d}t} + \nu \int _0^{t_1}\Vert \varvec{u}_\nu (t) \Vert _{H^2}^2{\mathrm {d}t} \le \mathcal Q_2,\nonumber \\&\qquad \Vert \partial _t\varvec{u}_\nu \Vert _{L^2(0, t_1; L^{3/2}(\Omega ))} + \Vert \partial _t\theta _\nu \Vert _{L^2(0, t_1; L^{2}(\Omega ))} \le \mathcal Q_2,\nonumber \\ \end{aligned}$$
(3.3)

where \(\mathcal Q_2\) is a positive function independent of \(\nu \) \((0<\nu \le 1\)) defined by

$$\begin{aligned} \mathcal Q_2:=\mathcal Q_2(t_1, \Vert \varvec{u}_0 \Vert _{H^1}, \Vert \theta _0 \Vert _{H^1}, \Vert S \Vert _{H^2}), \end{aligned}$$

which is increasing in all its arguments.

In the sequel, the symbol \(C\) denotes a generic positive constant, which may depend on the domain \(\Omega \) and vary from line to line.

Proof of Lemma 3.1

For the sake of simplicity, we write \((\varvec{u}, \theta )\) instead of \((\varvec{u}_\nu , \theta _\nu )\) by dropping the subscript \(\nu \) in the following proof. Multiplying (3.1)\(_1\) with \(\varvec{u}\), integrating in \(L^2(\Omega )\), and using the Cauchy–Schwarz inequality, we obtain

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert \varvec{u} \Vert ^2 + \nu \Vert \nabla \varvec{u} \Vert ^2 \le \Vert \varvec{u} \Vert ^2 + \frac{1}{2}\Vert \theta \Vert ^2 +\frac{1}{2}\Vert S \Vert ^2. \end{aligned}$$
(3.4)

Taking the inner product of (3.1)\(_3\) with \(\theta \) in \(L^2(\Omega )\) and using Hölder’s inequality and the Sobolev embedding, we find

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert \theta \Vert ^2 + \Vert \nabla \theta \Vert ^2\le & {} \Vert \varvec{u} \Vert \Vert \nabla S \Vert _{L^4}\Vert \theta \Vert _{L^4} +\Vert \Delta S \Vert \Vert \theta \Vert \\\le & {} C\Vert \varvec{u} \Vert \Vert S \Vert _{H^2}\Vert \nabla \theta \Vert + \Vert S \Vert _{H^2}\Vert \theta \Vert , \end{aligned}$$

which, by Young’s inequality, yields

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert \theta \Vert ^2 + \Vert \nabla \theta \Vert ^2 \le C\Vert S \Vert _{H^2}^2 \Vert \varvec{u} \Vert ^2 + \frac{1}{2}\Vert S \Vert _{H^2}^2 + \frac{1}{2}\Vert \theta \Vert ^2 + \frac{1}{2}\Vert \nabla \theta \Vert ^2, \end{aligned}$$
(3.5)

where the constant \(C>0\) only depends on the domain \(\Omega \).

Summing (3.4) and (3.5) together, we arrive at

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}(\Vert \varvec{u} \Vert ^2 + \Vert \theta \Vert ^2) + 2\nu \Vert \nabla \varvec{u} \Vert ^2 + \Vert \nabla \theta \Vert ^2 \le C\Vert S \Vert _{H^2}^2 + C(\Vert S \Vert _{H^2}^2 + 1)( \Vert \varvec{u} \Vert ^2 + \Vert \theta \Vert ^2 ). \end{aligned}$$

Applying the Gronwall lemma, we obtain

$$\begin{aligned}&\sup _{t\in [0, t_1]}(\Vert \varvec{u}(t) \Vert ^2+\Vert \theta (t) \Vert ^2) +2\nu \int _0^{t_1}\Vert \nabla \varvec{u}(t) \Vert ^2{\mathrm {d}t} + \int _0^{t_1}\Vert \nabla \theta (t) \Vert ^2{\mathrm {d}t}\nonumber \\&\quad \le e^{C t_1 (\Vert S \Vert _{H^2}^2+1)} \big (\Vert \varvec{u}_0 \Vert ^2+\Vert \theta _0 \Vert ^2 +C t_1\Vert S \Vert _{H^2}^2\big ). \end{aligned}$$
(3.6)

In order to find the uniform \(H^1\)-estimate, we need to use the vorticity formulation together with the Biot–Savart law. Let \(\omega ={\mathrm {curl}\,}\varvec{u}=\partial _x u_2 - \partial _y u_1\), then the vorticity \(\omega \) satisfies

$$\begin{aligned} \partial _t\omega -\nu \Delta \omega + \varvec{u}\cdot \nabla \omega = \partial _x\theta +\partial _x S, \end{aligned}$$
(3.7)

with the Dirichlet boundary condition

$$\begin{aligned} \omega = 0,\quad \text { on }\partial \Omega . \end{aligned}$$

That \(\omega \) satisfies the homogeneous Dirichlet boundary condition is from the boundary conditions (3.2)\(_2\) and the calculation:

$$\begin{aligned} \omega ={\mathrm {curl}\,}\varvec{u}={\mathrm {curl}\,}( (\varvec{u}\cdot \varvec{\tau })\varvec{\tau }+ (\varvec{u}\cdot \varvec{n})\varvec{n})=\frac{\partial (\varvec{u}\cdot \varvec{n})}{\partial \varvec{\tau }} - \frac{\partial (\varvec{u}\cdot \varvec{\tau })}{\partial \varvec{n}}=0,\quad \text { on }\partial \Omega . \end{aligned}$$

