A Numerical Approach for the Existence of Dissipative Weak Solutions to a Compressible Two-fluid Model

Abstract

As an extension of the recent work of Novotný et al. (J Elliptic Parabol Equ 7:537–570 2021), we study the dissipative weak solutions to a compressible two-fluid model system describing the time evolution of two fluid flows sharing the same velocity field in multi-dimensional spaces. We prove the existence of dissipative weak solutions alternatively via a finite volume approximation. Further, we apply the weak–strong uniqueness principle to show the convergence of the finite volume approximation towards the strong solution on the lifespan of the latter.

A Appendix

This appendix is dedicated to the proof of Proposition 1.1. To this end, we establish the relative energy inequality to (1.1)–(1.5) satisfied for any dissipative weak solutions and any suitable test functions. Then the proof is finished with the help of Gronwall-type argument. Note that we shall present the proof in case of Dirichlet boundary conditions. The case of periodic boundary conditions can be carried out analogously and the details are therefore omitted.

1.1 A.1 Relative energy inequality

Let $(\varrho ,n,{\varvec{u}})$ be a dissipative weak solution to (1.1)–(1.5) and $(r,b,{\mathbf {U}})$ belongs to

$$\begin{aligned} \left\{ \begin{aligned}&r,b\in C^1(\overline{Q_T}) ,\,\, r,b>0 \text { in }\overline{Q_T}, \\&{\mathbf {U}}\in C^1(\overline{Q_T} ;\mathbb {R}^d),\,\,{\mathbf {U}}|_{\partial \Omega }={\varvec{0}}.\\ \end{aligned}\right. \end{aligned}$$

(A.1)

Similar to [19], we introduce the relative energy as

$$\begin{aligned} {\mathcal {E}}\Big ((\varrho ,n,{\varvec{u}})\,\Big |\,(r,b,{\mathbf {U}})\Big )(\tau ) :=\int _{\Omega } \Big [ \frac{1}{2}{\mathfrak {r}}|{\varvec{u}}-{\mathbf {U}}|^2 \end{aligned}$$

$$\begin{aligned} +H(\varrho )-H(r)-H'(r)(\varrho -r)+G(n)-G(b)-G'(b)(n-b) \Big ] \mathrm{d} x. \end{aligned}$$

(A.2)

Notice that we may rewrite the relative energy in an equivalent form as follows

$$\begin{aligned} {\mathcal {E}}\Big ((\varrho ,n,{\varvec{u}})\,\Big |\,(r,b,{\mathbf {U}})\Big )(\tau )= & {} \int _{\Omega } \left( \frac{1}{2}{\mathfrak {r}}|{\varvec{u}}|^2+H(\varrho )+G(n) \right) \mathrm{d} x+\int _{\Omega }\left( \frac{1}{2}\varrho |{\mathbf {U}}|^2 -H'(r)\varrho \right) \mathrm{d} x\\&+\int _{\Omega }\left( \frac{1}{2}n |{\mathbf {U}}|^2 -G'(b)n \right) \mathrm{d} x\end{aligned}$$

$$\begin{aligned} -\int _{\Omega } {\mathfrak {r}}{\varvec{u}}\cdot {\mathbf {U}} \mathrm{d} x+\int _{\Omega } \left( r^{\gamma }+b^{\alpha } \right) \mathrm{d} x. \end{aligned}$$

(A.3)

The crucial observation is that the integrals on the right-hand side of (A.3) can be expressed through the weak formulations (1.7)–(1.10) with suitable choices of test functions. To handle the density-dependent terms, testing the continuity Eq. (1.7) by $\frac{1}{2}|{\mathbf {U}}|^2$ and $H'(r)$ gives

$$\begin{aligned}&\int _0^{\tau }\int _{\Omega } \Big ( \varrho {\mathbf {U}}\cdot \partial _t {\mathbf {U}}+ \varrho {\varvec{u}}\cdot \nabla _x {\mathbf {U}} \cdot {\mathbf {U}} \Big )\mathrm{d} x\mathrm{d} t = \left[ \int _{\Omega } \frac{1}{2} \varrho |{\mathbf {U}}|^2 \mathrm{d} x\right] _{t=0}^{t=\tau }; \end{aligned}$$

