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.