Local well-posedness of a quasi-incompressible two-phase flow

Abstract

We show well-posedness of a diffuse interface model for a two-phase flow of two viscous incompressible fluids with different densities locally in time. The model leads to an inhomogeneous Navier–Stokes/Cahn–Hilliard system with a solenoidal velocity field for the mixture, but a variable density of the fluid mixture in the Navier–Stokes type equation. We prove existence of strong solutions locally in time with the aid of a suitable linearization and a contraction mapping argument. To this end, we show maximal \(L^2\)-regularity for the Stokes part of the linearized system and use maximal \(L^p\)-regularity for the linearized Cahn–Hilliard system.

Introduction and main result

In this contribution, we study a thermodynamically consistent, diffuse interface model for two-phase flows of two viscous incompressible system with different densities in a bounded domain in two or three space dimensions. The model was derived by Garcke and Grün [6] and leads to the following inhomogeneous Navier–Stokes/Cahn–Hilliard system:

$$\begin{aligned}&\partial _t (\rho \mathbf{v} ) + \text {div}( \rho \mathbf{v} \otimes \mathbf{v} ) +\text {div}\Big ( \mathbf{v} \otimes \tfrac{\tilde{\rho }_1 - \tilde{\rho }_2}{2} m(\varphi ) \nabla (\tfrac{1}{\varepsilon } W'(\varphi ) - \varepsilon \varDelta \varphi ) \Big ) \nonumber \\&\qquad \qquad = \text {div}(- \varepsilon \nabla \varphi \otimes \nabla \varphi ) +\text {div}(2 \eta (\varphi ) D\mathbf{v} ) - \nabla q, \end{aligned}$$
(1)
$$\begin{aligned}&\text {div}\mathbf{v} = 0, \end{aligned}$$
(2)
$$\begin{aligned}&\partial _t \varphi +\mathbf{v} \cdot \nabla \varphi = \text {div}( m( \varphi ) \nabla \mu ), \end{aligned}$$
(3)
$$\begin{aligned}&\mu = - \varepsilon \varDelta \varphi + \frac{1}{\varepsilon } W' (\varphi ) \end{aligned}$$
(4)

in \(Q_T:= \varOmega \times (0,T)\) together with the initial and boundary values

$$\begin{aligned} \mathbf{v} _{|\partial \varOmega }&= \partial _n \varphi _{|\partial \varOmega } =\partial _n \mu _{|\partial \varOmega } = 0 \quad \text { on } (0,T) \times \partial \varOmega , \end{aligned}$$
(5)
$$\begin{aligned} \varphi (0)&= \varphi _0 , \mathbf{v} (0) = \mathbf{v} _0 \quad \text { in } \varOmega . \end{aligned}$$
(6)

Here, \(\varOmega \subseteq {\mathbb {R}}^d\), \(d= 2,3\), is a bounded domain with \(C^4\)-boundary. In this model, the fluids are assumed to be partly miscible and \(\varphi :\varOmega \times (0,T)\rightarrow {\mathbb {R}}\) denotes the volume fraction difference of the fluids. \(\mathbf{v} \), q, and \(\rho \) denote the mean velocity, the pressure and the density of the fluid mixture. It is assumed that the density is a given function of \(\varphi \), more precisely

$$\begin{aligned} \rho =\rho (\varphi ) = \frac{\tilde{\rho }_1+\tilde{\rho }_2}{2} +\frac{\tilde{\rho }_2-\tilde{\rho }_1}{2} \varphi \qquad \text {for all }\varphi \in {\mathbb {R}}. \end{aligned}$$

where \(\tilde{\rho }_1, \tilde{\rho }_2\) are the specific densities of the (non-mixed) fluids. Moreover, \(\mu \) is a chemical potential and \(W(\varphi )\) is a homogeneous free energy density associated with the fluid mixture, \(\varepsilon >0\) is a constant related to “thickness” of the diffuse interface, which is described by \(\{x\in \varOmega : |\varphi (x,t)|<1-\delta \}\) for some (small) \(\delta >0\), and \(m(\varphi )\) is a mobility coefficient, which controls the strength of the diffusion in the system. Finally, \(\eta (\varphi )\) is a viscosity coefficient and \(D\mathbf{v} = \frac{1}{2}(\nabla \mathbf{v} + \nabla \mathbf{v} ^T)\).

Existence of weak solution for this system globally in time was shown by Depner and Garcke [4] and [5] for non-degenerate and degenerate mobility in the case of a singular free energy density W. Moreover, Grün [13] convergence (of suitable subsequences) of a fully discrete finite-element scheme for this system to a weak solution in the case of a smooth \(W:{\mathbb {R}}\rightarrow {\mathbb {R}}\) with suitable polynomial growth. In the case of dynamic boundary conditions, which model moving contact lines, existence of weak solutions for this system was shown by Gal et al. [11]. In the case of non-Newtonian fluids of suitable p-growth, existence of weak solutions was proved by Abels and Breit [3]. For the case of a non-local Cahn–Hilliard equation and Newtonian fluids, the corresponding results was derived by Frigeri [10] and for a model with surfactants by Garcke and the authors in [7]. Recently, Giorgini [12] proved existence of local strong solutions in a two-dimensional bounded, sufficiently smooth domain and global existence of strong solutions in the case of a two-dimensional torus.

Remark 1

In [4], it is shown that the first equation is equivalent to

$$\begin{aligned}&\rho \partial _t \mathbf{v} + \Big (\rho \mathbf{v} + \tfrac{\tilde{\rho }_1 -\tilde{\rho }_2}{2} m(\varphi )&\nabla (\tfrac{1}{\varepsilon } W' (\varphi ) -\varepsilon \varDelta \varphi ) \Big ) \cdot \nabla \mathbf{v} + \nabla p -\text {div}(2 \eta (\varphi ) D\mathbf{v} ) \nonumber \\&\quad = - \varepsilon \varDelta \varphi \nabla \varphi . \end{aligned}$$
(7)

This reformulation will be useful in our analysis.

For the following, we assume:

Assumption 1

  1. 1.

    Let \(\varOmega \subseteq {\mathbb {R}}^d\) be a bounded domain with \(C^4\)-boundary and \(d = 2,3\).

  2. 2.

    Let \(\eta ,m \in C^5_b ({\mathbb {R}})\) be such that \(\eta (s) \ge \eta _0 > 0\) and \(m(s)\ge m_0\) for every \(s \in {\mathbb {R}}\) and some \(\eta _0,m_0 > 0\).

  3. 3.

    The density \(\rho :{\mathbb {R}}\rightarrow {\mathbb {R}}\) is given by

    $$\begin{aligned} \rho = \rho (\varphi ) = \frac{\tilde{\rho }_1 + \tilde{\rho }_2}{2} +\frac{\tilde{\rho }_2 - \tilde{\rho }_1}{2} \varphi \qquad \text { for all } \varphi \in {\mathbb {R}}. \end{aligned}$$
  4. 4.

    \(W:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is five times continuously differentiable.

With these assumptions, we will show our main existence result on short-time existence of strong solutions for (1)–(6):

Theorem 2

(Existence of strong solutions) Let \(\varOmega \), \(\eta \), m, \(\rho \) and W be as in Assumption 1. Moreover, let \(\mathbf{v}_0 \in H^{1}_0 (\varOmega )^d \cap L^2_\sigma (\varOmega ) \) and \(\varphi _0 \in (L^p (\varOmega ) , W^4_{p,N} (\varOmega ))_{1- \frac{1}{p}, p}\) be given with \(|\varphi _0(x)|\le 1\) for all \(x\in \varOmega \) and \(4< p < 6\). Then, there exists \(T > 0\) such that (1)–(6) has a unique strong solution

$$\begin{aligned} \mathbf{v}&\in W^1_2 (0,T; L^2_\sigma (\varOmega )) \cap L^2 (0,T; H^2 (\varOmega )^d \cap H^1_0 (\varOmega )^d), \\ \varphi&\in W^1_p (0,T; L^p (\varOmega )) \cap L^p (0,T; W^4_{p,N} (\varOmega )), \end{aligned}$$

where \(W^4_{p,N}(\varOmega )=\{u\in W^4_p(\varOmega ): \partial _n u|_{\partial \varOmega }= \partial _n \varDelta u|_{\partial \varOmega }=0\}\).

We will prove this result with the aid of a contraction mapping argument after a suitable reformulation, similar to [1]. But for the present system, the linearized system is rather different.

The structure of this contribution is as follows: in Sect. 2, we introduce some basic notation and recall some results used in the following. The main result is proved in Sect. 3. For its proof, we use suitable estimates of the nonlinear terms, which are shown in Sect. 4, and a result on maximal \(L^2\)-regularity of a Stokes-type system, which is shown in Sect. 5.

Preliminaries

For an open set \(U\subseteq {\mathbb {R}}^d\), \(m\in {\mathbb {N}}_0\) and \(1\le p \le \infty \), we denote by \(W^m_p(U)\) the \(L^p\)-Sobolev space of order m and \(W^m_p(U;X)\) its X-valued variant, where X is a Banach space. In particular, \(L^p(U)=W^0_p(U)\) and \(L^p(U;X)= W^0_p(U;X)\). Moreover, \(B^s_{pq}(\varOmega )\) denotes the standard Besov space, where \(s\in {\mathbb {R}}\), \(1\le p,q\le \infty \), and \(L^2_\sigma (\varOmega )\) is the closure of \(C^\infty _{0,\sigma }(\varOmega )= \{\mathbf{u }\in C^\infty _0(\varOmega )^d: \mathrm{div}\, \mathbf{u} =0\}\) in \(L^2(\varOmega )^d\) and \({\mathbb {P}}_\sigma :L^2(\varOmega )^d\rightarrow L^2_\sigma (\varOmega )\) the orthogonal projection onto it, i.e., the Helmholtz projection.

We will frequently use:

Theorem 3

(Composition with Sobolev functions) Let \(\varOmega \subseteq {\mathbb {R}}^d\) be a bounded domain with \(C^1\)-boundary, \(m,n\in {\mathbb {N}}\) and let \(1\le p <\infty \) such that \(m - dp > 0\). Then, for every \(f\in C^m({\mathbb {R}}^N)\) and every \(R>0\) there exists a constant \(C>0\) such that for all \(u\in W^m_p(\varOmega )^N\) with \(\Vert u\Vert _{W^m_p(\varOmega )^N}\le R\), we have \(f(u)\in W^m_p(\varOmega )\) and \(\Vert f(u)\Vert _{W^m_p(\varOmega )}\le C\). Moreover, if \(f\in C^{m+1}({\mathbb {R}}^N)\), then for all \(R>0\) there exists a constant \(L>0\) such that

$$\begin{aligned} \Vert f(u)-f(v)\Vert _{W^m_p(\varOmega )}\le L \Vert u-v\Vert _{W^m_p(\varOmega )^N} \end{aligned}$$

for all \(u, v\in W^m_p(\varOmega )^N\) with \(\Vert u\Vert _{W^m_p(\varOmega )^N}, \Vert v\Vert _{W^m_p(\varOmega )^N}\le R\).

Proof

The first part follows from [14, Chapter 5, Theorem 1 and Lemma]. The second part can be easily reduced to the first part. \(\square \)

In particular, we have \(uv\in W^m_p(\varOmega )\) for all \(u,v\in W^m_p(\varOmega )\) under the assumptions of the theorem.

Let \(X_0,X_1\) be Banach spaces such that \(X_1\hookrightarrow X_0\) densely. It is well known that

$$\begin{aligned} W^1_p(I;X_0) \cap L^p(I;X_1) \hookrightarrow BUC(I;(X_0,X_1)_{1-\frac{1}{p},p}), \qquad 1\le p <\infty , \end{aligned}$$
(8)

continuously for \(I=[0,T]\), \(0<T<\infty \), and \(I=[0,\infty )\), cf. Amann [8, Chapter III, Theorem 4.10.2]. Here, \((X_0,X_1)_{\theta ,p}\) denotes the real interpolation space of \((X_0,X_1)\) with exponent \(\theta \) and summation index p. Moreover, BUC(IX) is the space of all bounded and uniformly continuous \(f:I\rightarrow X\) equipped with the supremum norm, where X is a Banach space.

Moreover, we will use:

Lemma 1

Let \(X_0 \subseteq Y \subseteq X_1\) be Banach spaces such that

$$\begin{aligned} \Vert x\Vert _Y \le C \Vert x\Vert ^{1 - \theta }_{X_0} \Vert x\Vert ^\theta _{X_1} \end{aligned}$$

for every \(x \in X_0\) and a constant \(C > 0\), where \(\theta \in (0,1)\). Then,

$$\begin{aligned} C^{0, \alpha } ([0,T]; X_1) \cap L^\infty (0,T; X_0) \hookrightarrow C^{0, \alpha \theta } ([0,T] ; Y) . \end{aligned}$$

continuously.

The result is well-known and can be proved in a straight forward manner.

Proof of the main result

We prove the existence of a unique strong solution \((\mathbf{v} , \varphi ) \in X_T\) for small \(T > 0\), where the space \(X_T\) will be specified later. The idea for the proof is to linearize the highest order terms in the equations above at the initial data and then to split the equations in a linear and a nonlinear part such that

$$\begin{aligned} {\mathcal {L}} (\mathbf{v} , \varphi ) = {\mathcal {F}} (\mathbf{v} , \varphi ) , \end{aligned}$$

where we still have to specify in which sense this equation has to hold. To linearize it formally at the initial data, we replace \(\mathbf{v} \), p and \(\varphi \) by \(\mathbf{v} _0 + \varepsilon \mathbf{v} \), \(p_0 + \varepsilon p\) and \(\varphi _0 + \varepsilon \varphi \) and then differentiate with respect to \(\varepsilon \) at \(\varepsilon = 0\). In (1) and the equivalent equation (7), the highest order terms with respect to t and x are \(\rho \partial _t \mathbf{v} \), \( \text {div}(2 \eta (\varphi ) D\mathbf{v} )\) and \(\nabla p\). Hence, the linearizations are given by

$$\begin{aligned} \frac{ d}{{ d} \varepsilon } \left( \rho (\varphi _0 + \varepsilon \varphi ) \partial _t (\mathbf{v} _0 + \varepsilon \mathbf{v} ) \right) _{| \varepsilon = 0}&= \rho ' (\varphi _0) \varphi \partial _t \mathbf{v} _0 +\rho (\varphi _0) \partial _t \mathbf{v} = \rho _0 \partial _t \mathbf{v} , \\ \frac{d}{{ d} \varepsilon } \left( \text {div}(2 \eta (\varphi _0 + \varepsilon \varphi ) D(\mathbf{v} _0 + \varepsilon \mathbf{v} ) ) \right) _{| \varepsilon = 0}&= \text {div}(2 \eta ' (\varphi _0) \varphi D\mathbf{v} _0) + \text {div}(2 \eta (\varphi _0) D\mathbf{v} ),\\ \frac{d}{{ d} \varepsilon } \nabla (p_0 + \varepsilon p)_{| \varepsilon = 0}&= \nabla p , \end{aligned}$$

where \(\rho _0 := \rho (\varphi _0)\) and \(\rho _0 ' := \rho ' (\varphi _0)\). Moreover, we omit the term \(\text {div}(2 \eta ' (\varphi _0) \varphi D\mathbf{v} _0) \) in the second linearization since it is of lower order. For the last equation, we get the linearization