By the Biot–Savart law (2.5) (see also [38, 45]), we have

$$\begin{aligned} \Vert \varvec{u} \Vert _{H^1}^2 \le C\Vert \omega \Vert _{L^2}^2,\qquad \Vert \varvec{u} \Vert _{H^2}^2 \le C\Vert \omega \Vert _{H^1}^2\le C\Vert \nabla \omega \Vert ^2, \end{aligned}$$
(3.8)

where the Poincaré inequality is employed for the last inequality.

Taking the inner product of (3.7) with \(\omega \) in \(L^2(\Omega )\) and using the Cauchy–Schwarz inequality gives

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert \omega \Vert ^2 + \nu \Vert \nabla \omega \Vert ^2\le \Vert \omega \Vert ^2 + \frac{1}{2}\Vert \partial _x\theta \Vert ^2 + \frac{1}{2}\Vert \partial _xS \Vert ^2. \end{aligned}$$
(3.9)

Taking the inner product of (3.1)\(_3\) with \(-\Delta \theta \) in \(L^2(\Omega )\) and using Hölder’s and Ladyzhenskaya’s inequalities, we arrive at

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert \nabla \theta \Vert ^2 + \Vert \Delta \theta \Vert ^2\le & {} \Vert \varvec{u} \Vert _{L^4}\Vert \nabla \theta \Vert _{L^4}\Vert \Delta \theta \Vert + \Vert \varvec{u} \Vert _{L^4}\Vert \nabla S \Vert _{L^4}\Vert \Delta \theta \Vert + \Vert \Delta S \Vert \Vert \Delta \theta \Vert \\\le & {} C\Vert \varvec{u} \Vert ^{1/2}\Vert \varvec{u} \Vert _{H^1}^{1/2}\Vert \nabla \theta \Vert ^{1/2}\Vert \Delta \theta \Vert ^{3/2}\\&\quad +\,C\Vert \varvec{u} \Vert _{H^1}\Vert S \Vert _{H^2}\Vert \Delta \theta \Vert + \Vert S \Vert _{H^2}\Vert \Delta \theta \Vert , \end{aligned}$$

which, by Young’s inequality, yields

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert \nabla \theta \Vert ^2 + \Vert \Delta \theta \Vert ^2\le & {} C\Vert \varvec{u} \Vert ^{2} \Vert \nabla \theta \Vert ^2\Vert \varvec{u} \Vert _{H^1}^2 \nonumber \\&+\, C\Vert S \Vert _{H^2}^2\Vert \varvec{u} \Vert _{H^1}^2+C\Vert S \Vert ^2_{H^2}+ \frac{1}{2}\Vert \Delta \theta \Vert ^2. \end{aligned}$$
(3.10)

Combining the estimates (3.9) and (3.10) and using (3.8), we see that

$$\begin{aligned}&\frac{\mathrm {d}}{\mathrm {d}t}(\Vert \omega \Vert ^2 + \Vert \nabla \theta \Vert ^2) + 2\nu \Vert \nabla \omega \Vert ^2 + \Vert \Delta \theta \Vert ^2\\&\quad \le C(\Vert \varvec{u} \Vert ^2\Vert \nabla \theta \Vert ^2 + \Vert S \Vert ^2_{H^2} + 1 )(\Vert \omega \Vert ^2 + \Vert \nabla \theta \Vert ^2) + C\Vert S \Vert _{H^2}^2. \end{aligned}$$

Applying the Gronwall lemma, we obtain

$$\begin{aligned}&\sup _{t\in [0, t_1]}(\Vert \omega (t) \Vert ^2 + \Vert \nabla \theta (t) \Vert ^2) + 2\nu \int _0^{t_1}\Vert \nabla \omega (t) \Vert ^2{\mathrm {d}t} + \int _0^{t_1}\Vert \Delta \theta (t) \Vert ^2{\mathrm {d}t}\nonumber \\&\quad \le (\Vert \omega _0 \Vert ^2 + \Vert \nabla \theta _0 \Vert ^2 + Ct_1\Vert S \Vert _{H^2}^2)\exp \big \{ C( \sup _{t\in [0, t_1]}\Vert \varvec{u}(t) \Vert ^2\int _0^{t_1}\Vert \nabla \theta \Vert ^2{\mathrm {d}t} \nonumber \\&\qquad +\,t_1\Vert S \Vert ^2_{H^2} + t_1) \big \}, \end{aligned}$$
(3.11)

which implies the first inequality in (3.3) by taking the estimate (3.6) and the Biot–Savart law (3.8) into consideration.

