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Global well-posedness for the 2D Cahn-Hilliard-Boussinesq and a related system on bounded domains

Boundary Value Problems20172017:119

https://doi.org/10.1186/s13661-017-0850-5

Received: 17 April 2017

Accepted: 3 August 2017

Published: 18 August 2017

Abstract

This paper proves the global well-posedness for the 2D Cahn-Hilliard-Boussinesq and a related system with partial viscous terms on bounded domains.

Keywords

Cahn-HilliardBoussinesqglobal regularitybounded domain

MSC

35Q3576D0535M10

1 Introduction

Let \(\Omega\subset\mathrm{R}^{2}\) be a bounded domain with smooth boundary Ω and ν be the unit outward normal vector to Ω. First, we consider the following inviscid Cahn-Hilliard-Boussinesq system [1]:
$$\begin{aligned}& \partial_{t}u+u\cdot\nabla u+\nabla\pi=\mu\nabla\phi+\theta e_{2}, \end{aligned}$$
(1.1)
$$\begin{aligned}& \partial_{t}\theta+u\cdot\nabla\theta=\Delta\theta, \end{aligned}$$
(1.2)
$$\begin{aligned}& \operatorname {div}u=0, \end{aligned}$$
(1.3)
$$\begin{aligned}& \partial_{t}\phi+u\cdot\nabla\phi=\Delta\mu, \end{aligned}$$
(1.4)
$$\begin{aligned}& \mu=-\Delta\phi+f'(\phi), \quad f(\phi)=\frac{1}{4}\bigl(1-\phi ^{2}\bigr)^{2}, \end{aligned}$$
(1.5)
in \(\Omega\times(0,\infty)\) with the boundary and initial conditions
$$\begin{aligned}& u\cdot\nu=0,\qquad \frac{\partial\theta}{\partial\nu}=0,\qquad \frac {\partial\phi}{\partial\nu}=\frac{\partial\mu}{\partial\nu}=0 \quad\text{on } \partial\Omega\times(0,\infty), \end{aligned}$$
(1.6)
$$\begin{aligned}& (u,\theta,\phi) (\cdot,0)=(u_{0},\theta_{0}, \phi_{0}) \quad\text{in } \Omega. \end{aligned}$$
(1.7)
Here u, π, and θ denote the velocity, pressure and temperature of the fluid, respectively. ϕ is the order parameter and μ is a chemical potential and \(e_{2}:=\bigl( {\scriptsize\begin{matrix}{} 0\cr 1 \end{matrix}} \bigr)\).

Zhao [2] proved the global existence and uniqueness of smooth solutions to problem (1.1)-(1.7) with smooth initial data \(u_{0},\theta_{0}\in H^{3}\) and \(\phi_{0}\in H^{5}\). Zhou and Fan [3] considered the vanishing limit for a 2D Cahn-Hilliard-Navier-Stokes system with a slip boundary condition. We refer the readers to [2, 4, 5] and the references therein for more discussions in this direction.

When \(\phi=0\), the system reduces to the well-known Boussinesq system. Very recently, Zhou and Li [6] proved the global well-posedness of the 2D Boussinesq system with zero viscosity (1.1)-(1.3) and (1.6), (1.7) for rough initial data \(u_{0}\in L^{2}\), \(\operatorname {rot}u_{0}\in L^{\infty}\) and \(\theta_{0}\in B_{q,r}^{2-\frac{2}{r}}\) with \(1< r<\infty\) and \(2< q<\infty\), which improves the results in [7, 8] with smooth initial data \(u_{0},\theta_{0}\in H^{3}\). Several results for the related models can be found in [9, 10].

The first aim of this paper is to prove a similar result for problem (1.1)-(1.7), we will prove the following.

Theorem 1.1

Let \(\phi_{0}\in H^{4}\), \(u_{0}\in L^{2}\), \(\operatorname {rot}u_{0}\in L^{\infty}\) and \(\theta _{0}\in B_{q,r}^{2-\frac{2}{r}}\) with \(1< r<\infty\) and \(2< q<\infty\). Then problem (1.1)-(1.7) has a unique solution \((u,\theta ,\phi)\) satisfying
$$\begin{aligned}& u\in L^{\infty}\bigl(0,T;L^{2}\bigr),\qquad \operatorname {rot}u\in L^{\infty}\bigl(0,T;L^{\infty}\bigr), \\& \theta\in C\bigl([0,T];B_{q,r}^{2-\frac{2}{r}}\bigr)\cap L^{r}\bigl(0,T;W^{2,q}\bigr),\qquad\theta _{t}\in L^{r}\bigl(0,T;L^{q}\bigr), \\& \phi\in L^{\infty}\bigl(0,T;H^{4}\bigr)\cap L^{2} \bigl(0,T;H^{5}\bigr),\qquad\phi_{t}\in L^{\infty}\bigl(0,T;L^{2}\bigr)\cap L^{2}\bigl(0,T;H^{2}\bigr) \end{aligned}$$
(1.8)
for any fixed \(T>0\).
Next, we consider the following Cahn-Hilliard-Boussinesq system:
$$\begin{aligned}& \partial_{t}u+u\cdot\nabla u+\nabla\pi-\Delta u=\mu\nabla\phi + \theta e_{2}, \end{aligned}$$
(1.9)
$$\begin{aligned}& \partial_{t}\theta+u\cdot\nabla\theta=0, \end{aligned}$$
(1.10)
$$\begin{aligned}& \operatorname {div}u=0, \end{aligned}$$
(1.11)
$$\begin{aligned}& \partial_{t}\phi+u\cdot\nabla\phi=\Delta\mu, \end{aligned}$$
(1.12)
$$\begin{aligned}& \mu=-\Delta\phi+f'(\phi),\quad f(\phi):=\frac{1}{4}\bigl(1- \phi^{2}\bigr)^{2}, \end{aligned}$$
(1.13)
$$\begin{aligned}& u=0,\qquad\frac{\partial\phi}{\partial\nu}=\frac{\partial\mu }{\partial\nu}=0 \quad\text{on } \partial\Omega\times(0, \infty ), \end{aligned}$$
(1.14)
$$\begin{aligned}& (u,\theta,\phi) (\cdot,0)=(u_{0},\theta_{0}, \phi_{0}) \quad\text{in } \Omega. \end{aligned}$$
(1.15)
When \(\phi=0\), Zhou [11] showed the global well-posedness of the problem with rough initial data
$$u_{0}\in\mathcal{D}_{A_{q}}^{1-\frac{1}{r},r}\cap H_{0}^{1} \quad\text{with } 1< r< \infty,2< q< \infty \text{ and } \theta_{0}\in H^{1}, $$
which improved the results in [12] for \((u_{0},\theta_{0})\in H^{3}\times H^{2}\) and in [13] for \((u_{0},\theta_{0})\in H^{2}\times H^{1}\).
Here the space \(\mathcal{D}_{A_{q}}^{1-\frac{1}{r},r}\) denotes some fractional domain of the Stokes operator in \(L^{q}\) with \(2-\frac{2}{r}\) derivatives (see Danchin [14]); moreover, we have
$$ \mathcal{D}_{A_{q}}^{1-\frac{1}{r},r}\hookrightarrow B_{q,r}^{2-\frac{2}{r}} \cap L^{q}. $$
(1.16)