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d} \tilde{\varepsilon }} \text {div}(m (\varphi _0 +\tilde{\varepsilon } \varphi ) \nabla ( - \varepsilon \varDelta (\varphi _0 + \tilde{\varepsilon } \varphi )))_{|\tilde{\varepsilon } = 0} \\&\quad = - \varepsilon \text {div}( m ' (\varphi _0) \varphi \nabla \varDelta \varphi _0 ) -\varepsilon \text {div}( m (\varphi _0) \nabla \varDelta \varphi ) . \end{aligned}$$

We can omit the first term since it is of lower order. The second term can formally be reformulated as

$$\begin{aligned} - \varepsilon \text {div}(m (\varphi _0) \nabla \varDelta \varphi ) =-\varepsilon m' (\varphi _0) \nabla \varphi _0 \cdot \nabla \varDelta \varphi - \varepsilon m (\varphi _0) \varDelta ( \varDelta \varphi ). \end{aligned}$$

Here, the first summand is of lower order again. Hence, the linearization is given by \(- \varepsilon m (\varphi _0) \varDelta ^2 \varphi \) upto terms of lower order. Due to these linearizations, we define the linear operator \({\mathcal {L}} :X_T \rightarrow Y_T\) by

$$\begin{aligned} {\mathcal {L}} (\mathbf{v} , \varphi ) =\begin{pmatrix} {\mathbb {P}}_\sigma ( \rho _0 \partial _t \mathbf{v} ) - {\mathbb {P}}_\sigma (\text {div}(2 \eta (\varphi _0) D\mathbf{v} )) \\ \partial _t \varphi + \varepsilon m(\varphi _0) \varDelta ^2 \varphi \end{pmatrix}, \end{aligned}$$

where \({\mathcal {L}}\) consists of the principal part of the lionization’s, i.e., of the terms of the highest order. Furthermore, we define the nonlinear operator \({\mathcal {F}} :X_T \rightarrow Y_T\) by

$$\begin{aligned} {\mathcal {F}} ( \mathbf{v} , \varphi ) =\begin{pmatrix} {\mathbb {P}}_\sigma F_1 (\mathbf{v} , \varphi ) \\ - \nabla \varphi \cdot \mathbf{v} + \text {div}( \tfrac{1}{\varepsilon } m(\varphi ) \nabla W' (\varphi )) + \varepsilon m (\varphi _0) \varDelta ^2 \varphi -\varepsilon \text {div}( m (\varphi ) \nabla \varDelta \varphi ) \end{pmatrix} , \end{aligned}$$

where

$$\begin{aligned} F_1 (\mathbf{v} , \varphi )&= ( \rho _0 - \rho ) \partial _t \mathbf{v} - \text {div}(2 \eta (\varphi _0) D\mathbf{v} ) + \text {div}( 2 \eta (\varphi ) D\mathbf{v} ) -\varepsilon \varDelta \varphi \nabla \varphi \\&- \left( \left( \rho \mathbf{v} + \tfrac{\tilde{\rho }_1 -\tilde{\rho }_2}{2} m(\varphi ) \nabla (\tfrac{1}{\varepsilon } W' (\varphi )-\varepsilon \varDelta \varphi )\right) \cdot \nabla \ \right) \mathbf{v} . \end{aligned}$$

It still remains to define the spaces \(X_T\) and \(Y_T\). To this end, we set

$$\begin{aligned} Z^1_T&:= L^2 (0,T; H^2 (\varOmega )^d \cap H^1_0 (\varOmega )^d) \cap W^1_2 (0,T; L^2_\sigma (\varOmega )) , \\ Z^2_T&:= L^p (0,T; W^4_{p,N} (\varOmega )) \cap W^1_p (0,T; L^p (\varOmega )) \end{aligned}$$

with \(4< p < 6\), where

$$\begin{aligned} W^4 _{p,N} (\varOmega ) := \{ \varphi \in W^4_p (\varOmega ) | \ \partial _n \varphi = \partial _n (\varDelta \varphi ) = 0 \} . \end{aligned}$$

We equip \(Z^1_T\) and \(Z^2_T\) with the norms \(\Vert \cdot \Vert _{Z^1_T} '\) and \(\Vert \cdot \Vert _{Z^2_T} '\) defined by

$$\begin{aligned} \Vert \mathbf{v} \Vert _{Z^1_T} '&:= \Vert \mathbf{v} '\Vert _{L^2 (0,T; L^2 (\varOmega ))} +\Vert \mathbf{v} \Vert _{L^2 (0,T; H^2 (\varOmega ))} + \Vert \mathbf{v} (0)\Vert _{(L^2 (\varOmega ), H^2 (\varOmega ))_{\frac{1}{2}, 2}} , \\ \Vert \varphi \Vert _{Z^2_T} '&:= \Vert \varphi '\Vert _{L^p (0,T; L^p (\varOmega ))} +\Vert \varphi \Vert _{L^p (0,T; W^4_{p,N} (\varOmega ))} + \Vert \varphi (0) \Vert _{(L^p (\varOmega ), W^4_p (\varOmega ))_{1 - \frac{1}{p}, p}} . \end{aligned}$$

We use these norms since they guarantee that for all embeddings we will study later the embedding constant C does not depend on T, cf. Lemma 2. To this end, we use:

Lemma 2

Let \(0< T_0 < \infty \) be given and \(X_0\), \(X_1\) be some Banach spaces such that \(X_1 \hookrightarrow X_0\) densely. For every \(0< T < \frac{ T_0}{2}\), we define

$$\begin{aligned} X_T := L^p (0,T; X_1) \cap W^1_p (0,T; X_0), \end{aligned}$$

where \(1 \le p < \infty \), equipped with the norm

$$\begin{aligned} \Vert u\Vert _{X_T}:= \Vert u\Vert _{L^p(0,T;X_1)}+\Vert u\Vert _{W^1_p(0,T;X_0)} +\Vert u(0)\Vert _{(X_0,X_1)_{1-\frac{1}{p},p}}. \end{aligned}$$

Then, there exists an extension operator \(E : X_T \rightarrow X_{T_0}\) and some constant \(C > 0\) independent of T such that \(Eu_{|(0,T)} = u\) in \(X_T\) and

$$\begin{aligned} \Vert Eu\Vert _{X_{T_0}} \le C \Vert u\Vert _{X_T} \end{aligned}$$

for every \(u \in X_T\) and every \(0< T < \frac{T_0}{2}\). Moreover, there exists a constant \(\tilde{C} (T_0) > 0\) independent of T such that

$$\begin{aligned} \Vert u\Vert _{BUC ([0,T]; (X_0, X_1)_{1 - \frac{1}{p},p})} \le \tilde{C} (T_0) \Vert u\Vert _{X_T} \end{aligned}$$

for every \(u \in X_T\) and every \(0<T < \frac{ T_0}{2}\).

Proof

The result is well-known. In the case \(u(0)=0\), one can prove the result with the aid of the extension operator defined by

$$\begin{aligned} (Eu) (t) :={\left\{ \begin{array}{ll} u(t) &{} \text { if } t \in [0,T], \\ u(2T - t) &{} \text { if } t \in (T, 2T], \\ 0 &{} \text { if } t \in (2T, T_0] . \end{array}\right. } \end{aligned}$$

The case \(u(0)\ne 0\) can be easily reduced to the case \(u(0)=0\) by substracting a suitable extension of \(u_0\) to \([0,\infty )\). We refer to [16, Lemma 5.2] for the details. \(\square \)

The last preparation before we can start with the existence proof is the definition of the function spaces \(X_T := X_T^1 \times X_T^2\) and \(Y_T\) by

$$\begin{aligned} X_T^1&:= \{ \mathbf{v} \in Z^1_T | \ \mathbf{v} _{|t=0} = \mathbf{v} _0 \}, \\ X_T^2&:= \{ \varphi \in Z^2_T | \ \varphi _{| t = 0} = \varphi _0 \}, \\ Y_T&:= Y^1_T \times Y^2_T := L^2 (0,T; L^2_\sigma (\varOmega )) \times L^p (0,T; L^p (\varOmega )) , \end{aligned}$$

where

$$\begin{aligned} \mathbf{v} _0 \in (L^2_\sigma (\varOmega ), H^2 (\varOmega )^d \cap H^1_0 (\varOmega )^d \cap L^2_\sigma (\varOmega ))_{\frac{1}{2}, 2} = H^1_0 (\varOmega )^d \cap L^2_\sigma (\varOmega ) \end{aligned}$$

and

$$\begin{aligned} \varphi _0 \in (L^p (\varOmega ), W^4_{p,N} (\varOmega ) )_{1 - \frac{1}{p}, p} \end{aligned}$$

are the initial values from (6). Note that in the space \(X^2_T\) we have to ensure that \(\varphi _{|t=0} = \varphi _0 \in [-1,1]\) since we will use this property to show the Lipschitz continuity of \({\mathcal {F}} : X_T \rightarrow Y_T\) in Proposition 1. Moreover, we note that \(X_T\) is not a vector space due to the condition \(\varphi _{|t=0} = \varphi _0\). It is only an affine linear subspace of \(Z_T := Z^1_T \times Z^2_T\).

Proposition 1

Let the Assumptions 1 hold and \(\varphi _0\) be given as in Theorem 2. Then, there is a constant \(C (T,R) > 0\) such that

$$\begin{aligned} \Vert {\mathcal {F}} (\mathbf{v}_1 , \varphi _1 ) - {\mathcal {F}} (\mathbf{v}_2 , \varphi _2) \Vert _{Y_T} \le C (T, R) \Vert (\mathbf{v}_1 - \mathbf{v}_2, \varphi _1 -\varphi _2 )\Vert _{X_T} \end{aligned}$$
(9)

for all \((\mathbf{v}_i, \varphi _i) \in X_T\) with \(\Vert (\mathbf{v}_i, \varphi _i)\Vert _{X_T} \le R\) and \(i = 1,2\). Moreover, it holds \(C(T,R) \rightarrow 0\) as \(T \rightarrow 0\).

The proposition is proved in Sect. 4 below.

Theorem 4

Let \(T>0\) and \({\mathcal {L}}\), \(X_T\) and \(Y_T\) be defined as before. Then, \({\mathcal {L}}:X_T\rightarrow Y_T\) is invertible. Moreover, for every \(T_0>0\) there is a constant \(C(T_0)>0\) such that

$$\begin{aligned} \Vert {\mathcal {L}}^{-1}\Vert _{{\mathcal {L}}(Y_T,X_T)}\le C(T_0) \qquad \text {for all }T\in (0,T_0]. \end{aligned}$$

This theorem is proved in Sect. 5 below.

Proof of Theorem 2

First of all we note that (1)–(4) is equivalent to

$$\begin{aligned} (\mathbf{v} , \varphi ) = {\mathcal {L}}^{-1} ({\mathcal {F}} (\mathbf{v} , \varphi )) \quad \text {in } X_T . \end{aligned}$$
(10)

The fact that \({\mathcal {L}}\) is invertible will be proven later. Equation (10) implies that we have rewritten the system to a fixed-point equation which we want to solve by using the Banach fixed-point theorem.

To this end, we consider some \((\tilde{\mathbf{v }}, \tilde{\varphi }) \in X_T\) and define

$$\begin{aligned} M := \Vert {\mathcal {L}}^{-1 } \circ {\mathcal {F}} (\tilde{\mathbf{v }}, \tilde{\varphi })\Vert _{X_T} < \infty . \end{aligned}$$

Now, let \(R > 0\) be given such that \((\tilde{\mathbf{v }}, \tilde{\varphi }) \in \overline{B_R^{X_T} (0)}\) and \(R > 2M\). Then, it follows from Proposition 1 that there exists a constant \(C = C(T, R) > 0\) such that

$$\begin{aligned} \Vert {\mathcal {F}} (\mathbf{v} _1 , \varphi _1 ) - {\mathcal {F}} (\mathbf{v} _2 , \varphi _2) \Vert _{Y_T} \le C (T, R) \Vert (\mathbf{v} _1, \varphi _1) - ( \mathbf{v} _2, \varphi _2)\Vert _{X_T} \end{aligned}$$

for all \((\mathbf{v} _i, \varphi _i) \in X_T\) with \(\Vert (\mathbf{v} _i, \varphi _i)\Vert _{X_T} \le R\), \(j = 1,2\), where it holds \(C(T,R) \rightarrow 0\) as \(T \rightarrow 0\). Furthermore, we choose T so small that

$$\begin{aligned} \Vert {\mathcal {L}}^{-1}\Vert _{{\mathcal {L}} (Y_T, X_T)} C (T,R) < \frac{1}{2} . \end{aligned}$$

Here, we have to ensure that \(\Vert {\mathcal {L}}^{-1}\Vert _{{\mathcal {L}} (Y_T, X_T)} \) does not converge to \(+ \infty \) as \(T \rightarrow 0\). But, since Lemmas 7 and 9 below yield \(\Vert {\mathcal {L}}^{-1}\Vert _{{\mathcal {L}} (Y_T, X_T)} < C (T_0)\) for every \(0< T < T_0\) and for a constant that does not depend on T, this is not the case and we can choose \(T > 0\) in such a way that the previous estimate holds. Note that T depends on R and in general T has to become smaller, the larger we choose R.

Since we want to apply the Banach fixed-point theorem on \(\overline{B^{X_T}_R (0)} \subseteq X_T\) as we only consider functions \((\mathbf{v} , \varphi ) \in X_T\) which satisfy \(\Vert (\mathbf{v} , \varphi )\Vert _{X_T} \le R\), we have to show that \({\mathcal {L}}^{-1} \circ {\mathcal {F}}\) maps from \(\overline{B^{X_T}_R (0)}\) to \(\overline{B^{X_T}_R (0)}\).

