Weak and very weak solutions to the viscous Cahn–Hilliard–Oberbeck–Boussinesq phase-field system on two-dimensional bounded domains

Abstract

In this paper, we consider weak and very weak solutions to the viscous Cahn–Hilliard–Oberbeck–Boussinesq system for non-isothermal, viscous and incompressible binary fluid flows in two-dimensional bounded domains. The source functions have low spatial regularities, and the initial data belong to some interpolation spaces. The essential tools employed in the analysis are the extended maximal parabolic regularity for the associated linearized system and the well-posedness of the nonlinear part with the solution of the linearized dynamics as the frozen coefficients. We resolve the linear system by decomposition into the viscous biharmonic heat, Stokes, and heat equations. A spectral Faedo–Galerkin framework shall be pursued for the nonlinear part. Higher integrability with respect to time will be established using interpolation and compactness methods.

Appendix

1.1 A space–time version of de Rham’s theorem

We prove a space–time version of the classical de Rham’s theorem. The following proposition is an extension of the one stated in [27, Lemma 72.8], in particular, the case where $p = r = 2$ and $k=1$.

Proposition 7.1

Let $p, r \in (1, \infty )$ and k be a positive integer. Then ${\mathfrak {L}} \in W^{-1, r}(I; \varvec{W}^{-k,p}(\Omega ))$ satisfies

$$\begin{aligned} \langle {\mathfrak {L}}, \varvec{\varrho } \rangle _{ W^{-1, r}(I; \varvec{W}^{-k,p}(\Omega )), W_0^{1, r'}(I; \varvec{W}_0^{k,p'}(\Omega ))} = 0 \qquad \forall \, \varvec{\varrho } \in W_0^{1, r'}(I; \varvec{W}_0^{k,p'}(\Omega ) \cap \varvec{L}_\sigma ^{p'}(\Omega )) \end{aligned}$$

if and only if there exists a unique ${\mathfrak {p}} \in W^{-1,r}(I; {\widehat{W}}^{1-k,p}(\Omega ))$ such that ${\mathfrak {L}} = \nabla {\mathfrak {p}}$ in the distributional sense, that is,

$$\begin{aligned}&\langle {\mathfrak {L}}, \varvec{\rho } \rangle _{ W^{-1, r}(I; \varvec{W}^{-k,p}(\Omega )), W_0^{1, r'}(I; \varvec{W}_0^{k,p'}(\Omega ))} \\&\quad = - \langle {\mathfrak {p}}, \mathrm{div\,}\varvec{\rho } \rangle _{ W^{-1,r}(I; {\widehat{W}}^{1-k,p}(\Omega )), W_0^{1, r'}(I; {\widehat{W}}_0^{k-1, p'}(\Omega ))} \end{aligned}$$

for every $\varvec{\rho } \in W_0^{1, r'}(I; \varvec{W}_{0}^{k,p'}(\Omega ))$. In this case, there exists a constant $c > 0$ such that

$$\begin{aligned} \Vert {\mathfrak {p}}\Vert _{W^{-1,r}(I; {\widehat{W}}^{1-k,p}(\Omega ))} \le c \Vert {\mathfrak {L}}\Vert _{W^{-1, r}(I; \varvec{W}^{-k,p}(\Omega ))}. \end{aligned}$$

Proof

We proceed by a duality argument. First, let us note that the linear operator

$$\begin{aligned} \mathrm{div\,}: L^{r'}(I; \varvec{W}_{0}^{k,p'}(\Omega )) \rightarrow L^{r'}(I; {\widehat{W}}_0^{k-1, p'}(\Omega )) \end{aligned}$$

is bounded and surjective, see for instance Lemma II.2.1.1 and Lemma II.2.3.1 in [62] for the time-independent case. We claim that the restriction

$$\begin{aligned} \widetilde{\mathrm{div}} := \mathrm{div\,}: W_0^{1, r'}(I; \varvec{W}_{0}^{k,p'}(\Omega )) \rightarrow W_0^{1, r'}(I; {\widehat{W}}^{k-1,p'}_0(\Omega )) \end{aligned}$$

(7.1)

is also bounded and surjective. It is clear that (7.1) is well-defined, linear and bounded.