The second aim of this paper is to prove a similar result to problem (1.9)-(1.15), we will prove the following.

Theorem 1.2

Let \(u_{0}\in\mathcal{D}_{A_{q}}^{1-\frac{1}{r},r}\cap H_{0}^{1}\) with \(1< r<\infty\), \(2< q<\infty\) and \(\theta_{0}\in L^{q}\), \(\phi_{0}\in H^{4}\). Then problem (1.9)-(1.15) has a unique solution \((u,\theta,\phi)\) satisfying
$$ \begin{gathered} u\in L^{\infty}\bigl(0,T;H_{0}^{1}\bigr)\cap L^{2}\bigl(0,T;H^{2}\bigr),\\ u\in C\bigl([0,T];\mathcal{D}_{A_{q}}^{1-\frac{1}{r},r}\bigr)\cap L^{r}\bigl(0,T;W^{2,q}\bigr),\qquad u_{t}\in L^{r}\bigl(0,T;L^{q}\bigr),\\ \theta\in L^{\infty}\bigl(0,T;L^{q}\bigr),\qquad \phi\in L^{\infty}\bigl(0,T;H^{4}\bigr)\cap L^{2}\bigl(0,T;H^{5}\bigr),\\\phi_{t}\in L^{\infty}\bigl(0,T;L^{2}\bigr)\cap L^{2}\bigl(0,T;H^{2}\bigr) \end{gathered} $$
(1.17)
for any fixed \(T>0\).
Finally, we consider the following model in electrohydrodynamics [15]:
$$\begin{aligned}& \partial_{t}u+u\cdot\nabla u+\nabla\pi=(n-p)\nabla \psi, \end{aligned}$$
(1.18)
$$\begin{aligned}& \operatorname {div}u=0, \end{aligned}$$
(1.19)
$$\begin{aligned}& \partial_{t}n+u\cdot\nabla n-\operatorname {div}(\nabla n-n\nabla \psi)=0, \end{aligned}$$
(1.20)
$$\begin{aligned}& \partial_{t}p+u\cdot\nabla p-\operatorname {div}(\nabla p+p\nabla \psi)=0, \end{aligned}$$
(1.21)
$$\begin{aligned}& -\Delta\psi=p-n+D(x) \end{aligned}$$
(1.22)
in \(\Omega\times(0,\infty)\) with the boundary and initial conditions
$$\begin{aligned}& u\cdot\nu=0,\qquad\frac{\partial n}{\partial\nu}=\frac{\partial p}{\partial\nu}=\frac{\partial\psi}{\partial\nu}=0 \quad\text{on } \partial\Omega\times(0,\infty), \end{aligned}$$
(1.23)
$$\begin{aligned}& (u,n,p) (\cdot,0)=(u_{0},n_{0},p_{0})\quad \text{in } \Omega. \end{aligned}$$
(1.24)

Here n, p and ψ denote the anion concentration, cation concentration and electric potential, respectively. \(D(x)\) is the doping profile.

Equations (1.20), (1.21) and (1.22) appear in the context as the Nernst-Plank equation in astronomy [16] and as the Van Roosbroeck system in semiconductor devices [17].

The third aim of this paper is to prove a similar result to problem (1.18)-(1.24), we will prove the following.

Theorem 1.3

Let \(u_{0}\in L^{2}\), \(\operatorname {rot}u_{0}\in L^{\infty}\) and \(n_{0},p_{0}\in B_{q,r}^{2-\frac{2}{r}}\) with \(1< r<\infty\) and \(2< q<\infty\) and \(n_{0}, p_{0}\geq0\) in Ω and \(D\in L^{\infty}(\Omega)\). Then problem (1.18)-(1.24) has a unique solution \((u,n,p,\psi)\) satisfying
$$ \begin{gathered} u\in L^{\infty}\bigl(0,T;L^{2}\bigr),\qquad \operatorname {rot}u\in L^{\infty}\bigl(0,T;L^{\infty}\bigr), \\ 0\leq n,p\in C\bigl([0,T];B_{q,r}^{2-\frac{2}{r}}\bigr)\cap L^{r}\bigl(0,T;W^{2,q}\bigr),\qquad n_{t},p_{t} \in L^{r}\bigl(0,T;L^{q}\bigr), \\ \psi\in C\bigl([0,T];W^{2,q}\bigr)\cap L^{r} \bigl(0,T;W^{4,q}\bigr),\qquad\psi_{t}\in L^{r} \bigl(0,T;W^{2,q}\bigr) \end{gathered}$$
(1.25)
for any fixed \(T>0\).