(A.4)

$$\begin{aligned}&\int _0^{\tau }\int _{\Omega } \Big ( \varrho \partial _t H'(r)+ \varrho {\varvec{u}}\cdot \nabla _x H'(r) \Big )\mathrm{d} x\mathrm{d} t = \left[ \int _{\Omega } \varrho H'(r) \mathrm{d} x\right] _{t=0}^{t=\tau } ; \end{aligned}$$

(A.5)

In the same manner, we test the continuity Eq. (1.8) by $\frac{1}{2}|{\mathbf {U}}|^2$ and $G'(b)$ to obtain

$$\begin{aligned} \int _0^{\tau }\int _{\Omega } \Big ( n {\mathbf {U}}\cdot \partial _t {\mathbf {U}}+ n {\varvec{u}}\cdot \nabla _x {\mathbf {U}} \cdot {\mathbf {U}} \Big )\mathrm{d} x\mathrm{d} t = \left[ \int _{\Omega } \frac{1}{2} n |{\mathbf {U}}|^2 \mathrm{d} x\right] _{t=0}^{t=\tau }; \end{aligned}$$

(A.6)

$$\begin{aligned} \int _0^{\tau }\int _{\Omega } \Big ( n \partial _t G'(b)+ n{\varvec{u}}\cdot \nabla _x G'(b) \Big )\mathrm{d} x\mathrm{d} t = \left[ \int _{\Omega } n G'(b) \mathrm{d} x\right] _{t=0}^{t=\tau } . \end{aligned}$$

(A.7)

Upon choosing ${\mathbf {U}}$ as a test function in the momentum Eq. (1.9),

$$\begin{aligned}&\int _0^{\tau }\int _{\Omega } \Big ( {\mathfrak {r}}{\varvec{u}}\cdot \partial _t {\mathbf {U}} + {\mathfrak {r}}{\varvec{u}}\otimes {\varvec{u}}: \nabla _x {\mathbf {U}}+p(\varrho ,n)\mathrm{div}_x {\mathbf {U}} -{\mathbb {S}}(\nabla _x{\varvec{u}}):\nabla _x {\mathbf {U}} \Big ) \mathrm{d} x\mathrm{d} t \nonumber \\&\quad +\int _0^{\tau }\int _{ \overline{\Omega }}\nabla _x {\mathbf {U}} : \mathrm{d} \mu _c(t) \mathrm{d} t =\left[ \int _{\Omega } {\mathfrak {r}}{\varvec{u}}\cdot {\mathbf {U}} \mathrm{d} x\right] _{t=0}^{t=\tau } . \end{aligned}$$

(A.8)

Taking (A.4)–(A.8) and the balance of total energy (1.10) into account, we may estimate (A.3), in agreement with the compressible Navier-Stokes system [11], to arrive at

