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.
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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]:
in \(\Omega\times(0,\infty)\) with the boundary and initial conditions
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
for any fixed \(T>0\).
Next, we consider the following Cahn-Hilliard-Boussinesq system:
When \(\phi=0\), Zhou [11] showed the global well-posedness of the problem with rough initial data
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
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
for any fixed \(T>0\).
Finally, we consider the following model in electrohydrodynamics [15]:
in \(\Omega\times(0,\infty)\) with the boundary and initial conditions
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
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
has a unique solution θ satisfying the following inequality for any fixed \(T>0\):
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
has a unique solution \((u,\pi)\) satisfying the following estimate for any fixed \(T>0\):
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
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
which gives
Taking ∇ to (1.5) and testing by \(\nabla\Delta\phi\), we infer that
which leads to
Here we used the Gagliardo-Nirenberg inequality
It follows from (2.2), (2.3), (1.6) and the \(H^{3}\)-regularity of the Poisson equation that
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
Testing (2.6) by ω and using (1.3), we get
whence
Integrating the above inequality, we observe that
Similarly, testing (2.6) by \(|\omega|^{s-2}\omega\) and using (1.3), we derive
whence
Integrating the above inequality, one has
Taking \(s\rightarrow+\infty\), we have
Using Lemma 1.1 with \(f:=-u\cdot\nabla\theta\), we have
which yields
Here we used the interpolation inequality
for any \(0<\epsilon<1\).
It follows from (2.10) that
Testing (1.4) by \(\Delta^{2}\phi\), using (2.2) and (2.7), we obtain that
which implies
Here we used the Gagliardo-Nirenberg inequalities
It follows from (2.8), (2.12) and (2.13) that
Testing (1.1) by \(u_{t}\) and using (1.3), (2.1), (2.13) and (2.16), we have
Applying \(\partial_{t}\) to (1.4), testing by \(\phi_{t}\), using (1.3), (2.13), and (2.17), we reach
which gives
Here we used the inequality
due to the inequality
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
whence
It follows from (2.9), (2.12) and (2.20) that
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
Next, we still have (2.2) and (2.5).
In the following proofs, we will use the Gagliardo-Nirenberg inequalities
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
which gives
Here we used the \(H^{2}\)-estimates of the Stokes system
We still have (2.13).
It follows from (1.9), (3.1), (3.4) and (2.13) that
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
which gives
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
and define ξ satisfying
Then \((\delta u, \delta\theta, \delta\phi)\) satisfy
Testing (3.9) by δu and using (1.17) and (1.11), we derive
Testing (3.10) by ξ and using (1.17) and (1.11), we obtain
Testing (3.11) by δϕ and using (1.17) and (1.11), we have
Summing up (3.12), (3.13) and (3.14), and using the Gronwall inequality, we conclude that
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)
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)
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)
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.
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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).
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Ma, C., Gu, W. & Sun, J. Global well-posedness for the 2D Cahn-Hilliard-Boussinesq and a related system on bounded domains. Bound Value Probl 2017, 119 (2017). https://doi.org/10.1186/s13661-017-0850-5
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DOI: https://doi.org/10.1186/s13661-017-0850-5