Since the proof of Theorem 1.3 is very similar to that of Theorem 1.1 and that of [6], we omit the details here.

Now we recall the maximal regularity for the heat equation [18] and the Stokes system [14], which are critical to the proof of our main theorems.

Lemma 1.1

[18]

Assume that \(\theta_{0}\in B_{q,r}^{2-\frac{2}{r}}\) and \(f\in L^{r}(0,T;L^{q})\) with \(1< r,q<\infty\). Then the problem
$$ \left \{ \textstyle\begin{array}{l} \partial_{t}\theta-\Delta\theta=f,\\ \frac{\partial\theta}{\partial n}=0 \quad\textit{on } \partial\Omega\times (0,T),\\ \theta(\cdot,0)=\theta_{0} \quad\textit{in } \Omega, \end{array}\displaystyle \right . $$
(1.26)
has a unique solution θ satisfying the following inequality for any fixed \(T>0\):
$$ \begin{aligned}[b] &\|\theta\|_{C([0,T];B_{q,r}^{2-\frac{2}{r}})}+\|\theta\| _{L^{r}(0,T;W^{2,q})}+\|\theta_{t} \|_{L^{r}(0,T;L^{q})} \\ &\quad\leq C\bigl(\|\theta_{0}\|_{B_{q,r}^{2-\frac{2}{r}}}+\|f\| _{L^{r}(0,T;L^{q})}\bigr), \end{aligned}$$
(1.27)
with \(C:=C(r,q,\Omega)\).

Lemma 1.2

[14]

Assume that \(u_{0}\in\mathcal {D}_{A_{q}}^{1-\frac{1}{r},r}\) and \(g\in L^{r}(0,T;L^{q})\) with \(1< r,q<\infty\). Then the problem
$$ \left \{ \textstyle\begin{array}{l} \partial_{t}u-\Delta u+\nabla\pi=g,\\ \operatorname {div}u=0,\\ u=0 \quad\textit{on } \partial\Omega\times(0,T),\\ u(\cdot,0)=u_{0} \quad \textit{in } \Omega, \end{array}\displaystyle \right . $$
(1.28)
has a unique solution \((u,\pi)\) satisfying the following estimate for any fixed \(T>0\):
$$ \begin{aligned}[b] &\|u\|_{C([0,T];\mathcal{D}_{A_{q}}^{1-\frac{1}{r},r})}+\|u\| _{L^{r}(0,T;W^{2,q})}+\|u_{t} \|_{L^{r}(0,T;L^{q})}+\|\nabla\pi\| _{L^{r}(0,T;L^{q})} \\ &\quad\leq C\bigl(\|u_{0}\|_{\mathcal{D}_{A_{q}}^{1-\frac{1}{r},r}}+\|g\| _{L^{r}(0,T;L^{q})}\bigr), \end{aligned}$$
(1.29)
with \(C:=C(r,q,\Omega)\).

2 Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1. To prove the existence part, we only need to show a priori estimates (1.8). The uniqueness can be proved by the standard energy method of Yudovich [19], and thus we omit the details here.