From the considerations above, we know that there exists \((\tilde{\mathbf{v }}, \tilde{\varphi }) \in \overline{B^{X_T}_R (0)}\) such that

$$\begin{aligned} \Vert {\mathcal {L}}^{-1} \circ {\mathcal {F}} (\tilde{\mathbf{v }}, \tilde{\varphi }) \Vert _{X_T} = M < \frac{R}{2} . \end{aligned}$$
(11)

Then, a direct calculation shows

$$\begin{aligned} \Vert {\mathcal {L}} ^{-1} \circ {\mathcal {F}} (\mathbf{v} , \varphi ) \Vert _{X_T}&\le \Vert {\mathcal {L}} ^{-1} \circ {\mathcal {F}} ( \mathbf{v} , \varphi ) - {\mathcal {L}} ^{-1} \circ {\mathcal {F}} (\tilde{\mathbf{v }}, \tilde{\varphi }) \Vert _{X_T} + \Vert {\mathcal {L}}^{-1} \circ {\mathcal {F}} (\tilde{\mathbf{v }}, \tilde{\varphi }) \Vert _{X_T} \\&< \Vert {\mathcal {L}}^{-1} \Vert _{{\mathcal {L}} (Y_T, X_T)} \Vert {\mathcal {F}} (\mathbf{v} , \varphi ) - {\mathcal {F}} (\tilde{\mathbf{v }} , \tilde{\varphi }) \Vert _{Y_T} + \frac{R}{2} \\&\le \Vert {\mathcal {L}}^{-1} \Vert _{{\mathcal {L}} (Y_T, X_T)} C (R,T) \Vert (\mathbf{v} , \varphi ) - (\tilde{\mathbf{v }}, \tilde{\varphi })\Vert _{X_T} + \frac{R}{2} < R \end{aligned}$$

for every \((\mathbf{v} , \varphi ) \in \overline{B^{X_T}_R (0)}\), where we used the estimate for the Lipschitz continuity of \({\mathcal {F}}\). This shows that \({\mathcal {L}} ^{-1} \circ {\mathcal {F}} (\mathbf{v} , \varphi )\) is in \( \overline{B^{X_T}_R (0)}\) for every \((\mathbf{v} , \varphi ) \in \overline{ B^{X_T} _R (0)}\), i.e.,

$$\begin{aligned} {\mathcal {L}}^{-1 } \circ {\mathcal {F}} : \overline{ B^{X_T}_R (0) } \rightarrow \overline{ B^{X_T} _R (0) }. \end{aligned}$$

For applying the Banach fixed-point theorem, it remains to show that the mapping \({\mathcal {L}}^{-1 } \circ F :B^{X_T}_R (0) \rightarrow B^{X_T} _R (0) \) is a contraction. To this end, let \((\mathbf{v} _i, \varphi _i) \in B^{X_T}_R (0)\) be given for \(i = 1,2\). Then, it holds

$$\begin{aligned}&\Vert {\mathcal {L}}^{-1} \circ {\mathcal {F}} (\mathbf{v} _1, \varphi _1) -{\mathcal {L}}^{-1} \circ {\mathcal {F}} (\mathbf{v} _2 , \varphi _2) \Vert _{X_T} \\&\quad \le \Vert {\mathcal {L}}^{-1}\Vert _{{\mathcal {L}} (Y_T, X_T)} C(R,T) \Vert (\mathbf{v} _1, \varphi _1) - (\mathbf{v} _2, \varphi _2) \Vert _{X_T} \\&\quad < \frac{1}{2} \Vert (\mathbf{v} _1, \varphi _1) - (\mathbf{v} _2, \varphi _2) \Vert _{X_T} , \end{aligned}$$

which shows the statement. Hence, the Banach fixed-point theorem can be applied and yields some \((\mathbf{v} , \varphi ) \in \overline{B^{X_T}_R (0)} \subseteq X_T\) such that the fixed-point equation (10) holds, which implies that \((\mathbf{v} , \varphi )\) is a strong solution for Eqs. (1)–(4).

Finally, in order to show uniqueness in \(X_T\), let \((\hat{\mathbf{v }}, \hat{\varphi })\in X_T\) be another solution. Choose \(\hat{R}\ge R\) such that \((\hat{\mathbf{v }}, \hat{\varphi })\in \overline{B_{\hat{R}}^{X_T} (0)}\). Then by the previous arguments, we can find some \(\hat{T}\in (0,T]\) such that (10) has a unique solution. This implies \((\hat{\mathbf{v }}, \hat{\varphi })|_{[0, \hat{T}]}= (\mathbf{v} , \varphi )|_{[0,\hat{T}]}\). A standard continuation argument shows that the solutions coincide for all \(t\in [0,T]\). \(\square \)

Lipschitz continuity of \({\mathcal {F}}\)

Before we continue, we study in which Banach spaces \(\mathbf{v} \), \(\varphi \), \(\nabla \varphi \), \(m (\varphi )\) and so on are bounded.

Note that in the definition of \(X^2_T\), p has to be larger than 4 because we will need to estimate terms like \(\nabla \varDelta \varphi \cdot \nabla \mathbf{v} \), where \(p = 2\) is not sufficient for the analysis and therefore we need to choose \(p > 2\). But for most terms in the analysis \(p=2\) would be sufficient and \(4< p < 6\) would not be necessary. Nevertheless, for consistency all calculations are done for the case \(4< p < 6\).

Due to (8), it holds

$$\begin{aligned} \mathbf{v} \in X^1_T \hookrightarrow BUC ([0,T]; B^1_{22} (\varOmega )) = BUC ([0,T]; H^1 (\varOmega )) , \end{aligned}$$
(12)

where we used \(B^s_{22} (\varOmega ) = H^s_2 (\varOmega )\) for every \(s \in {\mathbb {R}}\). In particular, this implies

$$\begin{aligned}&\nabla \mathbf{v} \in L^\infty (0,T; L^2 (\varOmega )) \cap L^2 (0,T; L^6 (\varOmega )) \hookrightarrow L^{\frac{8}{3}} (0,T; L^4 (\varOmega )), \end{aligned}$$
(13)
$$\begin{aligned}&\nabla \mathbf {v} \in L^\infty (0,T; L^2 (\varOmega )) \cap L^2 (0,T; L^6 (\varOmega )) \hookrightarrow L^4 (0,T; L^3 (\varOmega )). \end{aligned}$$
(14)

Let \(\varphi \in X^2_T\) be given. From it (8) follows

$$\begin{aligned} \varphi \in L^p (0,T; W^4_{p,N} (\varOmega ) ) \cap W^1_p (0,T; L^p (\varOmega )) \hookrightarrow BUC \left( [0,T]; W^{4 - \frac{4}{p}}_p (\varOmega )\right) . \end{aligned}$$
(15)

This implies

$$\begin{aligned} \nabla \varDelta \varphi \in BUC \left( [0,T]; W^{1 - \frac{4}{p}}_p (\varOmega )\right) \end{aligned}$$
(16)

since \(p > 4\). Note that when we write “\(\varphi \) is bounded in Z” for some function space Z, we mean that the set of all functions \(\{\varphi \in X_T^2 : \ \Vert \varphi \Vert _{X^2_T} \le R\}\) is bounded in Z in such a way that the upper bound only depends on R and not on T, i.e., there exists \(C(R) > 0\) such that \(\Vert \varphi \Vert _{Z} \le C(R)\) for every \(\varphi \in X_T^2\) with \(\Vert \varphi \Vert _{X^2_T} \le R\).

First of all, we have

$$\begin{aligned} \varphi \in W^1_p (0,T; L^p (\varOmega )) \hookrightarrow C^{0, 1 - \frac{1}{p}} ([0,T]; L^p (\varOmega )) . \end{aligned}$$

Moreover, we already know that \(\varphi \in BUC([0,T]; W^{4 -\frac{4}{p}}_p (\varOmega ))\) and we have

$$\begin{aligned} \left( B^{4 - \frac{4}{p}}_{pp} (\varOmega ), L^p(\varOmega )\right) _{\theta , 2} = B^3_{p2} (\varOmega ) \hookrightarrow W^3_p (\varOmega ) \end{aligned}$$

together with the estimate

$$\begin{aligned} \Vert \varphi (t)\Vert _{W^3_p (\varOmega )} \le C \Vert \varphi (t) \Vert ^{1 - \theta }_{W^{4 - \frac{4}{p}}_p (\varOmega )} \Vert \varphi (t)\Vert ^\theta _{L^p (\varOmega )} \end{aligned}$$

for every \(t \in [0,T]\). Hence, Lemma 1 implies

$$\begin{aligned}&\varphi \in C^{0, 1 - \frac{1}{p}} ([0,T]; L^p (\varOmega )) \cap C\left( [0,T]; W^{4 - \frac{4}{p}}_p (\varOmega )\right) \nonumber \\&\quad \hookrightarrow C^{0, \left( 1 - \frac{1}{p}\right) \theta } ([0,T]; W^3_p (\varOmega )) . \end{aligned}$$
(17)

Because of \(W^3_p (\varOmega ) \hookrightarrow C^2 (\overline{\varOmega })\) for \(d=2,3\) due to \(4<p<6\), we obtain that

$$\begin{aligned} \varphi \text { is bounded in } C([0,T]; C^2 (\overline{\varOmega })) . \end{aligned}$$
(18)

In the nonlinear operator \({\mathcal {F}} :X_T \rightarrow Y_T\), the terms \(\eta (\varphi )\), \(\eta (\varphi _0)\), \(m (\varphi )\), \(m (\varphi _0)\) and \(W' (\varphi )\) appear. Hence, we need to know in which spaces these terms are bounded in the sense that there is a constant \(C(R) > 0\), which does not depend on T, such that the norms of these terms in a certain Banach space are bounded by C(R) for every \((\mathbf{v} , \varphi ) \in X_T\) with \(\Vert (\mathbf{v} , \varphi )\Vert _{X_T} \le R\).

Due to (17) and because the embedding constant only depends on R, it holds

$$\begin{aligned} \Vert \varphi (t)\Vert _{W^3_p (\varOmega )} \le C(R) \end{aligned}$$

for every \(t \in [0,T]\) and \(\varphi \in X_T^2\) with \(\Vert \varphi \Vert _{X^2_T} \le R\). Hence, Theorem 3 yields

$$\begin{aligned} \Vert f(\varphi (t))\Vert _{W^3_p (\varOmega )}, \Vert f (\varphi _0)\Vert _{W^3_p (\varOmega )}, \Vert W' (\varphi (t))\Vert _{W^3_p (\varOmega )} \le C(R) \end{aligned}$$

for every \(t \in [0,T]\) and every \(\varphi \in X^2_T\) with \(\Vert \varphi \Vert _{X^2_T} \le R\), where \( f \in \{\eta , m\}\). Thus,

$$\begin{aligned} f (\varphi ), f(\varphi _0), f'(\varphi ), W' (\varphi ) \text { are bounded in } L^\infty (0,T; W^3_p (\varOmega )) \end{aligned}$$
(19)

for \(f \in \{\eta , m\}\). Moreover, Theorem 3 yields the existence of \(L > 0\) such that

$$\begin{aligned} \Vert f (\varphi _1 (t)) - f (\varphi _2 (t))\Vert _{W^3_p (\varOmega )} \le L \Vert \varphi _1 (t) - \varphi _2 (t)\Vert _{W^3_p (\varOmega )} \end{aligned}$$
(20)

for every \(t \in [0,T]\), \(\varphi _1, \varphi _2 \in X^2_T\) and \(f \in \{\eta , m , W' \}\).

In the next step, we want to show that \(f(\varphi )\) is bounded in \(X_T^2\) and therefore, the same embeddings hold as for \(\varphi \), where \(f \in \{\eta , m , W'\}\). Note that from now on until the end of the proof of the interpolation result for \(f(\varphi )\), we always use some general \(f \in C^4_b ({\mathbb {R}})\). But all these embeddings are valid for \(f \in \{\eta , m, W'\}\). We want to prove that if it holds \(\varphi \in X^2_T\) with \(\Vert \varphi \Vert _{X^2_T} \le R\), then there exists a constant \(C(R) > 0\) such that \(\Vert f(\varphi )\Vert _{X^2_T} \le C(R)\). To this end, let \(\varphi \in X^2_T\) be given with \(\Vert \varphi \Vert _{X^2_T} \le R\). Since we already know \(\varphi \in C([0,T]; C^2 (\overline{\varOmega }))\), cf. (18), we can conclude

$$\begin{aligned} \Vert \varphi (t)\Vert _{C^2 (\overline{\varOmega })} \le C (R) \end{aligned}$$

for all \(t \in [0,T]\). Hence, it holds \(f (\varphi (t)) \in C^2 (\overline{\varOmega })\) for every \(t \in [0,T]\) and

$$\begin{aligned} \nabla f ( (\varphi (t)) = f' (\varphi (t)) \nabla \varphi (t) . \end{aligned}$$

Due to (19), \(f ' (\varphi )\) is bounded in \(L^\infty (0,T; W^3_p (\varOmega ))\). In particular, this implies \(\Vert f' (\varphi (t))\Vert _{W^3_p (\varOmega )} \le C(R)\) for a.e. \(t \in (0,T)\) and a constant \(C(R) > 0\). Since it holds \(\varphi \in L^p (0,T; W^4_p (\varOmega ))\), it follows \(\nabla \varphi (t) \in W^3_p (\varOmega )\) for a.e. \(t \in (0,T)\). Since \(W^3_p(\varOmega )\) is a Banach algebra, we obtain \(f' (\varphi (t)) \nabla \varphi (t) \in W^3_p (\varOmega )\) for a.e. \(t \in (0,T)\) together with the estimate

$$\begin{aligned} \Vert \nabla f (\varphi (t))\Vert _{W^3_p (\varOmega )} = \Vert f' (\varphi (t)) \nabla \varphi (t)\Vert _{W^3_p (\varOmega )} \le C \Vert f ' (\varphi (t)) \Vert _{W^3_p (\varOmega )} \Vert \nabla \varphi (t)\Vert _{W^3_p (\varOmega )} \end{aligned}$$

for a.e. \(t \in (0,T)\) and every \(\varphi \in X^2_T\) with \(\Vert \varphi \Vert _{X^2_T} \le R\). Since \(f' (\varphi )\) is bounded in \(L^\infty (0,T; W^3_p (\varOmega ))\) and \(\nabla \varphi \) is bounded in \(L^p (0,T; W^3_p (\varOmega ))\), the estimate above implies the boundedness of \(\nabla f (\varphi )\) in \(L^p (0,T; W^3_p (\varOmega ))\), i.e., there exists \(C(R) > 0\) such that

$$\begin{aligned} \Vert \nabla f (\varphi )\Vert _{L^p (0,T; W^3_p (\varOmega ))} \le C (R) \qquad \text { for all } \varphi \in X^2_T \text { with } \Vert \varphi \Vert _{X^2_T} \le R. \end{aligned}$$

Altogether this implies that

$$\begin{aligned} f(\varphi ) \text { is bounded in } L^p (0,T; W^4_p (\varOmega )) . \end{aligned}$$

Analogously, we can conclude from the boundedness of \(\varphi \) in \(W^1_p (0,T; L^p (\varOmega ))\) that \(f (\varphi ) \) is also bounded in \(W^1_p (0,T; L^p (\varOmega ))\) because of \(\frac{\mathrm{d}}{\mathrm{d}t} f (\varphi (t)) = f' (\varphi (t)) \partial _t \varphi (t),\) where \(f' (\varphi ) \) is bounded in \(C^0 (\overline{Q}_T) \). Thus, the same interpolation result holds as in (17), i.e.,

$$\begin{aligned} f(\varphi ) \text { is bounded in } C^{0, ( 1 - \frac{1}{p} ) \theta } ([0,T]; W^3_p (\varOmega )) , \end{aligned}$$
(21)

where \(\theta := \frac{\frac{4}{p} - 1}{ \frac{4}{p} - 4}\).