We now turn to the second inequality in (3.3) on the time derivatives of \((\varvec{u}, \theta )\). Applying the projection operator \(\mathrm {P}_\Omega \) to (3.1)\(_1\) gives the identity

$$\begin{aligned} \partial _t\varvec{u}=\mathrm {P}_\Omega (\nu \Delta \varvec{u} - \varvec{u}\nabla \varvec{u} + \varvec{f}), \end{aligned}$$

where \(\varvec{f}=\theta \varvec{e}_2 + S\varvec{e}_2\). Noticing that, by (3.8) and (3.11), \(\nu \Vert \Delta \varvec{u} \Vert _{L^2(0,t_1; L^2(\Omega ))}\) is uniformly bounded independently of \(\nu \) (\(0<\nu \le 1\)) and using the estimate (3.11) again, the arguments for (4.25) in the case when \(p=2\) and \(q=2\) tell that

$$\begin{aligned} \partial _t\varvec{u}\in L^2(0, t_1; L^s(\Omega )), \qquad \forall \, \frac{3}{2}\le s<2. \end{aligned}$$
(3.12)

Hence,

$$\begin{aligned} \partial _t\varvec{u} \in L^2(0, t_1; L^{3/2}(\Omega )). \end{aligned}$$

Regarding \(\partial _t\theta \), we take a test function \(\tilde{\theta }\in L^2(0, t_1; L^2(\Omega ))\) with norm at most \(1\) and find from (3.1)\(_3\) that

$$\begin{aligned} |\langle \partial _t\theta , \tilde{\theta } \rangle |\le \Vert \Delta \theta \Vert \Vert \tilde{\theta } \Vert + \Vert \varvec{u} \Vert _{L^4}\Vert \nabla \theta \Vert _{L^4}\Vert \tilde{\theta } \Vert + \Vert \varvec{u} \Vert _{L^4}\Vert \nabla S \Vert _{L^4}\Vert \tilde{\theta } \Vert + \Vert \Delta S \Vert \Vert \tilde{\theta } \Vert . \end{aligned}$$

Thanks to the uniform estimate (3.3)\(_1\) again, we obtain

$$\begin{aligned} \partial _t \theta \in L^2(0, t_1; L^{2}(\Omega ) ). \end{aligned}$$

Therefore, we finished proving the inequality (3.3). This ends the proof of Lemma 3.1. \(\square \)

4 Proof of Theorem 1.1

The goal here is to prove the main result of this article and we divide it to three parts. We first prove the existence of quasi-strong solution for the 2D Euler–Boussinesq system (1.3)–(1.4), then improve the regularity of the solution, and finally show the uniqueness of the solution.

4.1 Existence of a Quasi-Strong Solution

Thanks to the fact that the estimate (3.3) in Lemma 3.1 is independent of \(\nu \), we infer the existence of a couple \((\varvec{u}, \theta )\) such that

$$\begin{aligned}&\varvec{u}\in L^\infty (0, t_1; V),\qquad \partial _t\varvec{u}\in L^2(0, t_1; L^{3/2}(\Omega )),\\&\theta \in L^\infty (0, t_1; H^1_0(\Omega ))\cap L^2(0, t_1; H^2(\Omega )),\qquad \partial _t\theta \in L^2(0, t_1; L^{2}(\Omega )), \end{aligned}$$

for which the following convergences up to not relabeled subsequences are true.

  • \(\varvec{u}_\nu \rightarrow \varvec{u}\) weak-\(*\) in \(L^\infty (0, t_1; V)\) and \(\partial _t\varvec{u}_\nu \rightarrow \partial _t\varvec{u}\) weakly in \(L^2(0, t_1; L^{3/2}(\Omega ))\). As a consequence (see e.g., [46]), \(\varvec{u}_\nu \rightarrow \varvec{u}\) strongly in \(L^2(0,t_1; L^6(\Omega ))\).

  • \(\theta _\nu \rightarrow \theta \) weak-\(*\) in \(L^\infty (0, t_1; H^1_0(\Omega ))\) and weakly in \(L^2(0, t_1; H^2(\Omega ))\), and \(\partial _t\theta _\nu \rightarrow \partial _t\theta \) weakly in \(L^2(0, t_1; L^{2}(\Omega ))\). Therefore, \(\theta _\nu \rightarrow \theta \) strongly in \(L^2(0, t_1; H_0^1(\Omega ))\).

By interpolation (see e.g., [47]), we also have \(\varvec{u}\in \mathcal C([0, t_1]; L^2_{\varvec{\tau }}(\Omega ))\) and \(\theta \in \mathcal C([0, t_1]; H_0^1(\Omega ))\). The estimate (1.6) in Theorem 1.1 directly follows from the uniform estimate (3.3) which is independent of \(\nu \).