Testing (1.2) by θ and using (1.3), we see that
$$ \|\theta\|_{L^{2}}^{2}+2 \int_{0}^{T}\|\nabla\theta\|_{L^{2}}^{2} \,dt\leq\| \theta_{0}\|_{L^{2}}^{2}. $$
(2.1)
Testing (1.1) by u and (1.4) by μ, respectively, summing up the resulting equations and using (1.5), (1.3) and (2.1), we find that
$$\begin{gathered} \frac{1}{2}\frac{d}{dt} \int\bigl(|u|^{2}+|\nabla\phi|^{2}+2f(\phi)\bigr) \,dx+ \int |\nabla\mu|^{2} \,dx \\ \quad= \int\theta e_{2} u \,dx\leq\|\theta\|_{L^{2}}\|u \|_{L^{2}}\leq C\|u\| _{L^{2}}\leq C\|u\|_{L^{2}}^{2}+C, \end{gathered}$$
which gives
$$ \sup_{0\leq t\leq1} \int\bigl(|u|^{2}+|\nabla\phi|^{2}+f(\phi )\bigr) \,dx+ \int_{0}^{T} \int|\nabla\mu|^{2} \,dx \,dt\leq C. $$
(2.2)
Taking to (1.5) and testing by \(\nabla\Delta\phi\), we infer that
$$\begin{aligned} \int_{0}^{T} \int|\nabla\Delta\phi|^{2} \,dx \,dt={}&{-} \int_{0}^{T} \int\nabla\mu \cdot\nabla\Delta\phi x \,dt- \int_{0}^{T} \int\nabla\bigl(\phi-\phi^{3}\bigr)\cdot \nabla\Delta\phi \,dx \,dt \\ ={}&{-} \int_{0}^{T} \int\nabla\mu\cdot\nabla\Delta\phi \,dx \,dt- \int _{0}^{T} \int\nabla\phi\cdot\nabla\Delta\phi \,dx \,dt \\ &-3 \int_{0}^{T} \int\phi^{2}(\Delta\phi)^{2} \,dx \,dt-6 \int_{0}^{T} \int\phi |\nabla\phi|^{2}\Delta\phi \,dx \,dt \\ \leq{}&{-} \int_{0}^{T} \int\nabla\mu\cdot\nabla\Delta\phi \,dx \,dt- \int _{0}^{T} \int\nabla\phi\cdot\nabla\Delta\phi \,dx \,dt\\ &+C \int_{0}^{T} \int |\nabla\phi|^{4} \,dx \,dt \\ \leq{}&{-} \int_{0}^{T} \int\nabla\mu\cdot\nabla\Delta\phi \,dx \,dt- \int _{0}^{T} \int\nabla\phi\cdot\nabla\Delta\phi \,dx \,dt \\ &+C \int_{0}^{T}\|\nabla\phi\|_{L^{2}}^{3} \|\nabla\Delta\phi\| _{L^{2}}\,dt+C \int_{0}^{T}\|\nabla\phi\|_{L^{2}}^{4} \,dt \\ \leq{}&\frac{1}{2} \int_{0}^{T} \int|\nabla\Delta\phi|^{2} \,dx \,dt+C \int _{0}^{T} \int|\nabla\mu|^{2} \,dx \,dt \\ &+C \int_{0}^{T} \int|\nabla\phi|^{2} \,dx \,dt+C \int_{0}^{T}\|\nabla\phi\|_{L^{2}}^{6} \,dt+C \int_{0}^{T}\|\nabla\phi\| _{L^{2}}^{4} \,dt \\ \leq{}&\frac{1}{2} \int_{0}^{T} \int|\nabla\Delta\phi|^{2} \,dx \,dt+C, \end{aligned}$$
which leads to
$$ \int_{0}^{T} \int|\nabla\Delta\phi|^{2} \,dx \,dt\leq C. $$
(2.3)
Here we used the Gagliardo-Nirenberg inequality
$$ \|\nabla\phi\|_{L^{4}}\leq C\|\nabla\phi\|_{L^{2}}^{\frac{3}{4}} \|\nabla \Delta\phi\|_{L^{2}}^{\frac{1}{4}}+C\|\nabla\phi \|_{L^{2}}. $$
(2.4)
It follows from (2.2), (2.3), (1.6) and the \(H^{3}\)-regularity of the Poisson equation that
$$ \int_{0}^{T}\|\phi\|_{H^{3}}^{2} \,dt\leq C. $$
(2.5)

Denote the vorticity \(\omega:=\operatorname {rot}u:=\partial_{1}u_{2}-\partial_{2}u_{1}\) and \(a\times b:=a_{1}b_{2}-a_{2}b_{1}\) for vectors \(a:=(a_{1},a_{2})\) and \(b:=(b_{1},b_{2})\).

