Global existence and uniqueness of solutions to the three-dimensional Boussinesq equations

Abstract

In this paper, we study the three-dimensional Boussinesq equations and obtain the global existence and uniqueness of a suitable weak solution in unbounded exterior domain.

1 Introduction

Let $D \subset{\mathbf {R}^{3}}$ be a bounded domain with a connected $C^{2}$-smooth boundary S, and $D':={\mathbf {R}^{3}} \backslash D$ be the unbounded exterior domain. We consider following three-dimensional Boussinesq equations:

$$\begin{aligned}& u_{t}+u\cdot\nabla u-\nu\Delta u+\nabla p=\theta e_{3}, \quad x\in D', t\geq0, \end{aligned}$$

(1.1)

$$\begin{aligned}& \theta_{t}+u\cdot\nabla\theta-\mu\Delta\theta=0, \end{aligned}$$

(1.2)

$$\begin{aligned}& \nabla\cdot u=0, \end{aligned}$$

(1.3)

$$\begin{aligned}& u(0,x)=u_{0}(x), \qquad\theta(0,x)=\theta_{0}(x), \end{aligned}$$

(1.4)

$$\begin{aligned}& u|_{S}=0,\qquad\theta|_{S}=0, \end{aligned}$$

(1.5)

where $p=p(x,t)$ is the scalar pressure, $u=u(x,t)$ is the velocity vector field, and $\theta=\theta(x,t)$ is a scalar quantity such as the concentration of a chemical substance or the temperature variation in a gravity field, in which case $\theta e_{3}$ represents the buoyancy force. The nonnegative parameters μ and ν denote the molecular diffusion and the viscosity, respectively; $u\cdot\nabla u:=u_{a}\partial_{a} u$, $\partial_{a} u:=\frac{\partial u}{\partial x_{a}}:=u_{;a}$, $u\cdot\nabla\theta:=u_{a}\partial_{a} \theta$, $\nabla\cdot u:=u_{a;a}=0$, $\frac{\partial u^{2}}{\partial x_{a}}:=(u^{2})_{;a}$, $u^{2}=u_{b}u_{b}$. Over the repeated indices a and b summation is understood, $1\leq a$, $b\leq3$.

The Boussinesq system is one of the most commonly used fluid models since it has a vortex stretching effect similar to that in the 3D incompressible flow. The Boussinesq system has an important roles in the atmospheric sciences [1] and is a model in many geophysical applications [2]. For this reason, this system is studied systematically by scientists from different domains.

Before starting and proving our main results, let us first briefly recall the related results in the literature. For the three-dimensional case, the progress on the local well-posedness and regularity criteria on the inviscid Boussinesq equations has been made (see, e.g., [3–11]). Since there is no global well-posedness for the standard high-dimensional Boussinesq equations for large initial data, it is natural to consider the blow-up (or regularity) criterion. Various criteria for the Boussinesq equations have been obtained by considerable works (see, e.g., [3–5, 7–9, 12–17] and the references therein).

In this paper, we establish the global existence and uniqueness of a suitable weak solution to the three-dimensional Boussinesq equations in unbounded exterior domains.

We assume that $u\in G$ and $\theta\in J$, where

$$\begin{aligned}& G:= \bigl\{ f| f\in L^{2}\bigl(0,T; H_{0}^{1} \bigl(D'\bigr)\bigr)\cap f_{t}\in L^{2} \bigl(D'\times[0,T]\bigr); \nabla\cdot f=0 \bigr\} , \\& J:= \bigl\{ g| g\in L^{2}\bigl(0,T; H_{0}^{1} \bigl(D'\bigr)\bigr)\cap g_{t}\in L^{2} \bigl(D'\times [0,T]\bigr) \bigr\} , \end{aligned}$$

with arbitrary $T>0$.

In this paper, $(f,g):=\int_{D'}f_{a}g_{a}\,dx$ denotes the inner product in $L^{2}(D')$, $\Vert f\Vert :=(f,f)^{\frac{1}{2}}$, $f_{ja}$ denotes the ath component of the vector function $f_{j}$, and $f_{ja;b}$ is the derivative $\frac{\partial f_{ja}}{\partial x_{b}}$.