Proof of Proposition 1

Let \((\mathbf{v} _i, \varphi _i ) \in X_T\) with \(\Vert (\mathbf{v} _i, \varphi _i)\Vert _{X_T} \le R\), \(i = 1,2\), be given. Then, it holds

$$\begin{aligned}&\Vert {\mathcal {F}} (\mathbf{v} _1 , \varphi _1 ) - {\mathcal {F}} (\mathbf{v} _2 , \varphi _2) \Vert _{Y_T} = \Vert {\mathbb {P}}_\sigma ( F_1 (\mathbf{v} _1, \varphi _1 ) - F_1 (\mathbf{v} _2, \varphi _2)) \Vert _{L^2 (Q_T)} \nonumber \\&\quad + \Vert (\nabla \varphi _2 \cdot \mathbf{v} _2 - \nabla \varphi _1 \cdot \mathbf{v} _1 ) + \tfrac{1}{\varepsilon } \text {div}( m (\varphi _1) \nabla W' (\varphi _1) -m(\varphi _2) \nabla W' (\varphi _2)) \nonumber \\&\quad + \varepsilon m (\varphi _0) \varDelta ^2 (\varphi _1 - \varphi _2) +\varepsilon \text {div}( m (\varphi _2) \nabla \varDelta \varphi _2 -m(\varphi _1) \nabla \varDelta \varphi _2)\Vert _{L^p (Q_T)} . \end{aligned}$$
(22)

For the sake of clarity, we study both summands in (22) separately and begin with the first one. Recall that the operator \(F_1\) is defined by

$$\begin{aligned} F_1 (\mathbf{v} , \varphi )&= \rho _0 \partial _t \mathbf{v} - \rho \partial _t \mathbf{v} - \text {div}(2 \eta (\varphi _0) D\mathbf{v} ) + \text {div}( 2 \eta (\varphi ) D\mathbf{v} ) - \varepsilon \varDelta \varphi \nabla \varphi \\&\quad - \left( \left( \rho \mathbf{v} + \tfrac{\tilde{\rho }_1 -\tilde{\rho }_2}{2} m(\varphi ) \nabla (\tfrac{1}{\varepsilon } W' (\varphi ) -\varepsilon \varDelta \varphi ) \right) \cdot \nabla \ \right) \mathbf{v} \end{aligned}$$

and that it holds \(\Vert {\mathbb {P}}_\sigma \Vert _{{\mathcal {L}} (L^2 (\varOmega )^d, L^2_\sigma (\varOmega ))} \le 1\) for the Helmholtz projection \({\mathbb {P}}_\sigma \). We estimate \( \Vert {\mathbb {P}}_\sigma ( F_1 (\mathbf{v} _1, \varphi _1 ) - F_1 (\mathbf{v} _2, \varphi _2 ) )\Vert _{L^2 (Q_T)}\):

For the first two terms, we can calculate

$$\begin{aligned}&\Vert \rho _0 \partial _t \mathbf{v} _1 - \rho (\varphi _1) \partial _t \mathbf{v} _1 -\rho _0 \partial _t \mathbf{v} _2 + \rho (\varphi _2) \partial _t \mathbf{v} _2 \Vert _{L^2(Q_T)} \\&\quad \le \Vert (\rho _0 - \rho (\varphi _1)) \partial _t (\mathbf{v} _1 - \mathbf{v} _2) \Vert _{L^2 (Q_T)} + \Vert ( \rho (\varphi _1) -\rho (\varphi _2)) \partial _t \mathbf{v} _2 \Vert _{L^2 (Q_T)} . \end{aligned}$$

Since it holds \(\partial _t \mathbf{v} _i \in L^2 (0,T; L^2_\sigma (\varOmega ))\), \(i = 1,2\), we need to estimate every \(\rho \)-term in the \(L^\infty \)-norm. To this end, we use that \(\rho \) is affine linear and

$$\begin{aligned} \varphi _i \text { is bounded in } C^{0, (1 -\frac{1}{p})\theta } ([0,T]; W^3_p (\varOmega )) \hookrightarrow C^{0, (1 -\frac{1}{p})\theta } ([0,T]; C^2 ( \overline{\varOmega })) \end{aligned}$$

for \(i = 1,2\) and \(\theta = \frac{\frac{4}{p}-1}{\frac{4}{p}-4}\), cf. (17). Then, we obtain for the first summand

$$\begin{aligned} \Vert (\rho _0 - \rho (\varphi _1)) \partial _t (\mathbf{v} _1 - \mathbf{v} _2) \Vert _{L^2 (Q_T)}&\le \Vert \rho (\varphi _0) - \rho (\varphi _1)\Vert _{L^\infty (Q_T)} \Vert \partial _t (\mathbf{v} _1 - \mathbf{v} _2)\Vert _{L^2 (Q_T)} \\&\le C \underset{t \in [0,T]}{\sup } \Vert \varphi _1 (0) -\varphi _1 (t)\Vert _{L^\infty (\varOmega )} \Vert \mathbf{v} _1 - \mathbf{v} _2\Vert _{X^1_T} \\&\le C T^{ \left( 1 - \frac{1}{p}\right) \theta } \Vert \varphi _1\Vert _{C^{0, \left( 1 - \frac{1}{p}\right) \theta } ([0,T]; C^2 ( \overline{\varOmega }))} \Vert \mathbf{v} _1 - \mathbf{v} _2\Vert _{X^1_T} \\&\le C R T^{\left( 1 - \frac{1}{p}\right) \theta } \Vert \mathbf{v} _1 - \mathbf{v} _2\Vert _{X^1_T} . \end{aligned}$$

Analogously, the second term can be estimated by

$$\begin{aligned}&\Vert ( \rho (\varphi _1) - \rho (\varphi _2)) \partial _t \mathbf{v} _2 \Vert _{L^2 (Q_T)} \le \Vert \rho (\varphi _1) - \rho (\varphi _2)\Vert _{L^\infty (Q_T)} \Vert \mathbf{v} _2\Vert _{X^1_T} \\&\quad \le C \underset{t \in [0,T]}{\sup } \Vert (\varphi _1(t) - \varphi _2 (t)) -(\varphi _1 (0) - \varphi _2 (0))\Vert _{L^\infty (\varOmega )} \Vert \mathbf{v} _2\Vert _{X^1_T} \\&\quad \le C R T^{\left( 1 - \frac{1}{p}\right) \theta } \Vert \varphi _1 -\varphi _2\Vert _{C^{0, \left( 1 - \frac{1}{p}\right) \theta } ([0,T]; C^2 ( \overline{\varOmega }))} \\&\quad \le C R T^{\left( 1 - \frac{1}{p}\right) \theta } \Vert \varphi _1 - \varphi _2\Vert _{X^2_T} . \end{aligned}$$

Here, we used the fact that \(\varphi _1 (0) = \varphi _0 = \varphi _2 (0)\) for \(\varphi _i \in X^2_T\), \(i = 1,2\).

The next term of \( \Vert {\mathbb {P}}_\sigma ( F_1 (\mathbf{v} _1, \varphi _1 ) - F_1 (\mathbf{v} _2, \varphi _2 ) )\Vert _{L^2 (Q_T)}\) is given by

$$\begin{aligned}&|| ( \text {div}(2 \eta (\varphi _0) D\mathbf{v} _2) - \text {div}(2 \eta (\varphi _0) D\mathbf{v} _1)) +(\text {div}(2 \eta (\varphi _1) D\mathbf{v} _1) - \text {div}(2 \eta (\varphi _2) D\mathbf{v} _2))\Vert _{Y^1_T} \\&\quad \le \Vert \text {div}(2 (\eta (\varphi _0) - \eta (\varphi _1)) (D\mathbf{v} _2 - D\mathbf{v} _1 ) ) \Vert _{Y^1_T} + \Vert \text {div}(2 ((\eta (\varphi _1) - \eta (\varphi _2)) D\mathbf{v} _2 ) ) \Vert _{Y^1_T} . \end{aligned}$$

In the next step, we apply the divergence on the \(\eta (\varphi _i)\)- and \(D\mathbf{v} _i\)-terms, and for the sake of clarity, we study both terms in the previous inequality separately. For the first one, we use \(\eta (\varphi ) \in C^{0, (1 -\frac{1}{p}) \theta } ([0,T]; W^3_p (\varOmega ))\) with \(\theta = \frac{\frac{4}{p}-1}{\frac{4}{p} - 4}\), cf. (21), to obtain

$$\begin{aligned}&\Vert \text {div}(2 (\eta (\varphi _0) - \eta (\varphi _1) ) ( D\mathbf{v} _2 - D\mathbf{v} _1 ) ) \Vert _{Y^1_T} \\&\quad \le \Vert 2 \nabla (\eta (\varphi _0) - \eta (\varphi _1 ) ) \cdot (D\mathbf{v} _2 -D\mathbf{v} _1) \Vert _{Y^1_T} + \Vert (\eta (\varphi _0) - \eta (\varphi _1)) \varDelta (\mathbf{v} _2 - \mathbf{v} _1 ) \Vert _{Y^1_T} \\&\le C \underset{t \in [0,T]}{\sup } \Vert \nabla \eta (\varphi _1 (0)) -\nabla \eta (\varphi _1 (t)) \Vert _{C^1 ( \overline{\varOmega })} \Vert D\mathbf{v} _2 -D\mathbf{v} _1 \Vert _{L^2 (0,T; H^1 (\varOmega ))} \\&\qquad + C \underset{t \in (0,T)}{\sup } \Vert \eta (\varphi _1 (0)) -\eta (\varphi _1 (t)) \Vert _{C^2 ( \overline{\varOmega })} \Vert \varDelta (\mathbf{v} _2 - \mathbf{v} _1) \Vert _{L^2 (0,T; L^2 (\varOmega ))} \\&\quad \le C T^{\left( 1 - \frac{1}{p}\right) \theta } \Vert \nabla \eta (\varphi _1)\Vert _{C^{0, \left( 1 - \frac{1}{p}\right) \theta } ([0,T]; W^2_p (\varOmega ))} \Vert \mathbf{v} _1 - \mathbf{v} _2\Vert _{X^1_T} \\&\qquad + C T^{\left( 1 - \frac{1}{p}\right) \theta } \Vert \eta (\varphi _1)\Vert _{C^{0, \left( 1 - \frac{1}{p}\right) \theta } ([0,T]; W^3_p (\varOmega ))} \Vert \mathbf{v} _1 - \mathbf{v} _2\Vert _{X^1_T} \\&\quad \le C R \left( T^{\left( 1 - \frac{1}{p}\right) \theta } +T^{\left( 1 - \frac{1}{p}\right) \theta } \right) \Vert \mathbf{v} _1 - \mathbf{v} _2\Vert _{X^1_T} . \end{aligned}$$

Analogously, as before we can estimate the second summand by

$$\begin{aligned}&\Vert \text {div}(2 ((\eta (\varphi _1) - \eta (\varphi _2)) D\mathbf{v} _2 ) ) \Vert _{Y^1_T} \\&\quad \le 2 \Vert \eta ' (\varphi _1) ( \nabla \varphi _1 - \nabla \varphi _2) \cdot D\mathbf{v} _2 \Vert _{Y^1_T} + 2 \Vert ( \eta ' (\varphi _1) - \eta ' (\varphi _2) ) \nabla \varphi _2 \cdot D\mathbf{v} _2 \Vert _{Y^1_T} \\&\qquad + \Vert ( \eta (\varphi _1) - \eta (\varphi _2) ) \varDelta \mathbf{v} _2\Vert _{Y^1_T} . \end{aligned}$$

For the sake of clarity, we study these three terms separately again. Firstly,

$$\begin{aligned}&\Vert \eta ' (\varphi _1) ( \nabla \varphi _1 - \nabla \varphi _2) \cdot D\mathbf{v} _2 \Vert _{Y^1_T} \le C (R) \left| \left| \Vert D\mathbf{v} _2\Vert _{L^2 (\varOmega )} \Vert \nabla \varphi _1 - \nabla \varphi _2\Vert _{C^1 (\overline{\varOmega })} \right| \right| _{L^2 (0,T)} \\&\quad \le C (R) \underset{t \in [0,T]}{\sup } \Vert \nabla (\varphi _1 (t) - \varphi _2 (t)) - \nabla (\varphi _1 (0) - \varphi _2 (0))\Vert _{C^1 (\overline{\varOmega })} \Vert D\mathbf{v} _2\Vert _{L^2 (0,T; L^2 (\varOmega ))} \\&\quad \le C (R) T^{\left( 1 - \frac{1}{p}\right) \theta } \Vert \nabla \varphi _1 - \nabla \varphi _2\Vert _{C^{\left( 1 - \frac{1}{p}\right) \theta } ([0,T]; W^2_p (\varOmega ))} \Vert \mathbf{v} _2\Vert _{X^1_T} \\&\quad \le C(R) T^{\left( 1 - \frac{1}{p}\right) \theta } \Vert \varphi _1 - \varphi _2\Vert _{X^2_T} , \end{aligned}$$

where we used in the first step that \(\eta ' (\varphi ) \) is bounded in \( C([0,T]; C^2 (\overline{\varOmega }))\). Furthermore, (20) together with

$$\begin{aligned} \varphi \in C^{0, \left( 1 - \frac{1}{p}\right) \theta } ([0,T]; W^3_p (\varOmega )) \hookrightarrow C([0,T]; C^2 (\overline{\varOmega })) \end{aligned}$$

implies

$$\begin{aligned}&\Vert ( \eta ' (\varphi _1) - \eta ' (\varphi _2) ) \nabla \varphi _2 \cdot D\mathbf{v} _2 \Vert _{Y^1_T} \\&\quad \le \underset{t \in [0,T]}{\sup }\Vert \eta ' (\varphi _1) - \eta ' (\varphi _2) \Vert _{W^3_p (\varOmega )} \Vert \nabla \varphi _2 \Vert _{C([0,T];C^1 (\overline{\varOmega }))} \Vert D\mathbf{v} _2\Vert _{L^2 (Q_T)} \\&\quad \le C (R) \underset{t \in [0,T]}{\sup } \Vert \varphi _1 (t) -\varphi _2 (t) \Vert _{W^3_p (\varOmega )} \le C (R) T^{\left( 1- \frac{1}{p}\right) \theta } \Vert \varphi _1 - \varphi _2\Vert _{X^2_T} \end{aligned}$$

since \(\varphi _1(0)-\varphi _2(0)=0\). Analogously, to the second summand we can estimate the third one by

$$\begin{aligned} \Vert ( \eta (\varphi _1) - \eta (\varphi _2) ) \varDelta \mathbf{v} _2\Vert _{Y_T} \le C(R) T^{\left( 1 - \frac{1}{p}\right) \theta } \Vert \varphi _1 - \varphi _2\Vert _{X^2_T} , \end{aligned}$$

which shows the statement for the second term.