Applying rot to (1.1), we deduce that
$$ \partial_{t}\omega+u\cdot\nabla\omega=-\nabla\Delta\phi\times \nabla \phi+\partial_{1}\theta. $$
(2.6)
Testing (2.6) by ω and using (1.3), we get
$$ \begin{aligned}\|\omega\|_{L^{2}}\frac{d}{dt}\|\omega\|_{L^{2}}=& \int(-\nabla\Delta \phi\times\nabla\phi+\partial_{1}\theta) \omega \,dx \\ \leq&\|\omega\|_{L^{2}}\bigl(\|\nabla\Delta\phi\|_{L^{2}}\|\nabla\phi \| _{L^{\infty}}+\|\partial_{1}\theta\|_{L^{2}}\bigr), \end{aligned}$$
whence
$$\frac{d}{dt}\|\omega\|_{L^{2}}\leq\|\nabla\Delta\phi \|_{L^{2}}\| \nabla\phi\|_{L^{\infty}}+\|\partial_{1}\theta \|_{L^{2}}. $$
Integrating the above inequality, we observe that
$$ \sup_{0\leq t\leq T}\|\omega\|_{L^{2}}\leq\|\omega_{0} \| _{L^{2}}+ \int_{0}^{T}\bigl(\|\nabla\Delta\phi\|_{L^{2}}\| \nabla\phi\| _{L^{\infty}}+\|\partial_{1}\theta\|_{L^{2}}\bigr)\leq C. $$
(2.7)
Similarly, testing (2.6) by \(|\omega|^{s-2}\omega\) and using (1.3), we derive
$$\|\omega\|_{L^{s}}^{s-1}\frac{d}{dt}\|\omega \|_{L^{s}}\leq\bigl(\|\nabla \Delta\phi\|_{L^{s}}\|\nabla\phi \|_{L^{\infty}} +\|\partial_{1}\theta\| _{L^{s}}\bigr)\|\omega \|_{L^{s}}^{s-1}, $$
whence
$$\frac{d}{dt}\|\omega\|_{L^{s}}\leq\|\nabla\Delta\phi \|_{L^{s}}\| \nabla\phi\|_{L^{\infty}}+\|\partial_{1}\theta \|_{L^{s}}. $$
Integrating the above inequality, one has
$$ \sup_{0\leq t\leq T}\|\omega\|_{L^{s}}\leq\|\omega_{0} \| _{L^{s}}+ \int_{0}^{T}\bigl(\|\nabla\Delta\phi\|_{L^{s}}\| \nabla\phi\| _{L^{\infty}}+\|\partial_{1}\theta\|_{L^{s}}\bigr) \,dt. $$
(2.8)
Taking \(s\rightarrow+\infty\), we have
$$ \sup_{0\leq t\leq T}\|\omega\|_{L^{\infty}}\leq\|\omega_{0} \| _{L^{\infty}}+ \int_{0}^{T}\bigl(\|\nabla\Delta\phi\|_{L^{\infty}}\| \nabla\phi \|_{L^{\infty}} +\|\partial_{1}\theta\|_{L^{\infty}}\bigr) \,dt. $$
(2.9)
Using Lemma 1.1 with \(f:=-u\cdot\nabla\theta\), we have
$$\begin{gathered} \|\theta\|_{C([0,T];B_{q,r}^{2-\frac{2}{r}})}+\|\theta\| _{L^{r}(0,T;W^{2,q})}+\|\theta_{t} \|_{L^{r}(0,T;L^{q})} \\ \quad\leq C\|\theta_{0}\|_{B_{q,r}^{2-\frac{2}{r}}}+C\|u\cdot\nabla\theta\| _{L^{r}(0,T;L^{q})} \\ \quad\leq C+C\|u\|_{L^{\infty}(0,T;L^{q})}\|\nabla\theta\| _{L^{r}(0,T;L^{\infty})}\leq C+C\|\nabla \theta\|_{L^{r}(0,T;L^{\infty})} \\ \quad\leq C+C\epsilon\|\nabla^{2}\theta\|_{L^{r}(0,T;L^{q})}+C\|\theta\| _{L^{r}(0,T;L^{2})}, \end{gathered}$$
which yields
$$ \|\theta\|_{C([0,T];B_{q,r}^{2-\frac{2}{r}})}+\|\theta\| _{L^{r}(0,T;W^{2,q})}+\|\theta_{t} \|_{L^{r}(0,T;L^{q})}\leq C. $$
(2.10)
Here we used the interpolation inequality
$$ \|\nabla\theta\|_{L^{\infty}}\leq\epsilon\|\nabla^{2}\theta\| _{L^{q}}+C\|\theta\|_{L^{2}} $$
(2.11)
for any \(0<\epsilon<1\).
It follows from (2.10) that
$$ \|\nabla\theta\|_{L^{1}(0,T;L^{\infty})}\leq C. $$
(2.12)
Testing (1.4) by \(\Delta^{2}\phi\), using (2.2) and (2.7), we obtain that
$$\begin{gathered} \frac{1}{2}\frac{d}{dt} \int(\Delta\phi)^{2} \,dx+ \int\bigl(\Delta^{2}\phi\bigr)^{2} \,dx=- \int u\cdot\nabla\phi\Delta^{2}\phi \,dx+ \int\Delta f'(\phi )\Delta^{2}\phi \,dx \\ \quad\leq\|u\|_{L^{4}}\|\nabla\phi\|_{L^{4}}\|\Delta^{2} \phi\|_{L^{2}}+\big\| \Delta f'(\phi)\big\| _{L^{2}}\| \Delta^{2}\phi\|_{L^{2}} \\ \quad\leq C\|\nabla\phi\|_{L^{4}}\|\Delta^{2}\phi \|_{L^{2}}+C\bigl(\|\phi\| _{L^{\infty}}^{2}\|\Delta\phi \|_{L^{2}}+\|\phi\|_{L^{\infty}}\|\nabla\phi \|_{L^{4}}^{2} +\|\Delta\phi\|_{L^{2}}\bigr)\|\Delta^{2}\phi\|_{L^{2}} \\ \quad\leq C\|\nabla\phi\|_{L^{4}}\|\Delta^{2}\phi \|_{L^{2}}+C\bigl(\|\Delta\phi \|_{L^{2}}\|\Delta^{2}\phi \|_{L^{2}}^{\frac{1}{2}} +C\|\Delta\phi\|_{L^{2}}+1\bigr)\| \Delta^{2}\phi\|_{L^{2}} \\ \quad\leq\frac{1}{2}\|\Delta^{2}\phi\|_{L^{2}}^{2}+C \|\Delta\phi\|_{L^{2}}^{2}+C\| \Delta\phi\|_{L^{2}}^{4}+C, \end{gathered}$$
which implies
$$ \|\phi\|_{L^{\infty}(0,T;H^{2})}\leq C, \qquad\|\phi\|_{L^{2}(0,T;H^{4})}\leq C. $$
(2.13)
Here we used the Gagliardo-Nirenberg inequalities
$$\begin{aligned}& \|\phi\|_{L^{\infty}}\leq C\|\phi\|_{L^{2}}^{\frac{3}{4}}\| \Delta^{2}\phi\| _{L^{2}}^{\frac{1}{4}}+C\|\phi \|_{L^{2}}, \end{aligned}$$
(2.14)
$$\begin{aligned}& \|\nabla\phi\|_{L^{4}}^{2}\leq C\|\nabla\phi\|_{L^{2}} \|\Delta\phi\| _{L^{2}}+C\|\nabla\phi\|_{L^{2}}^{2}. \end{aligned}$$
(2.15)
It follows from (2.8), (2.12) and (2.13) that
$$ \|\omega\|_{L^{\infty}(0,T;L^{s})}\leq C \quad\text{for any } 2\leq s< \infty. $$
(2.16)
Testing (1.1) by \(u_{t}\) and using (1.3), (2.1), (2.13) and (2.16), we have
$$\begin{aligned}[b] \|u_{t}\|_{L^{2}}&\leq\|u\cdot\nabla u\|_{L^{2}}+\| \Delta\phi\nabla\phi \|_{L^{2}}+\|\theta\|_{L^{2}} \\ &\leq\|u\|_{L^{4}}\|\nabla u\|_{L^{4}}+\|\Delta\phi \|_{L^{2}}\|\nabla \phi\|_{L^{\infty}}+\|\theta\|_{L^{2}} \\ &\leq C+C\|\nabla\phi\|_{L^{\infty}}.\end{aligned} $$
(2.17)
Applying \(\partial_{t}\) to (1.4), testing by \(\phi_{t}\), using (1.3), (2.13), and (2.17), we reach
$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int\phi_{t}^{2} \,dx+ \int(\Delta\phi_{t})^{2} \,dx&= \int\partial_{t}f'(\phi)\Delta \phi_{t} \,dx- \int u_{t}\cdot\nabla\phi \phi_{t} \,dx \\ &= \int\bigl(3\phi^{2}\phi_{t}-\phi_{t} \bigr)\Delta\phi_{t} \,dx+ \int u_{t}\phi\nabla \phi_{t} \,dx \\ &\leq C\|\phi_{t}\|_{L^{2}}\|\Delta\phi_{t} \|_{L^{2}}+\|u_{t}\|_{L^{2}}\|\phi\| _{L^{\infty}}\| \nabla\phi_{t}\|_{L^{2}} \\ &\leq C\|\phi_{t}\|_{L^{2}}\|\Delta\phi_{t} \|_{L^{2}}+C\bigl(1+\|\nabla\phi\| _{L^{\infty}}\bigr)\|\nabla\phi_{t} \|_{L^{2}} \\ &\leq\frac{1}{2}\|\Delta\phi_{t}\|_{L^{2}}^{2}+C \|\phi_{t}\|_{L^{2}}^{2}+C+C\| \nabla\phi \|_{L^{\infty}}^{2}, \end{aligned}$$
which gives
$$ \|\phi_{t}\|_{L^{\infty}(0,T;L^{2})}\leq C,\qquad \|\phi_{t}\| _{L^{2}(0,T;H^{2})}\leq C. $$
(2.18)
Here we used the inequality
$$\|\nabla\phi_{t}\|_{L^{2}}\leq C\|\Delta\phi_{t} \|_{L^{2}} $$
due to the inequality
$$\|v\|_{L^{2}}\leq C\|\operatorname {div}v\|_{L^{2}}+C\|\operatorname {rot}v\|_{L^{2}} $$
for \(v=\nabla\phi_{t}\) and \(v\cdot n=0\) on Ω.
By the standard \(H^{s}\)-regularity theory of elliptic equations, it follows from (1.4), (1.5), (2.13), (2.16) and (2.18) that
$$ \|\phi\|_{L^{\infty}(0,T;H^{4})}+\|\phi\|_{L^{2}(0,T;H^{5})}\leq C, $$
(2.19)
whence
$$ \|\nabla\Delta\phi\|_{L^{2}(0,T;L^{\infty})}\leq C. $$
(2.20)
It follows from (2.9), (2.12) and (2.20) that
$$ \|\omega\|_{L^{\infty}(0,T;L^{\infty})}\leq C. $$
(2.21)