Let \(\tilde{\varvec{u}}\in L_{\varvec{\tau }}^3(\Omega )\), \(\tilde{\theta }\in L^2(\Omega )\) and \(\psi , \varphi \in \mathcal C^1([0, t_1])\) with \(\psi (t_1)=\varphi (t_1)=0\), we then take the \(L^2\)-inner product of (3.1) with \((\tilde{\varvec{u}}\psi (t), \tilde{\theta }\varphi (t))\), integrate in time from \(0\) to \(t_1\), and integrate by parts for the first term; we arrive at

$$\begin{aligned}&-\!\int _0^{t_1}\langle \varvec{u}_\nu (t), \tilde{\varvec{u}} \rangle \psi '(t){\mathrm {d}t} \!-\! \nu \int _0^{t_1}\langle \Delta \varvec{u}_\nu (t), \tilde{\varvec{u}} \rangle \psi (t){\mathrm {d}t} \!+\!\int _0^{t_1}\langle \varvec{u}_\nu (t)\cdot \nabla \varvec{u}_\nu (t), \tilde{\varvec{u}} \rangle \psi (t){\mathrm {d}t}\nonumber \\&\qquad =\langle \varvec{u}_0, \tilde{\varvec{u}} \rangle \psi (0) + \int _0^{t_1}\langle \varvec{\theta }_\nu (t)\varvec{e}_2 + S\varvec{e}_2, \tilde{\varvec{u}} \rangle \psi (t){\mathrm {d}t},\end{aligned}$$
(4.1)
$$\begin{aligned}&-\int _0^{t_1}\langle \theta _\nu (t), \tilde{\theta } \rangle \varphi (t){\mathrm {d}t} -\int _0^{t_1}\langle \Delta \theta _\nu (t), \tilde{\theta } \rangle \varphi (t){\mathrm {d}t}+\int _0^{t_1}\langle \varvec{u}_\nu (t)\cdot \nabla \theta _\nu (t), \tilde{\theta } \rangle \varphi (t){\mathrm {d}t}\nonumber \\&\quad +\int _0^{t_1}\langle \varvec{u}_\nu (t)\cdot \nabla S, \tilde{\theta } \rangle \varphi (t){\mathrm {d}t} =\langle \theta _0, \tilde{\theta } \rangle \varphi (0) + \int _0^{t_1}\langle \Delta S, \tilde{\theta } \rangle \varphi (t){\mathrm {d}t}. \end{aligned}$$
(4.2)

Thanks to the uniform estimate (3.3) in Lemma 3.1, the second term in (4.1) converges to zero, that is

$$\begin{aligned} \nu \int _0^{t_1}\langle \Delta \varvec{u}_\nu (t), \tilde{\varvec{u}} \rangle \psi (t){\mathrm {d}t} \rightarrow 0,\qquad \text { as }\nu \rightarrow 0. \end{aligned}$$

The other linear terms in (4.1)–(4.2) converge to their corresponding limits in a straightforward manner due to the above convergences. The nonlinear term in (4.1) can be written as

$$\begin{aligned} \int _0^{t_1}\langle (\varvec{u}_\nu (t) - \varvec{u}(t))\cdot \nabla \varvec{u}_\nu (t), \tilde{\varvec{u}} \rangle \psi (t){\mathrm {d}t} +\int _0^{t_1}\langle \varvec{u}(t)\cdot \nabla \varvec{u}_\nu (t), \tilde{\varvec{u}} \rangle \psi (t){\mathrm {d}t}, \end{aligned}$$

and the first term above converges to zero due to the strong convergence of \(\varvec{u}_\nu \rightarrow \varvec{u}\) in \(L^2(0, t_1; L^6(\Omega ))\) and the uniform boundedness of \(\varvec{u}_\nu \) in \(L^\infty (0, t_1; V)\), and the second term above converges to

$$\begin{aligned} \int _0^{t_1}\langle \varvec{u}(t)\cdot \nabla \varvec{u}(t), \tilde{\varvec{u}} \rangle \psi (t){\mathrm {d}t}, \end{aligned}$$

because of the weak-\(*\) convergence of \(\varvec{u}_\nu \rightarrow \varvec{u}\) in \(L^\infty (0, t_1; V)\). The convergence of the nonlinear term in (4.2) is similar and simpler since we have better convergence results for \(\theta _\nu \). Therefore, we completed the proof of existence part of Theorem 1.1.

4.2 Regularity

Now, if we assume additionally \(\omega _0={\mathrm {curl}\,}\varvec{u}_0\in L^{\infty }(\Omega )\) and \(\theta _0\in H^2(\Omega )\), \(S\in H^3(\Omega )\), then we are able to prove \(L^\infty \)-estimate for the vorticity \(\omega \) and hence the \(L^p\)-estimate for the velocity \(\varvec{u}\) for any \(1<p<\infty \) and the uniform \(H^2\) and the time average of \(H^3(\Omega )\)-estimate for \(\theta \).

For proving the \(L^\infty \)-estimate of the vorticity \(\omega \), we require the \(L^2(0, t_1; W^{1,\infty }(\Omega ))\)-estimate of the forcing term \(\theta + S\) for the Euler equations. Hence, we first need an \(L^2(0, t_1; W^{1,\infty }(\Omega ))\)-regularity of \(\theta \), which turns to be the \(L^2(0, t_1; H^3(\Omega ))\)-regularity for \(\theta \). To obtain the time average of \(H^3\)-regularity for \(\theta \), we at least need the uniform \(W^{1,4}(\Omega )\)-estimate for the velocity \(\varvec{u}\). In conclusion, the plan for this subsection is as follows. We first derive the uniform \(W^{1,4}\)-estimate for \(\varvec{u}\), then show the uniform \(H^2\) and the time average of \(H^3\)-estimates for \(\theta \), and finally prove the \(L^\infty \)-estimate for the vorticity \(\omega \).