Let $g \in W_0^{1, r'}(I; {\widehat{W}}_0^{k-1,p'}(\Omega )) \hookrightarrow C({\bar{I}}; {\widehat{W}}_0^{k-1,p'}(\Omega )) \hookrightarrow L^{r'}(I; {\widehat{W}}_0^{k-1, p'}(\Omega ))$. Then there is a $\varvec{v} \in L^{r'}(I; \varvec{W}_{0}^{k,p'}(\Omega ))$ such that $\mathrm{div\,}\varvec{v} = \partial _t g$ almost everywhere in Q. For each $t \in [0, T]$, let us define

$$\begin{aligned} \varvec{w}(t) := \frac{T-t}{T}\int \nolimits _0^T \varvec{v}(s) \mathrm{\,d}s - \int \nolimits _t^T \varvec{v}(s) \mathrm{\,d}s. \end{aligned}$$

It is easy to see that $\varvec{w} \in W_0^{1, r'}(I; \varvec{W}_{0}^{k,p'}(\Omega ))$, and for all $t \in [0, T]$ we have

$$\begin{aligned}&\widetilde{\mathrm{div}}\, \varvec{w}(t) \\&\quad = \frac{T-t}{T} \int \nolimits _0^T \partial _t g(s) \mathrm{\,d}s- \int \nolimits _t^T \partial _t g(s) \mathrm{\,d}s = g(t) \end{aligned}$$

since $g(0) = g(T) = 0$ in ${\widehat{W}}_0^{k-1,p'}(\Omega )$. We point out that the insertion of the divergence operator inside the integral is valid since $\mathrm{div}$ is linear and continuous, see for instance [48, Chap. III, Theorem 3.7.12]. This shows that the map (7.1) is surjective.

It follows from the closed range theorem [67, page 205] that the dual operator $-\nabla = \widetilde{\mathrm{div}}' : W^{-1, r}(I; {\widehat{W}}^{1-k, p}(\Omega )) \rightarrow W^{-1, r}(I; \varvec{W}^{-k,p}(\Omega ))$ has a trivial kernel and a range $\text {Ran}(-\nabla )$ that is closed with respect to the topology of $W^{-1, r}(I; \varvec{W}^{-k,p}(\Omega ))$. As a consequence, the inverse $(- \nabla )^{-1}$ is a well-defined, linear and bounded operator from $\text {Ran}(-\nabla )$ onto $W^{-1, r}(I; {\widehat{W}}^{1-k,p}(\Omega ))$. Thus, if ${\mathfrak {L}} \in W^{-1, r}(I; \varvec{W}^{-k,p}(\Omega ))$ vanishes on $W_0^{1, r'}(I; \varvec{W}_0^{k,p'}(\Omega ) \cap \varvec{L}_\sigma ^{p'}(\Omega )) = \mathrm{Ker}(\widetilde{\mathrm{div}})$, then ${\mathfrak {L}} \in \mathrm{Ker}(\widetilde{\mathrm{div}})^\perp = \text {Ran}(-\nabla )$. Therefore, we may take ${\mathfrak {p}} = - (-\nabla )^{-1}{\mathfrak {L}} = \nabla ^{-1}{\mathfrak {L}}\in W^{-1, r}(I; {\widehat{W}}^{1-k,p}(\Omega ))$ with norm

$$\begin{aligned}&\Vert {\mathfrak {p}}\Vert _{W^{-1, r}(I; {\widehat{W}}^{1-k,p}(\Omega ))} \le \Vert (-\nabla )^{-1}\Vert _{{\mathcal {L}}(\text {Ran}(-\nabla ), W^{-1, r}(I; {\widehat{W}}^{1-k,p}(\Omega )))} \Vert {\mathfrak {L}}\Vert _{W^{-1, r}(I; \varvec{W}^{-k,p}(\Omega ))}. \end{aligned}$$