This completes the proof. □

3 Proof of Theorem 1.2

This section is devoted to the proof of Theorem 1.2. To prove the existence part, we only need to show a priori estimates (1.17).

First, testing (1.10) by \(|\theta|^{q-2}\theta\) and using (1.11), we see that
$$ \|\theta\|_{L^{\infty}(0,T;L^{q})}\leq\|\theta_{0}\|_{L^{q}}. $$
(3.1)

Next, we still have (2.2) and (2.5).

In the following proofs, we will use the Gagliardo-Nirenberg inequalities
$$\begin{aligned}& \|\nabla\phi\|_{L^{4}}\leq C\|\nabla\phi\|_{L^{2}}^{\frac{3}{4}} \|\phi\| _{H^{3}}^{\frac{1}{4}}, \end{aligned}$$
(3.2)
$$\begin{aligned}& \|\Delta\phi\|_{L^{4}}\leq C\|\nabla\phi\|_{L^{2}}^{\frac{1}{4}} \|\phi\| _{H^{3}}^{\frac{3}{4}}. \end{aligned}$$
(3.3)
Denoting \(\tilde{\pi}:=\pi-f(\phi)\), testing (1.9) by \(\nabla \tilde{\pi}-\Delta u\), using (3.2), (3.3), (2.2), (2.5) and (3.1), we find that
$$\begin{gathered} \frac{1}{2}\frac{d}{dt} \int|\nabla u|^{2} \,dx+ \int|\nabla\tilde{\pi}-\Delta u|^{2} \,dx \\ \quad= \int(\Delta\phi\nabla\phi+\theta e_{2}-u\cdot\nabla u) (\nabla \tilde{\pi}-\Delta u)\,dx \\ \quad\leq\bigl(\|\Delta\phi\|_{L^{4}}\|\nabla\phi\|_{L^{4}}+\|\theta\| _{L^{2}}+\|u\|_{L^{4}}\|\nabla u\|_{L^{4}}\bigr)\|\nabla\tilde{\pi}-\Delta u\| _{L^{2}} \\ \quad\leq C\bigl(\|\phi\|_{H^{3}}+1+\|u\|_{L^{2}}^{\frac{1}{2}}\| \nabla u\|_{L^{2}}^{\frac{1}{2}}\cdot\|\nabla u\|_{L^{2}}^{\frac{1}{2}} \|\nabla\tilde{\pi}-\Delta u\| _{L^{2}}^{\frac{1}{2}}\bigr)\|\nabla \tilde{\pi}-\Delta u\|_{L^{2}} \\ \quad\leq\frac{1}{2}\|\nabla\tilde{\pi}-\Delta u\|_{L^{2}}^{2}+C \|\phi\| _{H^{3}}^{2}+C+C\|\nabla u\|_{L^{2}}^{4}, \end{gathered}$$
which gives
$$ \|u\|_{L^{\infty}(0,T;H^{1})}+\|u\|_{L^{2}(0,T;H^{2})}\leq C. $$
(3.4)
Here we used the \(H^{2}\)-estimates of the Stokes system
$$ \|u\|_{H^{2}}\leq C\|\nabla\tilde{\pi}-\Delta u\|_{L^{2}}. $$
(3.5)

We still have (2.13).