From the Ladyzhenskaya’s inequality

$$\begin{aligned} \Vert f \Vert _{L^4}^4 \le \Vert f \Vert _{L^2}^2\Vert f \Vert _{H^1}^2, \end{aligned}$$

we deduce that \(\theta \in L^4(0, t_1; W^{1,4}(\Omega ))\) and by the Sobolev embedding, \(S\in W^{1,4}(\Omega )\). Currently, if we only assume \(\omega _0={\mathrm {curl}\,}\varvec{u}_0\in L^{4}(\Omega )\), then applying Proposition 4.1 with \(\varvec{f}=\theta \varvec{e}_2 + S\varvec{e}_2\in L^4(0, t_1; W^{1,4}(\Omega ))\) and \(p=4\) shows that

$$\begin{aligned} \varvec{u}\!\in \! L^\infty (0, t_1; W^{1,4}(\Omega )),\quad \Vert \varvec{u} \Vert _{L^\infty (0, t_1; W^{1,4}(\Omega )) } \le \mathcal Q_3(t_1, \Vert \omega _0 \Vert _{L^4}, \Vert \theta _0 \Vert _{H^1}, \Vert S \Vert _{H^2}).\nonumber \\ \end{aligned}$$
(4.3)

Furthermore, choosing \(p=3\) and \(q=4\) in (4.25) gives

$$\begin{aligned} \partial _t\varvec{u}\in L^4(0, t_1; L^3(\Omega )),\qquad \Vert \partial _t\varvec{u} \Vert _{L^4(0, t_1; L^3(\Omega ))} \le \mathcal Q_3(t_1, \Vert \omega _0 \Vert _{L^4}, \Vert \theta _0 \Vert _{H^1}, \Vert S \Vert _{H^2}).\nonumber \\ \end{aligned}$$
(4.4)

To obtain the \(H^2\)-regularity of \(\theta \), we differentiate (1.3) in time \(t\) to find

$$\begin{aligned} \partial _t\theta _t - \Delta \theta _t + \varvec{u}\cdot \nabla \theta _t + \varvec{u}_t\cdot \nabla \theta + \varvec{u}_t\cdot \nabla S = 0. \end{aligned}$$

Applying the standard energy estimate, we arrive at

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t} \Vert \theta _t \Vert ^2 + \Vert \nabla \theta _t \Vert ^2 \le \Vert \varvec{u}_t \Vert \Vert \nabla \theta \Vert _{L^4}\Vert \theta _t \Vert _{L^4}+ \Vert \varvec{u}_t \Vert \Vert \nabla S \Vert _{L^4}\Vert \theta _t \Vert _{L^4}, \end{aligned}$$
(4.5)

and, by Ladyzhenskaya’s inequality and the Sobolev embedding, the right-hand side is bounded by

$$\begin{aligned} C\Vert \varvec{u}_t \Vert \Vert \nabla \theta \Vert ^{1/2}\Vert \Delta \theta \Vert ^{1/2}\Vert \theta _t \Vert ^{1/2}\Vert \nabla \theta _t \Vert ^{1/2} + C\Vert \varvec{u}_t \Vert \Vert S \Vert _{H^2}\Vert \theta _t \Vert ^{1/2}\Vert \nabla \theta _t \Vert ^{1/2}, \end{aligned}$$

which is further bounded by

$$\begin{aligned} C\Vert \varvec{u}_t \Vert ^4 + C\Vert \nabla \theta \Vert ^{2}\Vert \Delta \theta \Vert ^2 + C\Vert S \Vert _{H^2}^4 + \frac{1}{2}\Vert \theta _t \Vert ^2 + \frac{1}{2} \Vert \nabla \theta _t \Vert ^2, \end{aligned}$$

where we used Young’s inequality.

Now, we derive from (4.5) that

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} \Vert \theta _t \Vert ^2 + \Vert \nabla \theta _t \Vert ^2 \le C\Vert \varvec{u}_t \Vert ^4 + C\Vert \nabla \theta \Vert ^{2}\Vert \Delta \theta \Vert ^2 +C\Vert S \Vert _{H^2}^4 + \Vert \theta _t \Vert ^2. \end{aligned}$$

Thus, the Gronwall lemma implies

$$\begin{aligned}&\sup _{t\in [0, t_1]}\Vert \theta _t(t) \Vert ^2 + \int _0^{t_1} \Vert \nabla \theta _t(t) \Vert ^2 {\mathrm {d}t} \le e^{t_1}\big ( \Vert \theta _t(0) \Vert ^2 + C\int _0^{t_1}\big [\Vert \varvec{u}_t \Vert ^4 + \Vert \nabla \theta \Vert ^{2}\Vert \Delta \theta \Vert ^2\\&\quad +\,\Vert S \Vert _{H^2}^4 \big ]{\mathrm {d}t} \big ), \end{aligned}$$

and, from equation (1.3)\(_3\), one has

$$\begin{aligned} \Vert \theta _t(0) \Vert \le \Vert \Delta \theta _0 \Vert + \Vert \varvec{u}_0 \Vert _{L^4}\Vert \nabla \theta _0 \Vert _{L^4} + \Vert \varvec{u}_0 \Vert _{L^4}\Vert \nabla S \Vert _{L^4} + \Vert \Delta S \Vert . \end{aligned}$$