For the third term, we obtain

$$\begin{aligned}&\Vert \rho (\varphi _2) \mathbf{v} _2 \cdot \nabla \mathbf{v} _2 - \rho (\varphi _1) \mathbf{v} _1 \cdot \nabla \mathbf{v} _1\Vert _{Y^1_T} \\&\quad \le \Vert ( \rho (\varphi _2) - \rho (\varphi _1)) \mathbf{v} _2 \cdot \nabla \mathbf{v} _2 \Vert _{Y^1_T} + \Vert \rho (\varphi _1) (\mathbf{v} _2 - \mathbf{v} _1) \cdot \nabla \mathbf{v} _2 \Vert _{Y^1_T} \\&\qquad + \Vert \rho (\varphi _1) \mathbf{v} _1 \cdot (\nabla \mathbf{v} _2 - \nabla \mathbf{v} _1 ))\Vert _{Y^1_T} . \end{aligned}$$

We estimate these three terms separately again. For the first term, we use that \(\mathbf{v} _2\) is bounded in \(L^\infty (0,T; L^6 (\varOmega ))\), cf. (12), and \(\nabla \mathbf{v} _2\) is bounded in \(L^2 (0,T; L^6 (\varOmega ))\) together with (20). Thus,

$$\begin{aligned}&\Vert ( \rho (\varphi _2) - \rho (\varphi _1)) \mathbf{v} _2 \cdot \nabla \mathbf{v} _2 \Vert _{Y^1_T} \\&\quad \le C (R) T^{\left( 1- \frac{1}{p}\right) \theta } \Vert \varphi _2 - \varphi _1 \Vert _{C^{0, \left( 1 - \frac{1}{p}\right) \theta } ([0,T]; W^3_p (\varOmega ))} \Vert \mathbf{v} _2 \Vert _{L^\infty (0,T; L^6 (\varOmega ))} \Vert \nabla \mathbf{v} _2\Vert _{L^2 (0,T; L^6 (\varOmega ))} \\&\quad \le C(R) T^{\left( 1 - \frac{1}{p}\right) \theta } \Vert \varphi _2 - \varphi _1\Vert _{X^1_T}. \end{aligned}$$

For the second term, we use \(\rho (\varphi _1) \in C([0,T]; C^2 (\overline{\varOmega }))\), \(\mathbf{v} _i \in L^\infty (0,T; L^6 (\varOmega ))\) and \(\nabla \mathbf{v} _2 \in L^4 (0,T; L^3 (\varOmega ))\), cf. (12) and (14), \(i = 1,2\). Hence,

$$\begin{aligned} \Vert \rho (\varphi _1) (\mathbf{v} _2 - \mathbf{v} _1) \cdot \nabla \mathbf{v} _2 \Vert _{Y^1_T}&\le C (R) T^\frac{1}{4} \Vert \mathbf{v} _1 - \mathbf{v} _2\Vert _{L^\infty (0,T; L^6 (\varOmega ))} \Vert \nabla \mathbf{v} _2 \Vert _{L^4 (0,T; L^3 (\varOmega ))} \\&\le C (R) T^\frac{1}{4} \Vert \mathbf{v} _1 - \mathbf{v} _2\Vert _{X^1_T}. \end{aligned}$$

For the third term, we use the same function spaces. This implies

$$\begin{aligned} \Vert \rho (\varphi _1) \mathbf{v} _1 \cdot (\nabla \mathbf{v} _2 - \nabla \mathbf{v} _1 ))\Vert _{Y_T}&\le C (R) T^{\frac{1}{4}} \Vert \nabla \mathbf{v} _1 - \nabla \mathbf{v} _2\Vert _{L^4 (0,T; L^3 (\varOmega ))} \\&\le C (R) T^{\frac{1}{4}} \Vert \mathbf{v} _1 - \mathbf{v} _2\Vert _{X^1_T} . \end{aligned}$$

Since \(\frac{\tilde{\rho }_1 - \tilde{\rho }_2}{2} \) is a constant, we obtain

$$\begin{aligned}&\left| \left| \tfrac{\tilde{\rho }_1 - \tilde{\rho }_2}{2} m(\varphi _1) \nabla ( \varDelta \varphi _1 ) \cdot \nabla \mathbf{v} _1 - \tfrac{\tilde{\rho }_1 - \tilde{\rho }_2}{2} m(\varphi _2) \nabla ( \varDelta \varphi _2 ) \cdot \nabla \mathbf{v} _2 \right| \right| _{Y^1_T} \\&\quad \le C \left( \Vert m(\varphi _1) \nabla ( \varDelta \varphi _1 ) \cdot ( \nabla \mathbf{v} _1 - \nabla \mathbf{v} _2 ) \Vert _{Y^1_T} \right. \\&\qquad + \Vert m(\varphi _1) ( \nabla (\varDelta \varphi _1) -\nabla (\varDelta \varphi _2)) \cdot \nabla \mathbf{v} _2 \Vert _{Y^1_T} \\&\qquad + \left. \Vert (m(\varphi _1) - m (\varphi _2) )\nabla (\varDelta \varphi _2) \cdot \nabla \mathbf{v} _2 \Vert _{Y^1_T} \right) . \end{aligned}$$

For the sake of clarity, we study all three terms separately again. In the following, we use \(\nabla \varDelta \varphi _i \in L^\infty (0,T; L^4 (\varOmega ))\), cf. (17), \(\nabla \mathbf{v} _i \in L^{\frac{8}{3}} (0,T; L^4 (\varOmega ))\), cf. (13), for \(i = 1,2\), and \(m (\varphi _1) \in C([0,T];C^2 (\overline{\varOmega }))\). Altogether this implies

$$\begin{aligned}&\Vert m(\varphi _1) \nabla ( \varDelta \varphi _1 ) \cdot ( \nabla \mathbf{v} _1 - \nabla \mathbf{v} _2 ) \Vert _{Y^1_T} \\&\quad \le C T^\frac{1}{8} \Vert \nabla \varDelta \varphi _1 \Vert _{L^\infty (0,T; L^4 (\varOmega ))} \Vert \nabla \mathbf{v} _1 - \nabla \mathbf{v} _2\Vert _{L^{\frac{8}{3}} (0,T; L^4 (\varOmega ))} \\&\quad \le C (R) T^\frac{1}{8} \Vert \mathbf{v} _1 - \mathbf{v} _2\Vert _{X^1_T} . \end{aligned}$$

Analogously, the second summand yields

$$\begin{aligned} \Vert m(\varphi _1) ( \nabla (\varDelta \varphi _1) - \nabla (\varDelta \varphi _2)) \cdot \nabla \mathbf{v} _2 \Vert _{Y^1_T} \le C (R) T^\frac{1}{8} \Vert \varphi _1 - \varphi _2\Vert _{X^2_T} . \end{aligned}$$

For the last term, we use \(m (\varphi _i) \in C^{0, (1 - \frac{1}{p}) \theta } ([0,T]; W^3_p (\varOmega )) \hookrightarrow C^0 ([0,T]; C^2 (\overline{\varOmega }))\) together with (20) and obtain

$$\begin{aligned}&||(m(\varphi _1) - m (\varphi _2) )\nabla (\varDelta \varphi _2) \cdot \nabla \mathbf{v} _2 \Vert _{Y^1_T} \\&\quad \le C(R ) T^\frac{1}{8} \Vert \varphi _1 (t) - \varphi _2 (t) \Vert _{C^0([0,T]; C^2 (\overline{\varOmega }))} \Vert \nabla \varDelta \varphi _2 \Vert _{L^\infty (0,T; L^4 (\varOmega )} \Vert \nabla \mathbf{v} _2\Vert _{L^\frac{8}{3} (0,T; L^4 (\varOmega ))} \\&\quad \le C(R) T^{\frac{1}{8} } \Vert \varphi _1 - \varphi _2\Vert _{X^2_T} . \end{aligned}$$

The next term has the same structure as the one before and can be estimated as

$$\begin{aligned}&\left| \left| \tfrac{\tilde{\rho }_1 - \tilde{\rho }_2}{2} m(\varphi _1) \nabla ( W' (\varphi _1) ) \cdot \nabla \mathbf{v} _1 - \tfrac{\tilde{\rho }_1 - \tilde{\rho }_2}{2} m(\varphi _2) \nabla ( W' (\varphi _2) ) \cdot \nabla \mathbf{v} _2 \right| \right| _{Y^1_T} \nonumber \\&\quad \le C \left( \Vert m(\varphi _1) \nabla W' ( \varphi _1) \cdot ( \nabla \mathbf{v} _1 - \nabla \mathbf{v} _2 ) \Vert _{Y^1_T} \right. \nonumber \\&\qquad + \Vert m(\varphi _1) ( \nabla W'( \varphi _1 ) - \nabla W' (\varphi _2)) \cdot \nabla \mathbf{v} _2 \Vert _{Y^1_T} \nonumber \\&\qquad + \left. \Vert (m(\varphi _1) - m (\varphi _2) )\nabla W' (\varphi _2 ) \cdot \nabla \mathbf{v} _2 \Vert _{Y^1_T} \right) . \end{aligned}$$
(23)

For \(\nabla \mathbf{v} _i\), \(i = 1,2\), we use its boundedness in \(L^4 (0,T; L^3 (\varOmega ))\), cf. (14). Moreover, we know \(\nabla W' (\varphi ) \in C([0,T]; W^{3 - \frac{4}{p}}_p (\varOmega ))\) and \(m ( \varphi ) \in C([0,T]; C^2 (\overline{\varOmega }))\) for \(\varphi \in B_R^{X^2_T}\). Using all these bounds, we can estimate the three terms in (23) separately. For the first term, we obtain

$$\begin{aligned} \Vert m(\varphi _1) \nabla W' ( \varphi _1) \cdot ( \nabla \mathbf{v} _1 - \nabla \mathbf{v} _2 ) \Vert _{Y^1_T}&\le C (R) T^\frac{1}{4} \Vert \nabla \mathbf{v} _1 - \nabla \mathbf{v} _2\Vert _{L^4 (0,T; L^3 (\varOmega ))} \\&\le C (R) T^\frac{1}{4} \Vert \mathbf{v} _1 - \mathbf{v} _2\Vert _{X^1_T} . \end{aligned}$$

For the second summand in (23), we have to estimate the difference \(\nabla W' (\varphi _1) - \nabla W' (\varphi _2)\) in an appropriate manner. To this end, we use (17), (20) and \(W^2_p (\varOmega ) \hookrightarrow C^1 (\overline{\varOmega })\). Moreover, we use \(\nabla \mathbf{v} _2 \in L^4 (0,T; L^3 (\varOmega ))\), cf. (14), and \(m (\varphi ) \in C([0,T]; C^2 (\overline{\varOmega }))\). Then, it follows

$$\begin{aligned}&\Vert m(\varphi _1) (\nabla W'( \varphi _1 ) - \nabla W' ( \varphi _2)) \cdot \nabla \mathbf{v} _2 \Vert _{Y^1_T} \\&\quad \le C (R) T^\frac{1}{4} \underset{t \in [0,T]}{\sup } \Vert \nabla W' (\varphi _1 (t) ) - \nabla W' (\varphi _2 (t))\Vert _{W^2_p (\varOmega )} \\&\quad \le C (R) T^\frac{1}{4} \underset{t \in [0,T]}{\sup } \Vert \varphi _1 (t) -\varphi _2 (t)\Vert _{W^3_p (\varOmega )} \\&\quad \le C (R) T^{\frac{1}{4} + \left( 1 - \frac{1}{p}\right) \theta } \Vert \varphi _1 -\varphi _2 \Vert _{X^2_T} . \end{aligned}$$

So it remains to estimate the third term of (23). As before we get

$$\begin{aligned}&\Vert (m(\varphi _1) - m (\varphi _2) )\nabla W' (\varphi _2 ) \cdot \nabla \mathbf{v} _2 \Vert _{Y^1_T} \\&\quad \le C(R) T^{\frac{1}{4} + \left( 1 - \frac{1}{p}\right) \theta } \Vert \varphi _1 - \varphi _2\Vert _{C^{0, (1 - \frac{1}{p} ) \theta }([0,T]; W^3_p (\varOmega ))} \\&\qquad \Vert \nabla W' (\varphi _2)\Vert _{BUC([0,T]; C^1 (\overline{\varOmega }))} \Vert \nabla \mathbf{v} _2\Vert _{L^4 (0,T; L^3 (\varOmega ))} \\&\quad \le C(R) T^{\frac{1}{4} + \left( 1 - \frac{1}{p}\right) \theta } \Vert \varphi _1 - \varphi _2\Vert _{X^2_T} , \end{aligned}$$

which completes the estimate for (23). \(\square \)

Finally, we study the last term of \( \Vert {\mathbb {P}}_\sigma ( F_1 (\mathbf{v} _1, \varphi _1 ) - F_1 (\mathbf{v} _2, \varphi _2 ) )\Vert _{L^2 (Q_T)}\). It holds

$$\begin{aligned} \Vert \varDelta \varphi _2 \nabla \varphi _2 - \varDelta \varphi _1 \nabla \varphi _1\Vert _{Y_T} \le \Vert \varDelta \varphi _2 ( \nabla \varphi _2 -\nabla \varphi _1) \Vert _{Y_T} + \Vert ( \varDelta \varphi _2 -\varDelta \varphi _1) \nabla \varphi _1 \Vert _{Y_T}. \end{aligned}$$

Using \(\varDelta \varphi _i \in C([0,T]; C^0 (\overline{\varOmega }))\) and \(\nabla \varphi _i \in C^{0, \left( 1- \frac{1}{p}\right) \theta } ([0,T]; W^2_p (\varOmega ))\), \(i = 1,2\), cf. (17), the first term can be estimated by

$$\begin{aligned} \Vert \varDelta \varphi _2 ( \nabla \varphi _2 - \nabla \varphi _1) \Vert _{Y^1_T}&\le C (R) T^{\frac{1}{2} + \left( 1 - \frac{1}{p}\right) \theta )} \Vert \nabla \varphi _1 -\nabla \varphi _2\Vert _{C^{0, \left( 1 - \frac{1}{p}\right) \theta }([0,T]; W^2_p (\varOmega ))} \\&\le C (R) T^{\frac{1}{2} + \left( 1 - \frac{1}{p}\right) \theta )} \Vert \varphi _1 - \varphi _2\Vert _{X^2_T} . \end{aligned}$$

Analogously, the second term can be estimated by

$$\begin{aligned} \Vert ( \varDelta \varphi _2 - \varDelta \varphi _1) \nabla \varphi _1 \Vert _{Y_T} \le C (R) T^{\frac{1}{2} + \left( 1 - \frac{1}{p}\right) \theta )} \Vert \varphi _1 - \varphi _2\Vert _{X^2_T} . \end{aligned}$$

Hence, we obtain

$$\begin{aligned} \Vert {\mathbb {P}}_\sigma ( F_1 (\mathbf{v} _1, \varphi _1 ) -F_1 (\mathbf{v} _2, \varphi _2 ) )\Vert _{L^2 (Q_T)} \le C(R,T) \Vert (\mathbf{v} _1 - \mathbf{v} _2), (\varphi _1 - \varphi _2)\Vert _{X_T} \end{aligned}$$

for a constant \(C(R,T) > 0\) such that \(C(R,T) \rightarrow 0\) as \(T \rightarrow 0\).

Remember that we study the nonlinear operator \({\mathcal {F}} :X_T \rightarrow Y_T\) given by

$$\begin{aligned} {\mathcal {F}} (\mathbf{v} , \varphi ) =\begin{pmatrix} {\mathbb {P}}_\sigma F_1 (\mathbf{v} , \varphi ) \\ - \nabla \varphi \cdot \mathbf{v} + \text {div}\left( \frac{1}{\varepsilon } m(\varphi ) \nabla W' (\varphi )\right) + \varepsilon m (\varphi _0) \varDelta ^2 \varphi - \varepsilon \text {div}( m (\varphi ) \nabla \varDelta \varphi ) \end{pmatrix} \end{aligned}$$

and we want to show its Lipschitz continuity such that (9) holds. We already showed its Lipschitz continuity for the first part. Now, we continue to study the second one. This part has to be estimated in \(L^p (0,T; L^p (\varOmega ))\) for \(4< p < 6\).