It follows from (1.9), (3.1), (3.4) and (2.13) that
$$ \|u_{t}\|_{L^{2}(0,T;L^{2})}\leq C. $$
(3.6)

We still have (2.19).

Using Lemma 1.2 with \(g:=\theta e_{2}+\Delta\phi\nabla\phi -u\cdot\nabla u\) and \(\tilde{\pi}:=\pi-f(\phi)\), we have
$$\begin{gathered} \|u\|_{C([0,T];\mathcal{D}_{A_{q}}^{1-\frac{1}{r},r})}+\|u\| _{L^{r}(0,T;W^{2,q})}+\|u_{t}\|_{L^{r}(0,T;L^{q})} \\ \quad\leq C\bigl(\|u_{0}\|_{\mathcal{D}_{A_{q}}^{1-\frac{1}{r},r}}+\|u\cdot\nabla u\| _{L^{r}(0,T;L^{q})}+ \|\Delta\phi\cdot\nabla\phi\|_{L^{r}(0,T;L^{q})}+\| \theta\|_{L^{r}(0,T;L^{q})}\bigr) \\ \quad\leq C+C\|u\cdot\nabla u\|_{L^{r}(0,T;L^{q})} \\ \quad\leq C+C\|u\|_{L^{\infty}(0,T;L^{q})}\|\nabla u\|_{L^{r}(0,T;L^{\infty})} \\ \quad\leq C+C\|\nabla u\|_{L^{r}(0,T;L^{\infty})} \\ \quad\leq C+C\epsilon\|\nabla^{2}u\|_{L^{r}(0,T;L^{q})}+C\|u\|_{L^{r}(0,T;L^{q})}, \end{gathered}$$
which gives
$$ \|u\|_{C([0,T];\mathcal{D}_{A_{q}}^{1-\frac{1}{r},r})}+\|u\| _{L^{r}(0,T;W^{2,q})}+\|u_{t}\|_{L^{r}(0,T;L^{q})} \leq C. $$
(3.7)
Here we used inequality (2.11) for \(\theta=u\).

This completes the proof of (1.17).

Now we are in a position to prove the uniqueness part. To this end, let \((u_{i}, \pi_{i}, \theta_{i}, \phi_{i})\) (\(i=1,2\)) be two solutions to problem (1.9)-(1.15), set
$$\begin{gathered} \delta u:=u_{1}-u_{2},\qquad \delta\pi:=\pi_{1}- \pi_{2}, \qquad\delta\theta:=\theta _{1}-\theta_{2},\\ \delta\phi:=\phi_{1}-\phi_{2}, \qquad\tilde{\pi}_{i}=\pi _{i}+f(\phi_{i}), \qquad\delta\tilde{\pi}:=\tilde{\pi}_{1}- \tilde{\pi}_{2} \end{gathered}$$
and define ξ satisfying
$$ \begin{gathered} -\Delta\xi=\delta\theta,\\ \xi=0 \quad\text{on } \partial\Omega\times(0,\infty). \end{gathered} $$
(3.8)
Then \((\delta u, \delta\theta, \delta\phi)\) satisfy
$$\begin{aligned}& \partial_{t}\delta u+u_{1}\cdot\nabla\delta u+\delta u \nabla u_{2}+\nabla\delta\tilde{\pi}-\Delta\delta u=\Delta \phi_{1}\nabla \delta\phi+\Delta\delta\phi\nabla\phi_{2}+ \delta\theta e_{2}, \end{aligned}$$
(3.9)
$$\begin{aligned}& \partial_{t}\delta\theta+u_{1}\cdot\nabla\delta\theta+ \delta u\cdot\nabla\theta_{2}=0, \end{aligned}$$
(3.10)
$$\begin{aligned}& \partial_{t}\delta\phi+u_{1}\cdot\nabla\delta\phi+\delta u\cdot \nabla\phi_{2}=-\Delta^{2}\delta\phi+\Delta \bigl(f'(\phi_{1})-f'(\phi _{2}) \bigr). \end{aligned}$$
(3.11)
Testing (3.9) by δu and using (1.17) and (1.11), we derive
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int|\delta u|^{2} \,dx+ \int|\nabla\delta u|^{2}\,dx \\& \quad=- \int\delta u\cdot\nabla u_{2}\cdot\delta u \,dx+ \int\Delta\phi _{1}\cdot\nabla\delta\phi\cdot\delta u \,dx \\& \qquad{}+ \int\Delta\delta\phi\nabla\phi_{2}\cdot\delta u \,dx- \int\Delta \xi e_{2}\delta u \,dx \\& \quad\leq\|\nabla u_{2}\|_{L^{2}}\|\delta u\|_{L^{4}}^{2}+ \|\Delta\phi_{1}\| _{L^{\infty}}\|\nabla\delta\phi\|_{L^{2}}\| \delta u\|_{L^{2}} \\& \qquad{}+\|\nabla\phi_{2}\|_{L^{\infty}}\|\Delta\delta\phi \|_{L^{2}}\|\delta u\|_{L^{2}}+\|\nabla\xi\|_{L^{2}}\|\nabla \delta u\|_{L^{2}} \\& \quad\leq C\|\delta u\|_{L^{4}}^{2}+C\|\nabla\delta\phi \|_{L^{2}}\|\delta u\| _{L^{2}}+C\|\Delta\delta\phi\|_{L^{2}} \|\delta u\|_{L^{2}}+\|\nabla\xi\| _{L^{2}}\|\nabla\delta u \|_{L^{2}} \\& \quad\leq\frac{1}{8}\|\nabla\delta u\|_{L^{2}}^{2}+C\| \delta u\|_{L^{2}}^{2}+C\| \delta\phi\|_{L^{2}}^{2}+ \frac{1}{8}\|\Delta\delta\phi\|_{L^{2}}^{2}+C\| \nabla\xi \|_{L^{2}}^{2}. \end{aligned}$$
(3.12)
Testing (3.10) by ξ and using (1.17) and (1.11), we obtain
$$ \begin{aligned}[b] \frac{1}{2}\frac{d}{dt} \int|\nabla\xi|^{2} \,dx&= \int u_{1}\nabla\Delta \xi\cdot\xi \,dx- \int\delta u\nabla\theta_{2}\xi \,dx \\ &=- \int u_{1}\Delta\xi\nabla\xi \,dx+ \int\delta u\theta_{2}\nabla\xi \,dx \\ &=-\sum_{i,j} \int\partial_{j}u_{1i}\partial_{i}\xi \partial _{j}\xi \,dx+ \int\delta u\theta_{2}\nabla\xi \,dx \\ &\leq C\|\nabla u_{1}\|_{L^{\infty}}\|\nabla\xi\|_{L^{2}}^{2}+ \|\theta_{2}\| _{L^{q}}\|\delta u\|_{L^{\frac{2q}{q-2}}}\|\nabla\xi \|_{L^{2}} \\ &\leq C\|\nabla u_{1}\|_{L^{\infty}}\|\nabla\xi\|_{L^{2}}^{2}+C \|\delta u\| _{L^{2}}^{1-\frac{3}{q}}\|\nabla\delta u\|_{L^{2}}^{\frac{2}{q}} \|\nabla\xi\| _{L^{2}} \\ &\leq\frac{1}{8}\|\nabla\delta u\|_{L^{2}}^{2}+C\| \nabla u_{1}\|_{L^{\infty}}\| \nabla\xi\|_{L^{2}}^{2}+C \|\nabla\xi\|_{L^{2}}^{2}+C\|\delta u\| _{L^{2}}^{2}. \end{aligned}$$
(3.13)
Testing (3.11) by δϕ and using (1.17) and (1.11), we have
$$ \begin{aligned}[b] &\frac{1}{2}\frac{d}{dt} \int(\delta\phi)^{2} \,dx+ \int(\Delta\delta \phi)^{2} \,dx \\ &\quad=- \int\delta u\cdot\nabla\phi_{2}\cdot\delta\phi \,dx+ \int\bigl(f'(\phi _{1})-f'( \phi_{2})\bigr)\Delta\delta\phi \,dx \\ &\quad\leq\|\nabla\phi_{2}\|_{L^{\infty}}\|\delta u\|_{L^{2}}\| \delta\phi\| _{L^{2}}+C\|\delta\phi\|_{L^{2}}\|\Delta\delta\phi \|_{L^{2}} \\ &\quad\leq C\|\delta u\|_{L^{2}}^{2}+C\|\delta\phi \|_{L^{2}}^{2}+\frac{1}{8}\|\Delta \delta\phi \|_{L^{2}}^{2}. \end{aligned}$$
(3.14)
Summing up (3.12), (3.13) and (3.14), and using the Gronwall inequality, we conclude that
$$\delta u=0,\qquad\xi=0 \quad\text{and} \quad\delta\phi=0. $$