Therefore, together with (4.4), we find

$$\begin{aligned} \sup _{t\in [0, t_1]}\Vert \theta _t(t) \Vert ^2 + \int _0^{t_1} \Vert \nabla \theta _t(t) \Vert ^2 {\mathrm {d}t} \le \mathcal Q_3(t_1, \Vert \omega _0 \Vert _{L^4}, \Vert \theta _0 \Vert _{H^2}, \Vert S \Vert _{H^2}). \end{aligned}$$
(4.6)

From equation (1.3)\(_3\) again, we obtain

$$\begin{aligned} \Vert \Delta \theta \Vert \le \Vert \theta _t \Vert + \Vert \varvec{u} \Vert _{L^4}\Vert \nabla \theta \Vert _{L^4} + \Vert \varvec{u} \Vert _{L^4}\Vert \nabla S \Vert _{L^4} + \Vert \Delta S \Vert , \end{aligned}$$

which, together with the estimates (4.3) and (4.6), immediately gives

$$\begin{aligned} \sup _{t\in [0, t_1]}\Vert \Delta \theta (t) \Vert \le \mathcal Q_3(t_1, \Vert \omega _0 \Vert _{L^4}, \Vert \theta _0 \Vert _{H^2}, \Vert S \Vert _{H^2}). \end{aligned}$$
(4.7)

Taking the gradient \(\nabla \) on (1.3)\(_3\), we similarly have

$$\begin{aligned} \Vert \nabla \Delta \theta \Vert\le & {} \Vert \nabla \theta _t \Vert + \Vert \nabla (\varvec{u}\cdot \nabla \theta ) \Vert + \Vert \nabla (\varvec{u}\cdot \nabla S) \Vert + \Vert \nabla \Delta S \Vert \nonumber \\\le & {} \Vert \nabla \theta _t \Vert + \Vert \nabla \varvec{u} \Vert _{L^4}\Vert \nabla \theta \Vert _{L^4} + \Vert \varvec{u} \Vert _{L^4}\Vert \Delta \theta \Vert _{L^4} \nonumber \\&+\, \Vert \nabla \varvec{u} \Vert _{L^4}\Vert \nabla S \Vert _{L^4} +\,\Vert \varvec{u} \Vert _{L^4}\Vert \Delta S \Vert _{L^4} + \Vert S \Vert _{H^3}. \end{aligned}$$
(4.8)

The troublesome term in (4.8) is \(\Vert \varvec{u} \Vert _{L^4}\Vert \Delta \theta \Vert _{L^4}\), which can be estimated by Ladyzhenskaya’s and Young’s inequalities:

$$\begin{aligned} \Vert \varvec{u} \Vert _{L^4}\Vert \Delta \theta \Vert _{L^4} \le C\Vert \varvec{u} \Vert _{L^4}\Vert \Delta \theta \Vert ^{1/2}\Vert \nabla \Delta \theta \Vert ^{1/2} \le C\Vert \varvec{u} \Vert _{L^4}^2\Vert \Delta \theta \Vert + \frac{1}{4}\Vert \nabla \Delta \theta \Vert . \end{aligned}$$

Hence, by the Sobolev embedding, the inequality (4.8) becomes

$$\begin{aligned} \Vert \nabla \Delta \theta \Vert\le & {} C\big (\Vert \nabla \theta _t \Vert + \Vert \nabla \varvec{u} \Vert _{L^4}\Vert \theta \Vert _{H^1} + \Vert \varvec{u} \Vert _{L^4}^2\Vert \Delta \theta \Vert \\&+\, \Vert \nabla \varvec{u} \Vert _{L^4}\Vert S \Vert _{H^2} +\Vert \varvec{u} \Vert _{L^4}\Vert S \Vert _{H^2} + \Vert S \Vert _{H^3}\big ), \end{aligned}$$

which, by utilizing the estimates (4.3) and (4.6)–(4.7), shows

$$\begin{aligned} \int _0^{t_1}\Vert \nabla \Delta \theta (t) \Vert ^2{\mathrm {d}s}\le \mathcal Q_3(t_1, \Vert \omega _0 \Vert _{L^4}, \Vert \theta _0 \Vert _{H^2}, \Vert S \Vert _{H^3}). \end{aligned}$$
(4.9)

We thus proved the first two estimates in (1.7) and we now turn to the \(L^\infty \)-estimate of the vorticity \(\omega \).

By the Sobolev embedding, we have

$$\begin{aligned} \Vert \theta \varvec{e}_2 + S e_2 \Vert _{L^2(0, t_1; W^{1,\infty }(\Omega ))}\le \mathcal Q_3(t_1, \Vert \omega _0 \Vert _{L^4}, \Vert \theta _0 \Vert _{H^2}, \Vert S \Vert _{H^3}). \end{aligned}$$
(4.10)

At this point, applying Proposition (4.19) again, we read from (4.22) that

$$\begin{aligned} \Vert \omega \Vert _{L^\infty (0, t_1; L^p(\Omega ))} \le \Vert {\mathrm {curl}\,}\varvec{u}_0 \Vert _{L^p} + \int _0^{t_1}\Vert \theta + S \Vert _{W^{1,p}}{\mathrm {d}s},\quad p\ge 2, \end{aligned}$$

and letting \(p\rightarrow \infty \) and using (4.10) yield the last estimate in (1.7). This completes the proof of regularity part of Theorem 1.1.