The converse of the first statement in the proposition is trivial. $\square$

1.2 Analyticity of the semigroup for the linearized system

In the following, we prove that the linear operator $-\varvec{{\mathcal {A}}}$ generates a strongly continuous analytic semigroup on $\varvec{{\mathcal {H}}}_\omega$, where $\varvec{{\mathcal {A}}}$ is defined by (3.30). The sesqui-linear form associated with $\varvec{{\mathcal {A}}}$ is given by

$$\begin{aligned}&(\varvec{{\mathcal {A}}}(\phi , \gamma , \varvec{u}), (\psi , \eta , \varvec{v}))_{\varvec{{\mathcal {H}}}_\omega } =\varvec{{\mathfrak {a}}}((\phi , \gamma , \varvec{u}), (\psi , \eta , \varvec{v})) \end{aligned}$$

where $\varvec{{\mathfrak {a}}} = \varvec{{\mathfrak {a}}}_1 + \varvec{{\mathfrak {a}}}_2$ and

$$\begin{aligned} \varvec{{\mathfrak {a}}}_1((\phi , \gamma , \varvec{u}), (\psi , \eta , \varvec{v}))&:= \int \nolimits _\Omega \{\omega m^2\tau \epsilon \nabla \Delta \phi \cdot \nabla \Delta {\overline{\psi }} + \kappa \nabla \gamma \cdot \nabla {\overline{\eta }} + \nu \nabla \varvec{u} : \nabla \overline{\varvec{v}} \} \mathrm {\,d}x \\ \varvec{{\mathfrak {a}}}_2((\phi , \gamma , \varvec{u}), (\psi , \eta , \varvec{v}))&:= - \int \nolimits _\Omega \omega m[(\beta _1 - l_{\mathrm {c}} l_{\mathrm {h}})\Delta \phi - l_{\mathrm {c}}\Delta \gamma ] (m\tau \Delta {\overline{\psi }} - {\overline{\psi }}) \mathrm {\,d}x \\&\quad - \int \nolimits _{\Omega } \{ (\kappa l_{\mathrm {h}} \Delta \phi ) {\overline{\eta }} + (\alpha \varvec{g} \cdot \varvec{u}) {\overline{\eta }} + [(\alpha _1 + \alpha _2 l_{\mathrm {h}})\phi + \alpha _2 \gamma ] \varvec{g} \cdot \overline{\varvec{v}}\} \mathrm {\,d}x. \end{aligned}$$

Given $\beta \in (0, \pi )$, we denote the sector $\Sigma _\beta := \{\zeta \in {\mathbb {C}} \setminus \{0\} : |\text {arg}\, \zeta | < \pi - \beta \}$. First, we prove the following elementary inequality.

Lemma 7.2

For each $\beta \in (0, \pi )$, there exists $\tau _\beta > 0$ such that for every $a, b \ge 0$ and $\zeta \in \Sigma _\beta$ there holds $|a\zeta + b| \ge \tau _\beta (a|\zeta | + b)$.

Proof

Suppose $a, b > 0$. Setting $z = a\zeta /b$, it suffices to show that $|z + 1| \ge \tau _\beta (|z| + 1)$ for every $z \in \Sigma _\beta$. Write z in its polar form $z = re^{i\vartheta }$ where $|\vartheta | < \pi - \beta$ and $r > 0$. Let $\delta _\beta := \cos (\pi - \beta ) > -1$. Then

$$\begin{aligned} \frac{|z + 1|^2}{(|z| + 1)^2} = \frac{r^2 + 2r \cos \vartheta + 1}{(r + 1)^2} \ge 1 - \frac{2(1-\delta _\beta )r}{(r+1)^2} \ge 1 - \frac{1-\delta _\beta }{2} =: c_\beta \end{aligned}$$

for every $r > 0$, where $c_\beta > 0$. We may then take $\tau _\beta = \min \{1, c_\beta ^{1/2}\}$, and this clearly covers the case when $a = 0$ or $b = 0$. $\square$

Proposition 7.3

For small enough $\omega > 0$, the linear operator $-\varvec{{\mathcal {A}}} : D(\varvec{{\mathcal {A}}}) \subset \varvec{{\mathcal {H}}}_\omega \rightarrow \varvec{{\mathcal {H}}}_\omega$ generates an analytic $C_0$-semigroup on $\varvec{{\mathcal {H}}}_\omega$.