This completes the proof. □

4 Concluding remarks

The Cahn-Hilliard-Boussinesq system and a related system play an important role in the mathematical study of multi-phase flows. The applications of these systems cover a very wide range of physical objects, such as complicated phenomena in fluid mechanics involving phase transition, two-phase flow under shear through an order parameter formulation, the spinodal decomposition of binary fluid in a Hele-Shaw cell, tumor growth, cell sorting, and two phase flows in porous media.

In this paper, we have obtained the following global well-posedness results:
  1. (1)

    If initial data \(\phi_{0}\in H^{4}\), \(u_{0}\in L^{2}\), \(\operatorname {rot}u_{0}\in L^{\infty}\) and \(\theta_{0}\in B_{q,r}^{2-\frac{2}{r}}\) with \(1< r<\infty\) and \(2< q<\infty\), then problem (1.1)-(1.7) admits a unique global solution.

     
  2. (2)

    If initial data \(u_{0}\in\mathcal{D}_{A_{q}}^{1-\frac{1}{r},r}\cap H_{0}^{1}\) with \(1< r<\infty\), \(2< q<\infty\) and \(\theta_{0}\in L^{q}\), \(\phi_{0}\in H^{4}\), then problem (1.9)-(1.15) admits a unique global solution.

     
  3. (3)

    If initial data \(u_{0}\in L^{2}\), \(\operatorname {rot}u_{0}\in L^{\infty}\) and \(n_{0},p_{0}\in B_{q,r}^{2-\frac{2}{r}}\) with \(1< r<\infty\) and \(2< q<\infty\) and \(n_{0}, p_{0}\geq0\) in Ω and \(D\in L^{\infty}(\Omega)\), then problem (1.18)-(1.24) admits a unique global solution.

     

Declarations

Acknowledgements

CM is partially supported by NSFC (Grant No. 11661070) and the Scientific Research Foundation of the Higher Education Institutions of Gansu Province (Grant No. 2016B-077).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Tianshui Normal University, Tianshui, P.R. China
(2)
Department of Mathematics, Zhejiang Normal University, Jinhua, P.R. China
(3)
Center for Information, Nanjing Forestry University, Nanjing, P.R. China
(4)
Department of Applied Mathematics, Nanjing Forestry University, Nanjing, P.R. China

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