For the analysis, we use the boundedness of \(\nabla \varphi \) in \(C([0,T]; C^1 (\overline{\varOmega }))\) and of \(\mathbf{v} \) in \( L^\infty (0,T; L^6 (\varOmega ))\). Then, it holds

$$\begin{aligned}&\Vert (\nabla \varphi _1 \cdot \mathbf{v} _1 - \nabla \varphi _2 \cdot \mathbf{v} _2 ) \Vert _{L^p (Q_T)} \\&\quad \le \Vert \nabla \varphi _1 \cdot ( \mathbf{v} _1 - \mathbf{v} _2 )\Vert _{L^p (Q_T)} + \Vert ( \nabla \varphi _1 - \nabla \varphi _2 ) \cdot \mathbf{v} _2\Vert _{L^p (Q_T)} \\&\quad \le T^\frac{1}{p} \Vert \nabla \varphi _1\Vert _{L^\infty (Q_T)} \Vert \mathbf{v} _1 - \mathbf{v} _2\Vert _{L^\infty (0,T; L^6 (\varOmega ))} \\&\quad + T^\frac{1}{p} \Vert \nabla \varphi _1 - \nabla \varphi _2\Vert _{L^\infty (Q_T)} \Vert \mathbf{v} _2\Vert _{L^\infty (0,T; L^6 (\varOmega ))} \\&\quad \le T^\frac{1}{p} R \Vert \mathbf{v} _1 - \mathbf{v} _2\Vert _{X^1_T} + T^\frac{1}{p} R \Vert \varphi _1 - \varphi _2\Vert _{X^2_T} . \end{aligned}$$

Next, we study the term \(\text {div}(m (\varphi ) \nabla W' (\varphi ))\). We use the boundedness of \(f(\varphi ) \) in \(C([0,T]; C^2 (\overline{\varOmega })) \cap C^{0, (1 - \frac{1}{p}) \theta } ([0,T]; W^3_p (\varOmega ))\) for \(f \in \{ m , W' \}\) and \(\varphi \in X^2_T \) with \(\Vert \varphi \Vert _{X^2_T} \le R\). Then, it holds

$$\begin{aligned}&\Vert \text {div}(m(\varphi _1) \nabla W' (\varphi _1)) - \text {div}( m (\varphi _2) \nabla W' (\varphi _2))\Vert _{Y^2_T} \\&\quad \le C(R) \Vert m (\varphi _1) \nabla W' (\varphi _1)) -m (\varphi _2) \nabla W' (\varphi _2)\Vert _{L^p (0,T; W^1_p (\varOmega ))} \\&\quad \le C(R) T^\frac{1}{p} \underset{t \in [0,T]}{\sup } \Vert m (\varphi _1 (t)) - m (\varphi _2 (t))\Vert _{W^3_p (\varOmega )} \Vert \nabla W' (\varphi _1)\Vert _{C([0,T]; C^1 (\overline{\varOmega }))} \\&\qquad + C(R) T^\frac{1}{p} \Vert m (\varphi _2)\Vert _{C([0,T]; C^2 (\overline{\varOmega }))} \underset{t \in [0,T]}{\sup } \Vert W' (\varphi _1 (t)) - W' (\varphi _2 (t)) \Vert _{W^3_p (\varOmega )} \\&\quad \le C(R) T^\frac{1}{p} \left( \underset{t \in [0,T]}{\sup } \Vert \varphi _1 (t) - \varphi _2 (t)\Vert _{W^3_p (\varOmega )} + \underset{t \in [0,T]}{\sup } \Vert \varphi _1 (t) -\varphi _2 (t) \Vert _{W^3_p (\varOmega )} \right) \\&\quad \le C(R) T^\frac{1}{p} \underset{t \in [0,T]}{\sup } \Vert (\varphi _1 (t) - \varphi _2 (t)) - (\varphi _1 (0) - \varphi _2 (0)) \Vert _{W^3_p (\varOmega )} \\&\quad \le C(R) T^{\frac{1}{p} + \left( 1 - \frac{1}{p}\right) \theta } \Vert \varphi _1 - \varphi _2\Vert _{C^{0, \left( 1 - \frac{1}{p}\right) \theta } ([0,T]; W^3_p (\varOmega ))} \\&\quad \le C(R) T^{\frac{1}{p} + \left( 1 - \frac{1}{p}\right) \theta } \Vert \varphi _1 - \varphi _2\Vert _{X^2_T} . \end{aligned}$$

Here, we also used \(\varphi _1 (0) = \varphi _2 (0) = \varphi _0\) for \(\varphi _1, \varphi _2 \in X^2_T\) and (20).

Now, there remain two terms which we need to study together for the proof of the Lipschitz continuity. Due to the boundedness of \(m (\varphi ) \) in \( BUC([0,T]; W^3_p (\varOmega ))\) and of \(\nabla \varDelta \varphi \) in \( L^p (0,T; W^1_p (\varOmega ))\), Theorem 3 yields the boundedness of \(m (\varphi ) \nabla \varDelta \varphi \) in \( L^p (0,T; W^1_p (\varOmega ))\). Hence, this term is well-defined in the \(L^p (Q_T)\)-norm. We omit the prefactor \(\varepsilon \) for both terms again and estimate

$$\begin{aligned}&\Vert m (\varphi _0) \varDelta ^2 \varphi _1 - m (\varphi _0) \varDelta ^2 \varphi _2 + \text {div}(m (\varphi _2) \nabla \varDelta \varphi _2) - \text {div}(m (\varphi _1) \nabla \varDelta \varphi _1)\Vert _{L^p (Q_T)} \nonumber \\&\quad = \Vert (m(\varphi _0) - m (\varphi _1)) ( \varDelta ^2 \varphi _1 - \varDelta ^2 \varphi _2) + m (\varphi _1) \varDelta ^2 \varphi _1 - m (\varphi _1) \varDelta ^2 \varphi _2 + \nabla m (\varphi _2) \cdot \nabla \varDelta \varphi _2 \nonumber \\&\qquad + m ( \varphi _2) \varDelta ^2 \varphi _2 - \nabla m (\varphi _1) \cdot \nabla \varDelta \varphi _1 - m (\varphi _1) \varDelta ^2 \varphi _1 \Vert _{L^p (Q_T)} \nonumber \\&\quad \le \Vert ( m ( \varphi _1 (0) - m (\varphi _1)) (\varDelta ^2 \varphi _1 - \varDelta ^2 \varphi _2 )\Vert _{L^p (Q_T)} + \Vert (m (\varphi _2) - m (\varphi _1)) \varDelta ^2 \varphi _2 \Vert _{L^p (Q_T)} \nonumber \\&\qquad + \Vert \nabla m (\varphi _2) \cdot \nabla \varDelta \varphi _2 - \nabla m ( \varphi _1) \cdot \nabla \varDelta \varphi _1 \Vert _{L^p (Q_T)} \end{aligned}$$
(24)

For the sake of clarity, we estimate these three terms separately again. Due to the boundedness of \(m (\varphi _1)\) in \( C^{0, (1 - \frac{1}{p}) \theta } ([0,T] ; W^3_p (\varOmega ))\) we obtain for the first term

$$\begin{aligned}&\Vert (m( \varphi _1 (0) - m (\varphi _1)) (\varDelta ^2 \varphi _1 -\varDelta ^2 \varphi _2 ) \Vert _{L^p (Q_T)} \\&\quad \le \underset{t \in (0,T)}{\sup } \Vert m(\varphi _1 (0)) -m ( \varphi _1 (t)) \Vert _{C^0 (\overline{\varOmega })} \Vert \varDelta ^2 \varphi _1 - \varDelta ^2 \varphi _2\Vert _{L^p (Q_T)} \\&\quad \le C(R) T^{\left( 1 - \frac{1}{p}\right) \theta } \Vert m (\varphi _1) \Vert _{C^{0, \left( 1-\frac{1}{p}\right) \theta } ([0,T]; W^3_p (\varOmega ))} \Vert \varphi _1 - \varphi _2\Vert _{X^2_T} . \end{aligned}$$

Since \(m (\varphi _1)\) is bounded in \( C^{0, (1 - \frac{1}{p}) \theta } ([0,T] ; W^3_p (\varOmega ))\), we can estimate the second term in (24) by

$$\begin{aligned}&\Vert (m (\varphi _2) - m (\varphi _1)) \varDelta ^2 \varphi _2 \Vert _{L^p (Q_T)} \le \underset{t \in (0,T)}{\sup }\Vert m (\varphi _2 (t)) -m (\varphi _1 (t))\Vert _{C^2 (\overline{\varOmega })} \Vert \varDelta ^2 \varphi _2\Vert _{L^p (Q_T)} \\&\quad \le C(R) \underset{t \in (0,T)}{\sup }\Vert m (\varphi _2 (t)) - m (\varphi _1 (t))\Vert _{W^3_p (\varOmega )} \\&\quad \le C(R) \underset{t \in (0,T)}{\sup }\Vert (\varphi _2 (t) -\varphi _1 (t)) - (\varphi _2(0) - \varphi _1 (0))\Vert _{W^3_p (\varOmega )} \\&\quad \le C(R) T^{\left( 1 - \frac{1}{p}\right) \theta } \Vert \varphi _1 -\varphi _2\Vert _{C^{0, \left( 1 - \frac{1}{p}\right) \theta } ([0,T]; W^3_p (\overline{\varOmega }))} , \end{aligned}$$

where we used (20) again in the penultimate step. Finally, we study the last term in (24). Here, we get

$$\begin{aligned}&\Vert \nabla m (\varphi _2) \cdot \nabla \varDelta \varphi _2 -\nabla m ( \varphi _1) \cdot \nabla \varDelta \varphi _1 \Vert _{L^p (Q_T)} \nonumber \\&\quad \le \Vert (\nabla m (\varphi _2) - \nabla m (\varphi _1)) \cdot \nabla \varDelta \varphi _2 \Vert _{L^p (Q_T)} \nonumber \\&\qquad + \Vert \nabla m (\varphi _1) \cdot ( \nabla \varDelta \varphi _2 -\nabla \varDelta \varphi _1)\Vert _{L^p (Q_T)} . \end{aligned}$$
(25)

Since \(\nabla m (\varphi _1) \) is bounded in \(C([0,T]; C^1 (\overline{\varOmega }))\) and \(\nabla \varDelta \varphi _i\) is bounded in C([0, T]; \( L^p (\varOmega ))\) for \(i = 1,2\), we can estimate the second summand by

$$\begin{aligned}&\Vert \nabla m (\varphi _1) \cdot ( \nabla \varDelta \varphi _2 -\nabla \varDelta \varphi _1)\Vert _{L^p (Q_T)} \\&\quad \le C(R) T^{\frac{1}{p}} \Vert \nabla m (\varphi _1)\Vert _{C ([0,T]; C^1 (\overline{\varOmega }))} \Vert \nabla \varDelta \varphi _1 - \nabla \varDelta \varphi _2\Vert _{C([0,T]; L^p (\varOmega ))} \\&\quad \le C(R) T^\frac{1}{p} \Vert \varphi _1 - \varphi _2\Vert _{X^2_T} . \end{aligned}$$

Thus, it remains to estimate the first term of (25). Here, we get

$$\begin{aligned}&\Vert (\nabla m (\varphi _2) - \nabla m (\varphi _1)) \cdot \nabla \varDelta \varphi _2 \Vert _{L^p (Q_T)} \\&\quad \le C(R) T^\frac{1}{p} \underset{t \in [0,T]}{\sup } \Vert \nabla m (\varphi _2(t)) - \nabla m (\varphi _1 (t))\Vert _{C^0 (\overline{\varOmega })} \Vert \nabla \varDelta \varphi _2\Vert _{C([0,T]; L^p (\varOmega ))} \\&\quad \le C(R) T^\frac{1}{p} \underset{t \in [0,T]}{\sup } \Vert m (\varphi _2 (t)) - m (\varphi _1 (t))\Vert _{W^3_p (\varOmega )} \Vert \varphi _2\Vert _{C([0,T] ; W^3_p (\varOmega ))} \\&\quad \le C(R) T^\frac{1}{p} \underset{t \in [0,T]}{\sup } \Vert \varphi _1 (t) - \varphi _2 (t)\Vert _{W^3_p (\varOmega )} \\&\quad \le C(R) T^{\frac{1}{p} + \left( 1 - \frac{1}{p}\right) \theta } \Vert \varphi _1 - \varphi _2\Vert _{C^{0, \left( 1 - \frac{1}{p}\right) \theta } ([0,T] ; W^3_p (\varOmega ))} . \end{aligned}$$

Hence, (25) is Lipschitz continuous and therefore also the second part of \({\mathcal {F}}\) is Lipschitz continuous. Together with the Lipschitz continuity of the first part of \({\mathcal {F}}\), we have shown

$$\begin{aligned} \Vert {\mathcal {F}} (\mathbf{v} _1 , \varphi _1 ) - {\mathcal {F}} (\mathbf{v} _2 , \varphi _2) \Vert _{Y_T} \le C (T, R) \Vert (\mathbf{v} _1 - \mathbf{v} _2, \varphi _1 - \varphi _2 )\Vert _{X_T} \end{aligned}$$

for all \((\mathbf{v} _i, \varphi _i) \in X_T\) with \(\Vert (\mathbf{v} _i, \varphi _i)\Vert _{X_T} \le R\), \(i = 1,2\), and a constant \(C(T,R) > 0 \) such that \(C (T,R) \rightarrow 0\) as \(T \rightarrow 0\).

Existence and continuity of \({\mathcal {L}}^{-1}\)

In the following, we need:

Theorem 5

Let the linear, symmetric and monotone operator \({\mathcal {B}}\) be given from the real vector space E to its algebraic dual \(E'\), and let \(E' _b\) be the Hilbert space which is the dual of E with the seminorm

$$\begin{aligned} |x|_b = {\mathcal {B}} x (x) ^\frac{1}{2}, \qquad x \in E . \end{aligned}$$

Let \(A \subseteq E \times E' _b\) be a relation with domain \(D=\{x\in E: \ A(x) \ne \emptyset \}\). Let A be the subdifferential, \(\partial \varphi \), of a convex lower-semi-continuous function \(\varphi : E_b \rightarrow [0, \infty ]\) with \(\varphi (0) = 0\). Then, for each \(u_0\) in the \(E_b\)-closure of \(\mathrm {dom} (\varphi )\) and each \(f \in L^2 (0,T; E' _b)\) there is a solution \(u : [0,T] \rightarrow E\) with \({\mathcal {B}} u \in C([0,T], E' _b)\) of

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} ( {\mathcal {B}} u (t) ) + A ( u (t)) \ni f(t) , \qquad 0< t < T, \end{aligned}$$

with

$$\begin{aligned} \varphi \circ u \in L^1 (0,T), \sqrt{t} \frac{\mathrm{d}}{\mathrm{d}t} {\mathcal {B}} u (\cdot ) \in L^2 (0,T; E' _b), u(t) \in D, \text { a.e. } t \in [0,T] , \end{aligned}$$

and \({\mathcal {B}} u(0) = {\mathcal {B}} u_0\). If in addition \(u_0 \in \mathrm {dom} (\varphi )\), then

$$\begin{aligned} \varphi \circ u \in L^\infty (0,T), \qquad \frac{\mathrm{d}}{\mathrm{d}t} {\mathcal {B}} u \in L^2 (0,T; E' _b) . \end{aligned}$$

The proof of this theorem can be found in [15, Chapter IV, Theorem 6.1].

To prove Theorem 4, we need to show the existence of \((\tilde{\mathbf{v }}, \tilde{\varphi }) \in X_T\) such that (11) holds and to prove that \({\mathcal {L}} : X_T \rightarrow Y_T\) is invertible with uniformly bounded inverse, i.e., there exists a constant \(C > 0\) which does not depend on T such that \(\Vert {\mathcal {L}}^{-1} \Vert _{{\mathcal {L}} (Y_T, X_T)} \le C\). Recall that the linear operator \({\mathcal {L}} :X_T \rightarrow Y_T\) is defined by

$$\begin{aligned} {\mathcal {L}} (\mathbf{v} , \varphi ) =\begin{pmatrix} {\mathbb {P}}_\sigma ( \rho _0 \partial _t \mathbf{v} ) -{\mathbb {P}}_\sigma ( \text {div}(2 \eta (\varphi _0) D\mathbf{v} )) \\ \partial _t \varphi + \varepsilon m (\varphi _0) \varDelta ^2 \varphi \end{pmatrix} . \end{aligned}$$

We note that the first part only depends on \(\mathbf{v} \) while the second part only depends on \(\varphi \). Thus, both equations can be solved separately.