Let us define a suitable weak solution to problem (1.1)-(1.5) that satisfies the identity

$$\begin{aligned}& (u_{t}, v)+(\theta_{t}, w)+(\nabla u, \nabla v)+(\nabla \theta, \nabla w)+(u_{a}u_{b;a}, v_{b})+(u_{a} \theta_{b;a}, w_{b}) \\& \quad=(\theta e_{3}, v), \quad\forall v\in G, w\in J. \end{aligned}$$

(1.6)

Here we took into account that $-(\Delta u, v)=(\nabla u, \nabla v)$, $-(\Delta\theta, w)=(\nabla\theta, \nabla w)$, and $(\nabla p, v)=-(p, v_{a;a})=0$, if $v\in G$, $w\in J$.

Equation (1.6) is equivalent to the integrated equation

$$\begin{aligned}& \int_{0}^{t} \bigl[(u_{s}, v)+( \theta_{s}, w)+(\nabla u, \nabla v)+(\nabla\theta, \nabla w)+(u_{a}u_{b;a}, v_{b})+(u_{a} \theta _{b;a}, w_{b}) \bigr]\,ds \\& \quad= \int_{0}^{t}(\theta e_{3}, v)\,ds, \quad \forall v\in G, w\in J. \end{aligned}$$

(1.7)

We also assume that

$$ (u_{0},u_{0})\leq C,\qquad(\theta_{0}, \theta_{0})\leq C. $$

(1.8)

Now we are in position to state our main results.

Theorem 1.1

If assumption (1.8) holds, $u_{0}\in H_{0}^{1}(D)$ satisfies (1.3), and $\theta_{0}\in H_{0}^{1}(D)$, then for all $t\geq0$, there exist weak solutions $u\in G$, $\theta\in J$ to (1.6), and the solutions are unique, provided that $|\theta(\cdot,t)|$, $\Vert \nabla u\Vert ^{4}\in L_{\mathrm{loc}}^{1}(0,\infty)$.

Remark 1.1

The proof follows the method of [16, 18], and [19].

2 Proof of Theorem 1.1

In this section, we prove the global existence and uniqueness of suitable weak solutions to problem (1.1)-(1.5). We begin with the following lemma.

Lemma 2.1

Let $\theta_{0}\in L^{1}\cap L^{2}$ and $u_{0}\in L^{2}$. Then the weak solutions u, θ of problem (1.1)-(1.5) satisfy

$$ \bigl\Vert \theta(t)\bigr\Vert \leq \Vert \theta_{0}\Vert _{1}(1+Ct)^{-\frac {3}{4}},\qquad \bigl\Vert u(t)\bigr\Vert \leq \Vert u_{0}\Vert +C\Vert \theta_{0}\Vert _{1}t^{\frac{1}{4}}. $$

(2.1)

Proof

See, e.g., [20]. □

Lemma 2.2

Under assumption (1.8), the following estimate holds:

$$ \sup_{t\geq0}\bigl\Vert u(t)\bigr\Vert ^{2}+\bigl\Vert \theta(t)\bigr\Vert ^{2}+2 \int _{0}^{t} \bigl[\bigl\Vert \nabla u(s)\bigr\Vert ^{2}+\bigl\Vert \nabla\theta(s)\bigr\Vert ^{2} \bigr]\,ds\leq C, $$

(2.2)

where $C>0$ is a constant.

Proof

Take $v=u$ and $w=\theta$ in (1.6). Then

$$\begin{aligned}& (u_{a}u_{b;a}, u_{b})=-(u_{a}u_{b}, u_{b;a})=-\frac {1}{2}\bigl(u_{a}, \bigl(u^{2} \bigr)_{;a}\bigr)=\frac{1}{2}\bigl(u_{a;a}, u^{2}\bigr)=0, \\& (u_{a}\theta_{b;a}, \theta_{b})=- \frac{1}{2}\bigl(u_{a}, \bigl(\theta ^{2} \bigr)_{;a}\bigr)=\frac{1}{2}\bigl(u_{a;a}, \theta^{2}\bigr)=0. \end{aligned}$$

Thus, equation (1.6) with $v=u$ and $w=\theta$ implies

$$ \frac{1}{2}\partial_{t} \bigl[(u,u)+(\theta,\theta ) \bigr]+( \nabla u,\nabla u)+(\nabla\theta,\nabla\theta)=(\theta e_{3}, u). $$

(2.3)

Integrating (2.3) over $[0,t]$, by virtue of (1.8), we get the following estimate:

$$\begin{aligned}& \bigl\Vert u(t)\bigr\Vert ^{2}+\bigl\Vert \theta(t)\bigr\Vert ^{2}+2\int_{0}^{t} \bigl[\bigl\Vert \nabla u(s)\bigr\Vert ^{2}+\bigl\Vert \nabla\theta(s)\bigr\Vert ^{2} \bigr]\,ds \\& \quad \leq C+ \int_{0}^{t}(\theta e_{3}, u)\,ds \leq C+ \int_{0}^{t}\Vert \theta \Vert \Vert u\Vert \,ds \\& \quad \leq C+C\sup_{s\in[0,t]}\bigl\Vert u(s)\bigr\Vert \int_{0}^{t}(1+s)^{-\frac{5}{4}}\,ds \leq C+C\sup_{s\in[0,t]}\bigl\Vert u(s)\bigr\Vert . \end{aligned}$$