Proof

It is clear that $\varvec{{\mathcal {A}}}$ is a closed and densely defined linear operator. Let $\delta > 0$ be a constant to be chosen later. Applying integration by parts and Young’s inequality, it is not hard to see that for each $(\phi , \gamma , \varvec{u}) \in X^{3,2}(\Omega ) \times W_0^{1,2}(\Omega ) \times \varvec{X}_{\sigma }^{1,2}(\Omega )$, we have

$$\begin{aligned} \varvec{{\mathfrak {a}}}_1((\phi , \gamma , \varvec{u}), (\phi , \gamma , \varvec{u}))&\ge c \{\omega \Vert \phi \Vert ^2_{X^{3,2}(\Omega )} + \Vert \gamma \Vert ^2_{W_0^{1,2}(\Omega )} + \Vert \varvec{u}\Vert ^2_{\varvec{X}_{\sigma }^{1,2}(\Omega )} \}\\ |\varvec{{\mathfrak {a}}}_2((\phi , \gamma , \varvec{u}), (\phi , \gamma , \varvec{u}))|&\le c_{\omega }\Vert (\phi , \gamma , \varvec{u})\Vert ^2_{\varvec{{\mathcal {H}}}_\omega } + \omega \{\delta \Vert \phi \Vert _{X^{3,2}(\Omega )}^2 + c_\delta \Vert \gamma \Vert ^2_{W_0^{1,2}(\Omega )}\} \end{aligned}$$

where $c = \min \{m^2\tau \epsilon , \kappa , \nu \} > 0$ and $c_{\omega }, c_\delta > 0$ are independent of $(\phi , \gamma , \varvec{u})$. Let $\beta \in (0, \pi )$ be fixed. If $\varpi \ge 0$ and $\zeta \in \Sigma _\beta$, then by invoking the estimate in the previous lemma, we obtain

$$\begin{aligned}&|(\zeta + \varpi )\Vert (\phi , \gamma , \varvec{u})\Vert ^2_{\varvec{{\mathcal {H}}}_\omega } + \varvec{{\mathfrak {a}}}((\phi , \gamma , \varvec{u}), (\phi , \gamma , \varvec{u})) | \\&\quad \ge |(\zeta + \varpi )\Vert (\phi , \gamma , \varvec{u})\Vert ^2_{\varvec{{\mathcal {H}}}_\omega } + \varvec{{\mathfrak {a}}}_1((\phi , \gamma , \varvec{u}), (\phi , \gamma , \varvec{u})) | \\&\qquad - |\varvec{{\mathfrak {a}}}_2((\phi , \gamma , \varvec{u}), (\phi , \gamma , \varvec{u}))| \\&\quad \ge \tau _\beta \{(|\zeta | + \varpi )\Vert (\phi , \gamma , \varvec{u})\Vert ^2_{\varvec{{\mathcal {H}}}_\omega } + \varvec{{\mathfrak {a}}}_1((\phi , \gamma , \varvec{u}), (\phi , \gamma , \varvec{u}))\}\\&\qquad - |\varvec{{\mathfrak {a}}}_2((\phi , \gamma , \varvec{u}), (\phi , \gamma , \varvec{u}))| \\&\quad \ge \{\tau _\beta (|\zeta | + \varpi ) - c_{\omega }\}\Vert (\phi , \gamma , \varvec{u})\Vert ^2_{\varvec{{\mathcal {H}}}_\omega } + c_{\omega ,\delta ,\beta } \Vert (\phi , \gamma , \varvec{u})\Vert ^2_{ X^{3,2}(\Omega ) \times W_0^{1,2}(\Omega ) \times \varvec{X}_{\sigma }^{1,2}(\Omega )} \end{aligned}$$

where $c_{\omega , \delta ,\beta } = \min \{\omega (c \tau _\beta - \delta ), c\tau _\beta - \omega c_\delta \}$. Taking $0< \delta < c \tau _\beta$, $0< \omega < c\tau _\beta / c_\delta$, and $\varpi \ge c_{\omega }/\tau _\beta > 0$, we have $c_{\omega , \delta ,\beta } > 0$ and

$$\begin{aligned}&|(\zeta + \varpi )\Vert (\phi , \gamma , \varvec{u})\Vert ^2_{\varvec{{\mathcal {H}}}_\omega } + \varvec{{\mathfrak {a}}}((\phi , \gamma , \varvec{u}), (\phi , \gamma , \varvec{u}))| \nonumber \\&\quad \ge \tau _\beta |\zeta |\Vert (\phi , \gamma , \varvec{u})\Vert ^2_{\varvec{{\mathcal {H}}}_\omega } + c_{\omega , \delta ,\beta } \Vert (\phi , \gamma , \varvec{u})\Vert ^2_{ X^{3,2}(\Omega ) \times W_0^{1,2}(\Omega ) \times \varvec{X}_{\sigma }^{1,2}(\Omega )}. \end{aligned}$$

(7.2)

Thus, the sesqui-linear form $(\zeta + \varpi )(\cdot , \cdot )_{\varvec{{\mathcal {H}}}_\omega } + \varvec{{\mathfrak {a}}}$ is bounded and coercive on $X^{3,2}(\Omega ) \times W_0^{1,2}(\Omega ) \times \varvec{X}_{\sigma }^{1,2}(\Omega )$.