To show the existence of a unique solution \(\mathbf{v} \) for every right-hand side \(\mathbf{f}\) in the first equation, we use Theorem 5.

So we have to specify what E, \(E ' _b\), \(\varphi \) and so on are in the problem we study and show that the conditions of Theorem 5 are fulfilled. Then, Theorem 5 yields the existence of a solution. More precisely, we obtain the following lemma.

Lemma 3

Let Assumption 1 hold. Then, for every \(\mathbf{v}_0 \in H^1_0 (\varOmega )^d \cap L^2_\sigma (\varOmega )\), \(f \in L^2 (0,T; L^2_\sigma (\varOmega ))\), \(\varphi _0 \in W^{1}_r (\varOmega )\), \(r > d \ge 2\), and every \(0< T < \infty \) there exists a unique solution

$$\begin{aligned} \mathbf{v}\in W^1_2 (0,T; L^2_\sigma (\varOmega )) \cap L^\infty (0,T; H^1_0 (\varOmega )^d) \end{aligned}$$

such that

$$\begin{aligned} {\mathbb {P}}_\sigma (\rho _0 \partial _t \mathbf{v}) - {\mathbb {P}}_\sigma (\mathrm {div} (2 \eta (\varphi _0) D \mathbf{v}))&= f \quad \text { in } Q_T, \end{aligned}$$
(26)
$$\begin{aligned} \mathrm {div} (\mathbf{v})&= 0 \quad \text { in } Q_T , \end{aligned}$$
(27)
$$\begin{aligned} \mathbf{v}_{|\partial \varOmega }&= 0 \quad \text { on } (0, T) \times \partial \varOmega , \end{aligned}$$
(28)
$$\begin{aligned} \mathbf{v} (0)&= \mathbf{v}_0 \quad \text { in } \varOmega \end{aligned}$$
(29)

for a.e. (tx) in \((0,T) \times \varOmega \), where \(\mathbf{v} (t) \in H^2 (\varOmega )^d\) for a.e. \(t \in (0,T)\).

Proof

Since we want to solve (26)–(29) with Theorem 5, we define

$$\begin{aligned} {\mathcal {B}} u := {\mathbb {P}} _\sigma ( \rho _0 u) \end{aligned}$$

for \(u \in E\), where we still need to specify the real vector space E. But as we want to have \(\frac{\mathrm{d}}{\mathrm{d}t} {\mathcal {B}} u \in L^2 (0,T; L^2_\sigma (\varOmega ))\), the dual space \(E'_b\) has to coincide with \(L^2_\sigma (\varOmega )\). But this can be realized by choosing \(E = L^2_\sigma (\varOmega )\). Then, \(E_b' \cong L^2_\sigma (\varOmega )\) and with the notation in Theorem 5, we get the Hilbert space \(E'_b\) equipped with the seminorm

$$\begin{aligned} |\mathbf{u} |_b = {\mathcal {B}} \mathbf{u} ( \mathbf{u} ) ^\frac{1}{2}&= \left( \int _{\varOmega }{\mathbb {P}} _\sigma ( \rho _0 \mathbf{u} ) \cdot \mathbf{u} \mathrm{d}{} { x}\right) ^\frac{1}{2} = \left( \int _{\varOmega }\rho _0 \mathbf{u} \cdot {\mathbb {P}}_\sigma \mathbf{u} \mathrm{d}{} { x}\right) ^\frac{1}{2} \\&= \left( \int _{\varOmega }\rho _0 |\mathbf{u} |^2 \mathrm{d}{} { x}\right) ^\frac{1}{2} \cong \Vert \mathbf{u} \Vert _{L^2 (\varOmega )} . \end{aligned}$$

Thus, we obtain \(E '_b \cong L^2_\sigma (\varOmega ) = E_b\). Moreover, we define \(A: {\mathcal {D}} (A) \rightarrow L^2_\sigma (\varOmega ) ' \cong L^2_\sigma (\varOmega )\) by

$$\begin{aligned} (A \mathbf{u} )(\mathbf{v} ) :={\left\{ \begin{array}{ll} - \int _{\varOmega }{\mathbb {P}} _\sigma \text {div}(2 \eta (\varphi _0) D\mathbf{u} ) \cdot \mathbf{v} \mathrm{d}{} { x}&{} \text { if } \mathbf{u} \in \mathrm {dom} (A) \\ \emptyset &{} \text { if } \mathbf{u} \notin \mathrm {dom} (A) . \end{array}\right. } \end{aligned}$$
(30)

for every \( \mathbf{v} \in L^2_\sigma (\varOmega )\) and \({\mathcal {D}} (A) =H^2 (\varOmega )^d \cap H^1_0 (\varOmega )^d \cap L^2_\sigma (\varOmega )\). Thus, we get for the relation \({\mathcal {A}}\) defined by \({\mathcal {A}} := \{ (\mathbf{u} ,\mathbf{v} ) : \ \mathbf{v} = A\mathbf{u} , \ \mathbf{u} \in {\mathcal {D}} (A) \} \) the following inclusions

$$\begin{aligned} {\mathcal {A}} = \{ (\mathbf{u} , - {\mathbb {P}}_\sigma \text {div}(2 \eta (\varphi _0) D\mathbf{u} ) : \mathbf{u} \in H^2 (\varOmega )^d \cap H^1_0 (\varOmega )^d \cap L^2_\sigma (\varOmega ) \} \subseteq E \times E'_b , \end{aligned}$$

since the term \( {\mathbb {P}}_\sigma \text {div}(2 \eta (\varphi _0) D\mathbf{u} )\) is in \(L^2 _\sigma (\varOmega ) ' \cong L^2_\sigma (\varOmega )\). Now, we define \(\psi : L^2_\sigma (\varOmega ) \rightarrow [0, + \infty ]\) by

$$\begin{aligned} \psi (\mathbf{u} ) :={\left\{ \begin{array}{ll} \int _{\varOmega }\eta (\varphi _0) D\mathbf{u} : D\mathbf{u} \ dx &{} \text { if } \mathbf{u} \in H^1_0 (\varOmega )^d \cap L^2_\sigma (\varOmega ) = \mathrm {dom} (\psi ), \\ + \infty &{} \text { else. } \end{array}\right. } \end{aligned}$$
(31)

We note \(\psi (0) = 0\) and \(\mathbf{v} _0\) is in the \(L^2\)-closure of \(\mathrm {dom}(\psi )\), i.e., in \(L^2_\sigma (\varOmega )\). Hence, it remains to show that \(\psi \) is convex and lower-semi-continuous and that A is the subdifferential of \(\psi \). Then, we can apply Theorem 5. But the first two properties are obvious. Thus, it remains to show the subdifferential property, which is satisfied by Lemma 4 below. Hence, we are able to apply Theorem 5 which yields the existence. Moreover, the initial condition is also fulfilled as Theorem 5 yields

$$\begin{aligned} {\mathbb {P}}_\sigma (\rho _0 \mathbf{v} (0)) = {\mathcal {B}} \mathbf{v} (0) = {\mathcal {B}} \mathbf{v} _0 = {\mathbb {P}}_\sigma (\rho _0 \mathbf{v} _0) \qquad \text { in } L^2 (\varOmega ) . \end{aligned}$$

In particular, we can conclude

$$\begin{aligned} 0&= \int _\varOmega {\mathbb {P}}_\sigma ( \rho _0 \mathbf{v} (0) -\rho _0 \mathbf{v} _0 ) \cdot \varvec{\psi }\mathrm{d}{x} = \int _\varOmega (\rho _0 \mathbf{v} (0) - \rho _0 \mathbf{v} _0) \cdot \varvec{\psi }\mathrm{d}{x} \end{aligned}$$

for every \(\varvec{\psi }\in C^\infty _{0, \sigma } (\varOmega )\). By approximation, this identity also holds for \(\varvec{\psi }:=\mathbf{v} (0) - \mathbf{v} _0 \in L^2_\sigma (\varOmega )\) and we get

$$\begin{aligned} \int _\varOmega \rho _0 |\mathbf{v} (0) - \mathbf{v} _0|^2 \mathrm{d}{x} = 0 . \end{aligned}$$

This implies \(\mathbf{v} (0) = \mathbf{v} _0\) in \(L^2_\sigma (\varOmega )\).

For the uniqueness, we consider \(\mathbf{v} _1 , \mathbf{v} _2 \in W^1_2 (0,T; L^2_\sigma (\varOmega )) \cap L^\infty (0,T; H^1_0 (\varOmega ) \cap L^2_\sigma (\varOmega ))\) such that (26) holds for a.e. \((t,x) \in (0,T) \times \varOmega \). Then, \(\mathbf{v} := \mathbf{v} _1 - \mathbf{v} _2\) solves the homogeneous equation a.e. in \((0,T) \times \varOmega \). Testing this homogeneous equation with \(\mathbf{v} \), we get

$$\begin{aligned} \int _{\varOmega }\frac{1}{2} \rho _0 \mathbf{v} ^2_{|t=T} \mathrm{d}{} { x}+\int _0^T \int _{\varOmega }2 \eta (\varphi _0) D\mathbf{v} : D\mathbf{v} \mathrm{d}{} { x}\mathrm{d}{} { t}= 0. \end{aligned}$$

Hence, it follows \(\mathbf{v} \equiv 0\) and therefore, \(\mathbf{v} _1 = \mathbf{v} _2\), which yields the uniqueness. \(\square \)

In the proof above, we used that the mapping A coincides with the subdifferential \(\partial \varphi \). More precisely, we have the following lemma.

Lemma 4

Let \(\varOmega \subseteq {\mathbb {R}}^d\), \(d = 2,3\), be a domain and \(\psi :L^2_\sigma (\varOmega ) \rightarrow [0, + \infty ]\) be given as in (31). Moreover, we consider \(A : L^2_\sigma (\varOmega ) \rightarrow L^2_\sigma (\varOmega )\) to be given as in (30). Then, it holds

  1. 1.

    \({\mathcal {D}} (\partial \psi ) = {\mathcal {D}} (A)\).

  2. 2.

    \(\partial \psi (\mathbf{u} ) = \{ A \mathbf{u} \} \text { for all } \mathbf{u} \in {\mathcal {D}} (A) .\)

Proof

Remember that

$$\begin{aligned} {\mathcal {D}} (\partial \psi ) = \{ \mathbf{v} \in L^2_\sigma (\varOmega ) : \ \partial \psi (\mathbf{v} ) \ne \emptyset \} \end{aligned}$$

and \({\mathcal {D}} (A) = H^2 (\varOmega )^d \cap H^1_0 (\varOmega )^d \cap L^2_\sigma (\varOmega )\) by definition.

\(\mathbf{1} ^{\mathbf{st}}\) part: \({\mathcal {D}} (A) \subseteq {\mathcal {D}} (\partial \psi )\) and \(A\mathbf{u} \in \partial \psi (\mathbf{u} )\) for every \(\mathbf{u} \in {\mathcal {D}} (A)\).

To show the first part of the proof, let \(\mathbf{u} \in {\mathcal {D}} (A)\) be given. If it holds \(\mathbf{v} \in L^2_\sigma (\varOmega )\) but \(\mathbf{v} \notin H^1_0 (\varOmega )^d\), then the inequality

$$\begin{aligned} \langle {A\mathbf{u} , \mathbf{v} -\mathbf{u} }\rangle _{L^2 (\varOmega )} \le \psi (\mathbf{v} ) - \psi (\mathbf{u} ) \end{aligned}$$

is satisfied since it holds \(\psi (\mathbf{v} ) = + \infty \) in this case by definition.

So, let \(\mathbf{v} \in H^1_0 (\varOmega )^d \cap L^2_\sigma (\varOmega )\). Then, it holds

$$\begin{aligned} \langle {A\mathbf{u} , \mathbf{v} - \mathbf{u} }\rangle _{L^2 (\varOmega )}&= - \int _{\varOmega }{\mathbb {P}}_\sigma \text {div}(2 \eta (\varphi _0) D\mathbf{u} ) \cdot (\mathbf{v} - \mathbf{u} ) \mathrm{d}{} { x}\\&= \int _{\varOmega }2 \eta (\varphi _0) D\mathbf{u} : D\mathbf{v} \mathrm{d}{} { x}- \int _{\varOmega }2 \eta (\varphi _0) D\mathbf{u} : D \mathbf{u} \mathrm{d}{} { x}\\&\le \int _{\varOmega }\eta (\varphi _0) |D\mathbf{u} |^2 \mathrm{d}{} { x}+ \int _{\varOmega }\eta (\varphi _0) |D\mathbf{v} |^2 \mathrm{d}{} { x}- 2 \int _{\varOmega }\eta (\varphi _0) |D\mathbf{u} |^2 \mathrm{d}{} { x}\\&= \psi (\mathbf{v} ) - \psi (\mathbf{u} ) \end{aligned}$$

for every \(\mathbf{v} \in H^1_0 (\varOmega )^d \cap L^2_\sigma (\varOmega )\). This implies that \(A\mathbf{u} \) is a subgradient of \(\psi \) at \(\mathbf{u} \), i.e., \(A\mathbf{u} \in \partial \psi (\mathbf{u} )\), and \(\partial \psi (\mathbf{u} ) \ne \emptyset \), i.e., \(\mathbf{u} \in {\mathcal {D}} (A) \subseteq {\mathcal {D}} (\partial \psi )\). Hence, we have shown the first part of the proof.

\(\mathbf{2} ^{\mathbf{nd}}\) part: \({\mathcal {D}} (\partial \psi ) \subseteq {\mathcal {D}} (A) \) and \(\partial \psi (\mathbf{u} ) = \{A \mathbf{u} \}\).