(2.4)

Denote $\sup_{s\in[0,t]}\Vert u(s)\Vert =\alpha(t)$. Then (2.4) implies

$$ \alpha^{2}(t)\leq C+C\alpha(t). $$

(2.5)

Since $\alpha(t)\geq0$, inequality (2.5) yields

$$\sup_{t\geq0}\alpha(t)\leq C, $$

which, together with (2.4), gives (2.2). □

Remark 2.3

A priori estimate (2.2) implies that, for every $T\in [0,\infty)$, $u\in L^{\infty}(0,T; L^{2}(D'))\cap L^{2}(0,T; H_{0}^{1}(D'))$, $\theta\in L^{2}(0,T; H_{0}^{1}(D'))$. This and equation (1.6) also imply that $u_{t}, \theta_{t}\in L^{2}(D'\times[0,T])$.

Lemma 2.4

Under the assumptions of Theorem  1.1, there exist suitable weak solutions $u\in G$, $\theta\in J$ to problem (1.1)-(1.5).

Proof

The idea of the proof is to reduce the problem to the existence of the solution to a Cauchy problem for ordinary differential equations (ODE) of finite order and then to use an a priori estimate to establish the convergence of these solutions of ODE to the solutions of equations (1.6) and (1.7).

Let us look for the solutions to equation (1.6) of the form $u^{n}:=\sum_{j=1}^{n}c_{j}^{n}(t)\phi_{j}(x)$, $\theta^{n}:=\sum_{k=1}^{n}d_{k}^{n}(t)\phi _{k}(x)$, where $\{\phi_{i}(x)\}_{i=1}^{\infty}$ is an orthonormal basis of the space $L^{2}(D')$ of divergence-free vector functions belonging to $H_{0}^{1}(D')$, and in the expression $u^{n}$, $\theta^{n}$, the upper index n is not a power. If we substitute $u^{n}$, $\theta^{n}$ into (1.6), take $v=\phi_{v}$, $w=\phi_{w}$, and use the orthonormality of the system $\{\phi_{i}(x)\} _{i=1}^{\infty}$ and relation $(\nabla\phi_{j}, \nabla\phi _{v})=\lambda_{v}\delta_{jv}$, $(\nabla\phi_{k}, \nabla\phi_{w})=\lambda _{w}\delta_{kw}$, where $\lambda_{v}$, $\lambda_{w}$ are the eigenvalues of the vector Dirichlet Laplacian in D on the divergence-free vector fields, then we obtain a system of ODE for the unknown coefficients $c_{v}^{n}$, $d_{w}^{n}$:

$$\begin{aligned}& \partial_{t} c_{v}^{n}+\lambda_{v} c_{v}^{n}+\sum_{i,j=1}^{n}( \phi_{ia}\phi_{jb;a}, \phi_{vb})c_{i}^{n}c_{j}^{n}= \theta _{m},\quad c_{v}^{n}(0)=(u_{0}, \phi_{v}), \end{aligned}$$

(2.6)

$$\begin{aligned}& \partial_{t} d_{w}^{n}+\lambda_{w} d_{w}^{n}+\sum_{i,k=1}^{n}( \phi_{ia}\phi _{kb;a}, \phi_{wb})c_{i}^{n} d_{k}^{n}=0,\quad d_{w}^{n}(0)=( \theta_{0},\phi_{w}). \end{aligned}$$

(2.7)

Problem (2.4)-(2.5) has a global solution because of an a priori estimate that follows from (2.2) and Parseval’s relations:

$$ \bigl(u^{n}(t),u^{n}(t)\bigr)=\sum _{j=1}^{n}\bigl[c_{j}^{n}(t) \bigr]^{2}\leq C, \qquad\bigl(\theta^{n}(t), \theta^{n}(t)\bigr)=\sum_{k=1}^{n} \bigl[d_{k}^{n}(t)\bigr]^{2}\leq C. $$

(2.8)