For each $(\sigma , h, \varvec{f}) \in X^{2,2}(\Omega ) \times L^2(\Omega ) \times \varvec{L}_\sigma ^2(\Omega )$ the following variational equation for all $(\psi ,\eta , \varvec{v}) \in X^{3,2}(\Omega ) \times W_0^{1,2}(\Omega ) \times \varvec{X}_{\sigma }^{1,2}(\Omega )$

$$\begin{aligned}&(\zeta + \varpi )((\phi , \gamma , \varvec{u}), (\psi , \eta , \varvec{v}))_{\varvec{{\mathcal {H}}}_\omega } + \varvec{{\mathfrak {a}}}((\phi , \gamma , \varvec{u}), (\psi , \eta , \varvec{v})) \nonumber \\&\quad = ((\sigma , h, \varvec{f}), (\psi , \eta , \varvec{v}))_{\varvec{{\mathcal {H}}}_\omega } \end{aligned}$$

(7.3)

admits a unique solution $(\phi , \gamma , \varvec{u}) \in X^{3,2}(\Omega ) \times W_0^{1,2}(\Omega ) \times \varvec{X}_{\sigma }^{1,2}(\Omega )$ in virtue of the Lax–Milgram lemma. Moreover, it follows from the definition of $\varvec{{\mathcal {A}}}$ that $(\phi , \gamma , \varvec{u})$ is a weak solution to the following system of boundary value problems:

By classical elliptic regularity theory for the Poisson and stationary Stokes equations, we have $\gamma \in X^{2,2}(\Omega )$ and $\varvec{u} \in \varvec{X}_\sigma ^{2,2}(\Omega )$. Thus, we also have $\phi \in X^{4,2}(\Omega )$ for the solution of the above bi-Laplace equation since $\sigma - m\tau \Delta \sigma + ml_{\mathrm {c}} \Delta \gamma \in L^2(\Omega )$. Consequently, it holds that $(\phi , \gamma , \varvec{u}) \in D(\varvec{{\mathcal {A}}})$.

The variational equation (7.3) is equivalent to $[\zeta \varvec{I} + (\varpi \varvec{I} + \varvec{{\mathcal {A}}})](\phi , \gamma , \varvec{u}) = (\sigma , h, \varvec{f})$, and moreover, from (7.2) and the Cauchy–Schwarz inequality, one has

$$\begin{aligned} \tau _\beta |\zeta |\Vert (\phi , \gamma , \varvec{u})\Vert _{\varvec{{\mathcal {H}}}_\omega } \le \Vert (\sigma , h , \varvec{f})\Vert _{\varvec{{\mathcal {H}}}_\omega }. \end{aligned}$$

(7.4)

Hence, the sector $\Sigma _\beta$ lies in the resolvent set of $-(\varpi \varvec{I} + \varvec{{\mathcal {A}}})$, and for every $\zeta \in \Sigma _\beta$ the resolvent estimate $\Vert [\zeta \varvec{I} + (\varpi \varvec{I} + \varvec{{\mathcal {A}}})]^{-1}\Vert _{{\mathcal {L}}(\varvec{{\mathcal {H}}}_\omega )} \le \tau _\beta ^{-1}/|\zeta |$ holds due to (7.4). These show that $-(\varpi \varvec{I} + \varvec{{\mathcal {A}}})$ is sectorial, and hence, it generates an analytic $C_0$-semigroup on $\varvec{{\mathcal {H}}}_\omega$ by [26, Theorem 4.6]. Thanks to the bounded perturbation theorem in [54, Chapter 3, Corollary 2.2], we conclude that $-\varvec{{\mathcal {A}}} = -(\varpi \varvec{I} + \varvec{{\mathcal {A}}}) + \varpi \varvec{I}$ is also a generator of an analytic $C_0$-semigroup on $\varvec{{\mathcal {H}}}_\omega$. The proposition is now established.

$\square$