4.3 Uniqueness

Let \((\varvec{u}_1, \theta _1)\) and \((\varvec{u}_2, \theta _2)\) be two solutions of the 2D Euler–Boussinesq system (1.3)-(1.4) satisfying (1.7). Observe that, from (2.7) and (1.7),

$$\begin{aligned} \Vert \nabla \varvec{u}_j(t) \Vert _{L^{\gamma _{\mathrm {exp}}}} \le C_\Omega \Vert \varvec{\omega }_j \Vert _{L^{\infty }} \le \mathcal Q_4 ,\quad \omega _j={\mathrm {curl}\,} \varvec{u}_j,\qquad \forall \,t\in [0, t_1]. \end{aligned}$$
(4.11)

The differences \(\varvec{u}=\varvec{u}_2 - \varvec{u}_1\) and \(\theta =\theta _2 - \theta _1\) then satisfy the equations

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\varvec{u} + \varvec{u}_2\cdot \nabla \varvec{u} +\varvec{u}\cdot \nabla \varvec{u}_1+ \nabla \pi = \theta \varvec{e}_2,\quad \varvec{e}_2=(0,1),\\ \partial _t \theta - \Delta \theta + \varvec{u}_2\cdot \nabla \theta + \varvec{u}\cdot \nabla \theta _1 +\varvec{u}\cdot \nabla S = 0, \\ \end{array}\right. } \end{aligned}$$
(4.12)

for some pressure function \(\pi =\pi (t,x,y)\).

Taking the inner product of (4.12)\(_1\) with \(\varvec{u}\) in \(L^2(\Omega )\), (legitimately) integrating by parts, and applying the Orlicz space version of Hölder’s inequality (see Proposition 2.2), we arrive at

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert \varvec{u} \Vert ^2\le & {} \Vert \theta \Vert \Vert \varvec{u} \Vert + \Vert \nabla \varvec{u}_1 \Vert _{L^{\gamma _{\mathrm {exp}}}}\Vert |\varvec{u}|^2 \Vert _{L^{\gamma _{\mathrm {exp}}^*}}, \\\le & {} \frac{1}{4}\Vert \Delta \theta \Vert ^2 + C\Vert \varvec{u} \Vert ^2 + \mathcal Q_4 \epsilon ^{-1/(1+\epsilon )}\Vert |\varvec{u}|^2 \Vert _{L^{1+\epsilon }},\qquad \forall \,0<\epsilon \le 1, \end{aligned}$$

where we used the inequalities (2.6) and (4.11) and the Poincaré inequality for \(\theta \). Furthermore, by the interpolation inequality,

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert \varvec{u} \Vert ^2&\!\le \! \frac{1}{4}\Vert \Delta \theta \Vert ^2 + C\Vert \varvec{u} \Vert ^2 + \mathcal Q_4 \epsilon ^{-1/(1+\epsilon )}\Vert \varvec{u} \Vert ^{2/(1+\epsilon )}\Vert \varvec{u} \Vert _{L^\infty }^{\epsilon /(1+\epsilon )},\quad \forall \,0<\epsilon \le 1.\nonumber \\ \end{aligned}$$
(4.13)

Multiplying (4.12)\(_2\) by \(-\Delta \theta \) and integrating in \(\Omega \), we deduce from Ladyzhenskaya’s and Young’s inequalities and the Sobolev embedding that

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert \nabla \theta \Vert ^2+\Vert \Delta \theta \Vert ^2\le & {} C\Vert \varvec{u}_2 \Vert _{L^4}\Vert \nabla \theta \Vert ^{1/2}\Vert \Delta \theta \Vert ^{3/2}\nonumber \\&+\,(\Vert \nabla \theta _1 \Vert _{L^\infty }+\Vert \nabla S \Vert _{L^\infty })\Vert \varvec{u} \Vert \Vert \Delta \theta \Vert \nonumber \\\le & {} C\Vert \varvec{u}_2 \Vert _{L^4}^4\Vert \nabla \theta \Vert ^2 + C(\Vert \theta _1 \Vert _{H^3}^2 + \Vert S \Vert _{H^3}^2) \Vert \varvec{u} \Vert ^2 + \frac{1}{4}\Vert \Delta \theta \Vert ^2.\nonumber \\ \end{aligned}$$
(4.14)