Consider the set $\{u^{n}=u^{n}(t)\}_{n=1}^{\infty}$, $\{\theta^{n}=\theta ^{n}(t)\}_{n=1}^{\infty}$. Inequalities (2.2) and (2.8) for $u=u^{n}$, $\theta=\theta^{n}$ imply the existence of the weak limits $u^{n}\rightharpoonup u$, $\theta^{n}\rightharpoonup\theta$ in $L^{2}(0,T; H_{0}^{1}(D'))$. This allows us to pass to the limit in equation (1.7) in all the terms except the term $\int_{0}^{t} [(u_{s}, v)+(\theta_{s}, w) ]\,ds$. The weak limits of the terms $(u^{n}_{a}u_{b;a}^{n}, v_{b})$, $(u^{n}_{a} \theta^{n}_{b;a}, w_{b})$ exist and are equal to $(u_{a}u_{b;a}, v_{b})$, $(u_{a} \theta_{b;a}, w_{b})$, respectively, because

$$\begin{aligned}& \bigl(u^{n}_{a}u_{b;a}^{n}, v_{b}\bigr)=-\bigl(u^{n}_{a}u_{b}^{n}, v_{b;a}\bigr)\rightarrow-(u_{a}u_{b}, v_{b;a})= (u_{a}u_{b;a}, v_{b}), \\& \bigl(u^{n}_{a} \theta^{n}_{b;a}, w_{b}\bigr)=-\bigl(u^{n}_{a}\theta_{b}^{n}, w_{b;a}\bigr)\rightarrow-(u_{a}\theta_{b}, w_{b;a})= (u_{a}\theta_{b;a}, w_{b}). \end{aligned}$$

Note that $v_{b;a},w_{b;a}\in L^{2}(D')$ and $u^{n}_{a}u_{b}^{n}, u^{n}_{a}\theta _{b}^{n}\in L^{4}(D')$. Applying the interpolation inequality and the Schwarz inequality, we get

$$\begin{aligned}& \Vert u\Vert ^{2}_{L^{4}(D')}\leq\sqrt{2}\Vert u\Vert ^{\frac{1}{4}}\Vert \nabla u\Vert ^{\frac{3}{4}}\leq\epsilon \Vert \nabla u\Vert ^{2}+ \frac {27}{16\epsilon^{3}}\Vert u\Vert ^{2}, \quad u\in H_{0}^{1}\bigl(D'\bigr), \end{aligned}$$

(2.9)

$$\begin{aligned}& \Vert \theta \Vert ^{2}_{L^{4}(D')}\leq\sqrt{2}\Vert \theta \Vert ^{\frac{1}{4}}\Vert \nabla \theta \Vert ^{\frac{3}{4}}\leq\epsilon \Vert \nabla\theta \Vert ^{2}+ \frac {27}{16\epsilon^{3}}\Vert \theta \Vert ^{2}, \quad\theta\in H_{0}^{1} \bigl(D'\bigr), \end{aligned}$$

(2.10)

where $\epsilon>0$ is an arbitrary small number. We have $u^{n}_{a}u_{b}^{n} \rightharpoonup u_{a}u_{b}$ in $L^{2}(D')$ as $n\rightarrow\infty$ because bounded sets in a reflexive Banach space $L^{4}(D')$ are weakly compact. Consequently, $(u^{n}_{a}u_{b;a}^{n}, v_{b})\rightarrow(u_{a}u_{b;a}, v_{b})$ as $n\rightarrow\infty$. Similarly, we have $(u^{n}_{a}\theta_{b;a}^{n}, w_{b})\rightarrow(u_{a}\theta _{b;a}, w_{b})$ as $n\rightarrow\infty$.

The weak limit of the $\int_{0}^{t} [(\nabla u, \nabla v)+(\nabla \theta, \nabla w) ]\,ds$ exists because of the a priori estimate (2.2) and the weak compactness of the bounded sets in a Hilbert space. Since equation (1.7) holds and the limits of all its terms, except $\int_{0}^{t} [(u_{s}^{n}, v)+(\theta_{s}^{n}, w) ]\,ds$, do exist, there exists the limit $\int_{0}^{t} [(u_{s}^{n}, v)+(\theta _{s}^{n}, w) ]\,ds\rightarrow\int_{0}^{t} [(u_{s}, v)+(\theta_{s}, w) ]\,ds$ for all $v, w\in G$. By passing to the limit as $n\rightarrow \infty$ we prove that the limits u, θ satisfy (1.7). Differentiating (1.7) with respect to t yields (1.6) almost everywhere. □

Lemma 2.5

The global weak solutions u, θ from Lemma  2.4 are unique.