Let \( \mathbf{u} \in {\mathcal {D}} (\partial \psi ) \subseteq \mathrm {dom} (\psi ) \subseteq H^1_0 (\varOmega )^d \cap L^2_\sigma (\varOmega )\) be given. Then, by definition of the subgradient there exists \(\mathbf{w} \in \partial \psi (\mathbf{u} ) \subseteq L^2_\sigma (\varOmega )\) such that

$$\begin{aligned} \psi (\mathbf{u} ) - \psi (\mathbf{v} ) \le \langle {\mathbf{w} , \mathbf{u} - \mathbf{v} }\rangle _{L^2 (\varOmega )} \end{aligned}$$
(32)

for every \(\mathbf{v} \in L^2_\sigma (\varOmega )\). Now, we choose \(\mathbf{v} := \mathbf{u} +t \tilde{\mathbf{w }}\) for some \(\tilde{\mathbf{w }} \in H^1_0 (\varOmega )^d \cap L^2_\sigma (\varOmega )\) and \( t > 0\). Then, inequality (32) yields

$$\begin{aligned} \psi (\mathbf{u} ) - \psi (\mathbf{v} )&= \int _{\varOmega }\eta (\varphi _0) D\mathbf{u} : D\mathbf{u} \mathrm{d}{} { x}-\int _{\varOmega }\eta (\varphi _0) D(\mathbf{u} + t \tilde{\mathbf{w }}) : D ( \mathbf{u} + t \tilde{\mathbf{w }} ) \mathrm{d}{} { x}\\&= - 2 t \int _{\varOmega }\eta (\varphi _0) D\mathbf{u} : D \tilde{\mathbf{w }} \mathrm{d}{} { x}-t^2 \int _{\varOmega }\eta (\varphi _0) D\mathbf{u} : D \tilde{\mathbf{w }} \mathrm{d}{} { x}\\&\le -t \int _{\varOmega }\mathbf{w} \cdot \tilde{\mathbf{w }} \mathrm{d}{} { x}. \end{aligned}$$

Dividing this inequality by \( -t < 0\) and passing to the limit \(t\searrow 0\) yields

$$\begin{aligned} \int _{\varOmega }\mathbf{w} \cdot \tilde{\mathbf{w }} \mathrm{d}{} { x}\le \int _{\varOmega }2 \eta (\varphi _0) D\mathbf{u} : D \tilde{\mathbf{w }} \mathrm{d}{} { x}. \end{aligned}$$

When we replace \(\tilde{\mathbf{w }} \) by \(- \tilde{\mathbf{w }}\), we can conclude

$$\begin{aligned} \int _{\varOmega }\mathbf{w} \cdot \tilde{\mathbf{w }} \mathrm{d}{} { x}\ge \int _{\varOmega }2 \eta (\varphi _0) D\mathbf{u} : D \tilde{\mathbf{w }} \mathrm{d}{} { x}. \end{aligned}$$

Thus, it follows

$$\begin{aligned} \int _{\varOmega }\mathbf{w} \cdot \tilde{\mathbf{w }} \mathrm{d}{} { x}= \int _{\varOmega }2 \eta (\varphi _0) D\mathbf{u} : D \tilde{\mathbf{w }} \mathrm{d}{} { x}\end{aligned}$$
(33)

for every \(\tilde{\mathbf{w }} \in H^1_0 (\varOmega )^d \cap L^2_\sigma (\varOmega )\). Since we assumed \(\mathbf{w} \in L^2_\sigma (\varOmega )\), we can apply Lemma 5 below which yields \(\mathbf{u} \in H^2 (\varOmega )^d \cap H^1_0 (\varOmega )^d \cap L^2_\sigma (\varOmega )\). Using this regularity in (33), we can conclude

$$\begin{aligned} \int _{\varOmega }\mathbf{w} \cdot \tilde{\mathbf{w }} \mathrm{d}{} { x}= \int _{\varOmega }2 \eta (\varphi _0) D\mathbf{u} : D \tilde{\mathbf{w }} \mathrm{d}{} { x}= - \int _{\varOmega }{\mathbb {P}}_\sigma \text {div}(2 \eta (\varphi _0) D\mathbf{u} ) \cdot \tilde{\mathbf{w }} \mathrm{d}{} { x}\end{aligned}$$

for every \(\tilde{\mathbf{w }} \in H^1_0 (\varOmega )^d \cap L^2_\sigma (\varOmega )\). Therefore, we obtain \(\mathbf{w} = - {\mathbb {P}} _\sigma \text {div}(2 \eta (\varphi _0) D\mathbf{u} ) = A\mathbf{u} \) in \(L^2 (\varOmega )\), i.e., \(\mathbf{u} \in {\mathcal {D}} (A)\) and \(\partial \psi (\mathbf{u} ) = \{ A\mathbf{u} \}\). \(\square \)

For the regularity of the Stokes system with variable viscosity, we used the following lemma.

Lemma 5

Let \(\eta \in C^2 ({\mathbb {R}})\) be such that \(\eta (s) \ge s_0 >0\) for all \(s \in {\mathbb {R}}\) and some \(s_0 > 0\), \(\varphi _0 \in W^{1}_r (\varOmega )\), \(r > d \ge 2\), with \(\Vert \varphi _0\Vert _{W^{1}_r (\varOmega )} \le R\), and let \(\mathbf{u} \in H^1_0 (\varOmega )^d \cap L^2_\sigma (\varOmega )\) be a solution of

$$\begin{aligned} \langle {2 \eta (\varphi _0) D \mathbf{u}, D \tilde{\mathbf{w }}}\rangle _{L^2 (\varOmega )} = \langle {\mathbf{w} , \tilde{\mathbf{w }}}\rangle _{L^2 (\varOmega )} \qquad \text { for all }\tilde{\mathbf{w }} \in C^\infty _{0, \sigma } (\varOmega ) , \end{aligned}$$

where \( \mathbf{w} \in L^2 (\varOmega )^d\). Then, it holds \(\mathbf{u} \in H^2 (\varOmega )^d\) and

$$\begin{aligned} \Vert \mathbf{u}\Vert _{H^2 (\varOmega )} \le C(R) \Vert \mathbf{w}\Vert _{L^2 (\varOmega )} , \end{aligned}$$

where C(R) only depends on \(\varOmega \), \(\eta \), \(r > d\), and \(R > 0\).

The proof can be found in [2, Lemma 4].

Lemma 3 implies \(\mathbf{v} \in W_2^1 (0,T; L^2 _\sigma (\varOmega )) \cap L^\infty (0,T; H^1_0 (\varOmega )^d)\). But as we want to show that \(\mathbf{v} \) is in \(X^1_T\), it remains to show \(\mathbf{v} \in L^2 (0,T; H^2 (\varOmega )^d)\). To this end, we also use Lemma 5 above.

Lemma 6

For the unique solution \(\mathbf{v} \in W_2^1 (0,T; L^2_\sigma (\varOmega )) \cap L^\infty (0,T; H^1_0 (\varOmega )^d)\) of (26)–(29) from Lemma 3, it holds \(\mathbf{v} \in L^2 (0,T; H^2 (\varOmega )^d)\).

Proof

Let \(\mathbf{v} \in W_2^1 (0,T; L^2_\sigma (\varOmega )) \cap L^\infty (0,T; H^1_0 (\varOmega )^d)\) be the unique solution of (26) from Lemma 3, i.e.,

$$\begin{aligned} A ( \mathbf{v} (t)) = \mathbf{f} (t) - \frac{\mathrm{d}}{\mathrm{d}t} ( {\mathcal {B}} \mathbf{v} (t)) = \mathbf{f}(t) - {\mathbb {P}}_\sigma (\rho _0 \partial _t \mathbf{v} (t)) \quad \text {for all } 0< t < T . \end{aligned}$$

Since the right-hand side is not the empty set, we get by definition of A

$$\begin{aligned} {\mathbb {P}}_\sigma (\text {div}(2 \eta (\varphi _0) D\mathbf{v} (t))) ={\mathbb {P}}_\sigma (\rho _0 \partial _t \mathbf{v} (t)) - \mathbf{f} (t) \quad \text {for all } 0< t < T \end{aligned}$$

for given \(\mathbf{f} \in L^2 (0,T; L^2_\sigma (\varOmega ))\). From \(\partial _t \mathbf{v} \in L^2 (0,T; L^2_\sigma (\varOmega ))\), it follows

$$\begin{aligned} \langle { 2 \eta (\varphi _0) D\mathbf{v} (t) , D \mathbf{w} }\rangle _{L^2 (\varOmega )} = \langle {\rho _0 \partial _t \mathbf{v} (t) - \mathbf{f} (t) , \mathbf{w} }\rangle \qquad \text { for every } \mathbf{w} \in C^\infty _{0, \sigma } (\varOmega ) \end{aligned}$$

and a.e. \(t \in (0,T)\). Hence, we can apply Lemma 5 and obtain

$$\begin{aligned} \Vert \mathbf{v} (t)\Vert _{H^2 (\varOmega )}&\le C(R) \Vert \rho _0 \partial _t \mathbf{v} (t) -\mathbf{f} (t) \Vert _{L^2 (\varOmega )} \\&\le C(R) \left( \Vert \rho _0 \partial _t \mathbf{v} (t)\Vert _{L^2 (\varOmega )} +\Vert \mathbf{f} (t) \Vert _{L^2 (\varOmega )} \right) \end{aligned}$$

for a.e. \(t \in (0,T)\). Since the right-hand side of this inequality is bounded in \(L^2 (0,T)\), this shows the lemma. \(\square \)

We still need to ensure that \(\Vert {\mathcal {L}}^{-1}\Vert _{{\mathcal {L}} (Y_T, X_T)} \) remains bounded. This is shown in the next lemma.

Lemma 7

Let the assumptions of Lemma 3 hold and \(0<T_0 < \infty \) be given. Then,

$$\begin{aligned} \Vert {\mathcal {L}}^{-1}_{1, T}\Vert _{{\mathcal {L}} (Y^1_T, X^1_T)} \le \Vert {\mathcal {L}}^{-1}_{1, T_0}\Vert _{{\mathcal {L}} (Y^1_{T_0}, X^1_{T_0})}< \infty \qquad \text { for all } 0< T < T_0 . \end{aligned}$$

Proof

Let \( 0< T < T_0 \) be given. Lemma 3 together with Lemma 6 yields that the operator \({\mathcal {L}}_{1, T} : X_T \rightarrow Y_T\) is invertible for every \(0< T < \infty \) and every given \(\mathbf{f} \in L^2(0,T; L^2_\sigma (\varOmega ))\), \(\varphi _0 \in W^1_r (\varOmega )\), \(\mathbf{v} _0 \in H^1_0 (\varOmega )^d \cap L^2_\sigma (\varOmega )\). Then, we define \(\tilde{\mathbf{f}} \in L^2 (0,T_0, L^2_\sigma (\varOmega ))\) by

$$\begin{aligned} \tilde{\mathbf{f}} (t) :={\left\{ \begin{array}{ll} \mathbf{f}(t) &{} \text { if } t \in (0,T] , \\ 0 &{} \text { if } t \in (T, T_0 ) . \end{array}\right. } \end{aligned}$$

Due to Lemma 3 together with Lemma 6, there exists a unique solution \(\tilde{\mathbf{v }} \in X^1_{T_0}\) of

$$\begin{aligned} {\mathbb {P}}_\sigma (\rho _0 \partial _t \tilde{\mathbf{v }}) - {\mathbb {P}}_\sigma (\text {div}(2 \eta (\varphi _0) D \tilde{\mathbf{v }}))&= \tilde{\mathbf{f}} \quad \text { in } Q_{T_0}, \\ \text {div}( \tilde{\mathbf{v }})&= 0 \quad \text { in } Q_{T_0}, \\ \tilde{\mathbf{v }}_{|\partial \varOmega }&= 0 \quad \text { on } (0, T_0) \times \partial \varOmega , \\ \tilde{\mathbf{v }} (0)&= \mathbf{v} _0 \quad \text { in } \varOmega . \end{aligned}$$

So let \(\mathbf{v} \in X^1_T\) be the solution of the previous equations with \(T_0\) replaced by T. Then, \(\tilde{\mathbf{v }}\) and \(\mathbf{v} \) solve these equations on \((0,T) \times \varOmega \). Since the solution is unique, we can deduce \(\tilde{\mathbf{v }}_{|(0,T)} = \mathbf{v} \). Hence,

$$\begin{aligned} \Vert {\mathcal {L}}^{-1}_{1, T} (\mathbf{f}) \Vert _{X^1_T}&= \Vert \mathbf{v} \Vert _{X^1_{T}} \le \Vert \tilde{\mathbf{v }}\Vert _{X^1_{T_0}} =\Vert {\mathcal {L}}^{-1}_{1, T_0} (\tilde{\mathbf{f}})\Vert _{X^1_{T_0}} \\&\le \Vert {\mathcal {L}}^{-1}_{1, T_0}\Vert _{{\mathcal {L}} (Y^1_{T_0}, X^1_{T_0})} \Vert \tilde{\mathbf{f}}\Vert _{Y^1_{T_0}} = \Vert {\mathcal {L}}^{-1}_{1, T_0}\Vert _{{\mathcal {L}} (Y^1_{T_0}, X^1_{T_0})} \Vert \mathbf{f}\Vert _{Y^1_{T}} , \end{aligned}$$

which shows the statement since it holds \(\Vert {\mathcal {L}}^{-1}_{1, T_0}\Vert _{{\mathcal {L}} (Y^1_{T_0}, X^1_{T_0})} < \infty \) by the bounded inverse theorem. \(\square \)

Finally, we have to show invertibility of the second part of \({\mathcal {L}}\).

Lemma 8

Let Assumption 1 hold and \(\varphi _0 \in (L^p (\varOmega ) , W^4_{p,N} (\varOmega ))_{1 - \frac{1}{p}, p}\), \(f \in L^p (0,T; L^p (\varOmega )) \) with \(4< p < 6\) be given. Then, for every \(0< T < \infty \) there exists a unique

$$\begin{aligned} \varphi \in L^p (0,T; W^4_{p,N} (\varOmega )) \cap \{ u \in W^1_p(0,T; L^p (\varOmega )) : \ u_{|t=0} = \varphi _0 \} \end{aligned}$$

such that

$$\begin{aligned} \partial _t \varphi + \varepsilon m (\varphi _0) \varDelta ^2 \varphi&= f \quad \text { in } (0,T) \times \varOmega , \end{aligned}$$
(34)
$$\begin{aligned} \partial _n \varphi _{|\partial \varOmega }&= 0 \quad \text { on } (0,T) \times \partial \varOmega , \end{aligned}$$
(35)
$$\begin{aligned} \partial _n \varDelta \varphi _{|\partial \varOmega }&= 0 \quad \text { on } (0,T) \times \partial \varOmega , \end{aligned}$$
(36)
$$\begin{aligned} \varphi (0)&= \varphi _{0} \quad \text { in } \{ 0 \} \times \varOmega . \end{aligned}$$
(37)

Proof

The result follows from standard results on maximal regularity of parabolic equations, e.g., from [9, Theorem 8.2]. \(\square \)

Analogously, to the previous part, we need to ensure that \(\Vert {\mathcal {L}}^{-1}\Vert _{{\mathcal {L}} (Y_T, X_T)} \) remains bounded.

Lemma 9

Let the assumptions of Lemma 8 hold and \(0<T_0 < \infty \) be given. Then,

$$\begin{aligned} \Vert {\mathcal {L}}^{-1}_{2, T}\Vert _{{\mathcal {L}} (Y^2_T, X^2_T)} \le \Vert {\mathcal {L}}^{-1}_{2, T_0}\Vert _{{\mathcal {L}} (Y^2_{T_0}, X^2_{T_0})}< \infty \qquad \text { for all } 0< T < T_0 . \end{aligned}$$

This lemma can be proven analogously to Lemma 7.

From the results of this section, Theorem 4 follows immediately.

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Correspondence to Helmut Abels.

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Dedicated to Matthias Hieber on the occasion of his 60th birthday

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The authors acknowledge support by the SPP 1506 “Transport Processes at Fluidic Interfaces” of the German Science Foundation (DFG) through Grant GA695/6-1 and GA695/6-2. The results are part of the second author’s PhD-thesis [16].

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Abels, H., Weber, J. Local well-posedness of a quasi-incompressible two-phase flow. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028-020-00646-2

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Keywords

  • Two-phase flow
  • Navier–Stokes equation
  • Diffuse interface model
  • Mixtures of viscous fluids
  • Cahn–Hilliard equation

Mathematics Subject Classification

  • 76T99
  • 35Q30
  • 35Q35
  • 76D03
  • 76D05
  • 76D27
  • 76D45