Adding the inequalities (4.13)–(4.14) together and using

$$\begin{aligned} \Vert \varvec{u} \Vert _{L^\infty (\Omega \times (0, t_1))}^{\epsilon /(1+\epsilon )} \le \Vert \varvec{u} \Vert _{L^\infty (\Omega \times (0, t_1))} + 1, \end{aligned}$$

we obtain

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}( \Vert \varvec{u} \Vert ^2 + \Vert \nabla \theta \Vert ^2 )+\Vert \Delta \theta \Vert ^2\le & {} C\Vert \varvec{u} \Vert ^2 + \mathcal Q_4 \epsilon ^{-1/(1+\epsilon )}\Vert \varvec{u} \Vert ^{2/(1+\epsilon )}\Vert \varvec{u} \Vert _{L^\infty }^{\epsilon /(1+\epsilon )}\nonumber \\&+\,C\Vert \varvec{u}_2 \Vert _{L^4}^4\Vert \nabla \theta \Vert ^2 + C(\Vert \theta _1 \Vert _{H^3}^2 + \Vert S \Vert _{H^3}^2) \Vert \varvec{u} \Vert ^2\nonumber \\\le & {} \kappa _1 \epsilon ^{-1/(1+\epsilon )}( \Vert \varvec{u} \Vert ^2 + \Vert \nabla \theta \Vert ^2)^{1/(1+\epsilon )} + g(t)( \Vert \varvec{u} \Vert ^2\nonumber \\&+\,\Vert \nabla \theta \Vert ^2), \end{aligned}$$
(4.15)

where

$$\begin{aligned} \kappa _1:= \mathcal Q_4(\Vert \varvec{u} \Vert _{L^\infty (\Omega \times (0, t_1))} + 1),\qquad g(t):=C(1+\Vert \varvec{u}_2 \Vert _{L^4}^4 +\Vert \theta _1 \Vert _{H^3}^2 + \Vert S \Vert _{H^3}^2). \end{aligned}$$

From the estimate (1.7), we have

$$\begin{aligned} \kappa _1 < \infty ,\qquad g(t) \in L^1(0, t_1). \end{aligned}$$

We denote by \(Y(t)\) the sum \(\Vert \varvec{u}(t) \Vert ^2 + \Vert \nabla \theta (t) \Vert ^2\), then the differential inequality (4.15) yields

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} Y(t) \le \kappa _1 \epsilon ^{-1/(1+\epsilon )} Y(t)^{ 1/(1+\epsilon )} + g(t)Y(t), \end{aligned}$$

which, by letting \(\widetilde{Y}(t) = e^{-\int _0^t g(s){\mathrm {d}s}}Y(t)\), implies

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} \widetilde{Y}(t) \!\le \! \kappa _1 \epsilon ^{-1/(1+\epsilon )} e^{-\int _0^t g(s){\mathrm {d}s}} Y(t)^{ 1/(1+\epsilon )} \!\le \! \kappa _1 \epsilon ^{-1/(1+\epsilon )} \widetilde{Y}(t)^{ 1/(1+\epsilon )}, \quad \forall \,0\!<\!\epsilon \le 1. \end{aligned}$$
(4.16)

We compute

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\widetilde{Y}(t) = \frac{\mathrm {d}}{\mathrm {d}t}\bigg [\big (\widetilde{Y}(t)^{\epsilon /(1+\epsilon )}\big )^{(1+\epsilon )/\epsilon } \bigg ] =\frac{1+\epsilon }{\epsilon } \widetilde{Y}(t)^{1/(1+\epsilon )} \frac{\mathrm {d}}{\mathrm {d}t} \big (\widetilde{Y}(t)^{\epsilon /(1+\epsilon )}\big ), \end{aligned}$$

and deduce from (4.16) that

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} \big (\widetilde{Y}(t)^{\epsilon /(1+\epsilon )}\big ) \le \epsilon ^{\epsilon /(1+\epsilon )} \frac{\kappa _1}{1+\epsilon },\qquad \forall \,0<\epsilon \le 1. \end{aligned}$$
(4.17)

Due to the continuity \((\varvec{u}, \theta )\in \mathcal C([0, t_1]; L_{\varvec{\tau }}^2(\Omega ))\times \mathcal C([0, t_1]; H_0^1(\Omega ))\), the functional \(\widetilde{Y}\) is continuous on \([0, t_1]\). Noting that \(\widetilde{Y}(0)=0\) and integrating (4.17) in time on \((0, t)\) gives

$$\begin{aligned} \widetilde{Y}(t) \le \epsilon \bigg (\frac{\kappa _1 t}{1+\epsilon }\bigg )^{(1+\epsilon )/\epsilon },\qquad \forall t>0, \qquad \forall \,0<\epsilon \le 1. \end{aligned}$$
(4.18)

Choosing \(t^*>0\) small enough such that \(\kappa _1 t^*/(1+\epsilon )\le \kappa _1 t^*/2 < 1\) and letting \(\epsilon \) tend to \(0\) in (4.18) entails that \(\widetilde{Y}(t) \equiv 0\) on \([0, t^*]\). By the induction method, we can conclude that \(\widetilde{Y}(t) \equiv 0\) and hence \(Y(t)\equiv 0\) on \([0, t_1]\). This completes the proof of uniqueness part of Theorem 1.1.

Remark 4.1

We note that the proof of uniqueness shows the continuity of the solution semigroup in the topology of \(L_{\varvec{\tau }}^2(\Omega )\times H_0^1(\Omega )\) within the set \(\big \{\varvec{u}\in L_{\varvec{\tau }}^2(\Omega )\,:\,{\mathrm {curl}\,}\varvec{u}\in L^\infty (\Omega )\times H_0^1(\Omega )\cap H^2(\Omega )\big \}\). It is not clear whether the semigroup is continuous on \(L^p_{\varvec{\tau }}(\Omega )\times H^2(\Omega )\) for some \(p>2\).