Proof

Suppose that there are two solutions to (1.6), $u\in G$, $\theta\in J$ and $u'\in G$, $\theta'\in J$, and let $U=u-u'$, $\Theta=\theta-\theta'$. Then

$$\begin{aligned}& (U_{t}, v)+(\Theta_{t}, w)+(\nabla U, \nabla v)+(\nabla \Theta, \nabla w)+\bigl(u_{a}u_{b;a}-u'_{a}u'_{b;a}, v_{b}\bigr) \\& \quad{} +\bigl(u_{a} \theta_{b;a}-u'_{a} \theta'_{b;a}, w_{b}\bigr)=0. \end{aligned}$$

(2.11)

Since $U\in G$, $\Theta\in J$, we may set $v=U$, $w=\Theta$ in (2.11) and have

$$ \partial_{t}\bigl(\Vert U\Vert ^{2}+\Vert \Theta \Vert ^{2}\bigr)+2\bigl(\Vert \nabla U\Vert ^{2}+\Vert \nabla \Theta \Vert ^{2}\bigr)+2(U_{a}u_{b;a},U_{b})+2 \bigl(U_{a}\theta'_{b;a},\Theta_{b} \bigr)=0, $$

(2.12)

where we have used the following conclusions:

$$\begin{aligned}& \bigl(u_{a}u_{b;a}-u'_{a}u'_{b;a}, U_{b}\bigr)=(U_{a}u_{b;a},U_{b})+ \bigl(u'_{a}U_{b;a},U_{b}\bigr), \\& \bigl(u'_{a}U_{b;a},U_{b}\bigr)=- \frac {1}{2}\bigl(u'_{a;a},\bigl(U^{2} \bigr)_{b}\bigr)=0, \\& \bigl(u_{a}\theta_{b;a}-u'_{a} \theta'_{b;a}, \Theta_{b}\bigr)=(U_{a} \theta _{b;a},\Theta_{b})+\bigl(u'_{a} \Theta_{b;a},\Theta_{b}\bigr), \\& \bigl(u'_{a} \Theta _{b;a},\Theta_{b}\bigr)=-\frac{1}{2} \bigl(u'_{a;a},\bigl(\Theta^{2} \bigr)_{b}\bigr)=0. \end{aligned}$$

Thus, (2.12) implies

$$ \partial_{t}\bigl(\Vert U\Vert ^{2}+\Vert \Theta \Vert ^{2}\bigr)+2\bigl(\Vert \nabla U\Vert ^{2}+\Vert \nabla\Theta \Vert ^{2}\bigr)\leq2 \bigl[\bigl\vert (U_{a}u_{b;a},h_{b})\bigr\vert +\bigl\vert (U_{a}\theta _{b;a},\Theta_{b})\bigr\vert \bigr]. $$

(2.13)

From Young’s inequality we get the following estimates:

$$\begin{aligned}& \bigl\vert (U_{a}u_{b;a},U_{b})\bigr\vert \leq \Vert U\Vert ^{2}_{L^{4}(D')}\Vert \nabla u\Vert \leq \Vert \nabla u\Vert \biggl(\epsilon \Vert \nabla U\Vert ^{2}+ \frac{27}{16\epsilon ^{3}}\Vert U\Vert ^{2} \biggr), \end{aligned}$$

(2.14)

$$\begin{aligned}& \bigl\vert (U_{a}\theta_{b;a},\Theta_{b}) \bigr\vert =\bigl\vert (U_{a}\theta_{b}, \Theta_{b;a})\bigr\vert \leq9\Vert \nabla\Theta \Vert \bigl\Vert \vert U\vert \vert \theta \vert \bigr\Vert \leq\frac{1}{2}\Vert \nabla\Theta \Vert ^{2}+18 \bigl\Vert \vert U\vert \vert \theta \vert \bigr\Vert ^{2}. \end{aligned}$$

(2.15)

Denoting $z:=\Vert U\Vert ^{2}+\Vert \Theta \Vert ^{2}$ and choosing $\epsilon=\frac {1}{2\Vert \nabla u\Vert }$ in inequalities (2.14), we obtain

$$ \partial_{t} z \leq \biggl(\frac{27}{2}\Vert \nabla u\Vert ^{4}+18\bigl\vert \theta(\cdot,t)\bigr\vert \biggr)z,\quad z|_{t=0}=0. $$

(2.16)

Assume that $|\theta(\cdot,t)|$, $\Vert \nabla u\Vert ^{4}\in L^{1}_{\mathrm{loc}}(0,\infty)$. Then we get $z=0$ for all $t\geq0$. □

The proof of Theorem 1.1 is complete.