$$\begin{aligned}&\left[ {\mathcal {E}}\Big ((\varrho ,n,{\varvec{u}})\,\Big |\,(r,b,{\mathbf {U}})\Big )\right] _{t=0}^{t=\tau } +\int _0^{\tau }\int _{\Omega } \Big ( {\mathbb {S}}(\nabla _x{\varvec{u}}-\nabla _x {\mathbf {U}}):(\nabla _x {\varvec{u}}-\nabla _x {\mathbf {U}}) \Big )\mathrm{d} x\mathrm{d} t +\int _{ \overline{\Omega }} \mathrm{d} {\mathcal {D}}(\tau ) \nonumber \\&\quad \le \int _0^{\tau }\int _{\Omega } {\mathfrak {r}}({\mathbf {U}}-{\varvec{u}})\cdot \Big ( \partial _t {\mathbf {U}} +{\varvec{u}}\cdot \nabla _x {\mathbf {U}} \Big ) \mathrm{d} x\mathrm{d} t \nonumber \\&\quad +\int _0^{\tau }\int _{\Omega } {\mathbb {S}} (\nabla _x {\mathbf {U}}): \left( \nabla _x {\mathbf {U}}-\nabla _x {\varvec{u}}\right) \mathrm{d} x\mathrm{d} t -\int _0^{\tau }\int _{\Omega } p(\varrho ,n)\mathrm{div}_x {\mathbf {U}}\mathrm{d} x\mathrm{d} t \nonumber \\&\quad +\int _0^{\tau }\int _{\Omega } \Big [ \left( 1-\frac{\varrho }{r} \right) \gamma r ^{\gamma -1} \partial _t r -\varrho {\varvec{u}}\cdot \gamma r^{\gamma -2} \nabla _x r \Big ] \mathrm{d} x\mathrm{d} t \nonumber \\&\quad +\int _0^{\tau }\int _{\Omega } \Big [ \left( 1-\frac{n}{b} \right) \alpha b ^{\alpha -1} \partial _t b -n{\varvec{u}}\cdot \alpha b^{\alpha -2} \nabla _x b \Big ] \mathrm{d} x\mathrm{d} t \nonumber \\&\quad \le \int _0^{\tau }\int _{\Omega } {\mathfrak {r}}({\mathbf {U}}-{\varvec{u}})\cdot \Big ( \partial _t {\mathbf {U}} +{\varvec{u}}\cdot \nabla _x {\mathbf {U}} \Big ) \mathrm{d} x\mathrm{d} t +\int _0^{\tau }\int _{\Omega } {\mathbb {S}} (\nabla _x {\mathbf {U}}): \left( \nabla _x {\mathbf {U}}-\nabla _x {\varvec{u}}\right) \mathrm{d} x\mathrm{d} t \nonumber \\&\quad + \int _0^{\tau }\int _{\Omega } (r-\varrho )\partial _t \left( \frac{\gamma }{\gamma -1}r^{\gamma -1} \right) \mathrm{d} x\mathrm{d} t + \int _0^{\tau }\int _{\Omega } (b-n)\partial _t \left( \frac{\alpha }{\alpha -1}b^{\alpha -1} \right) \mathrm{d} x\mathrm{d} t \nonumber \\&\quad +\int _0^{\tau }\int _{\Omega } (r{\mathbf {U}}-\varrho {\varvec{u}})\cdot \nabla _x \left( \frac{\gamma }{\gamma -1}r^{\gamma -1} \right) \mathrm{d} x\mathrm{d} t +\int _0^{\tau }\int _{\Omega } (b{\mathbf {U}}-n {\varvec{u}}) \cdot \nabla _x \left( \frac{\alpha }{\alpha -1}b^{\alpha -1} \right) \mathrm{d} x\mathrm{d} t .\nonumber \\&\quad -\int _0^{\tau }\int _{\Omega }\Big [ p(\varrho ,n)-p(r,b) \Big ]\mathrm{div}_x {\mathbf {U}} \mathrm{d} x\mathrm{d} t -\int _0^{\tau }\int _{ \overline{\Omega }}\nabla _x {\mathbf {U}} : \mathrm{d} \mu _c(t) \mathrm{d} t . \end{aligned}$$

(A.9)

Remark A.1

Compared with [19], our relative energy inequality (A.9) holds for any test functions belonging to the class (A.1) and incorporates the phenomena of oscillations and concentrations.

1.2 A.2 Weak–strong uniqueness principle

Basically, the proof of weak–strong uniqueness principle consists of:

• choosing the classical solution $({\tilde{\varrho }},{\tilde{n}},{\tilde{{\varvec{u}}}})$ as the test function $(r,b,{\mathbf {U}})$ in the relative energy inequality (A.9);

• estimating each term on the right-hand side of the relative energy inequality in a suitable manner;

• application of Gronwall-type argument.

To do this, assume that $({\tilde{\varrho }},{\tilde{n}},{\tilde{{\varvec{u}}}})$ is a strong solution to (1.1)–(1.5) starting from the smooth initial data $({\tilde{\varrho }}_0,{\tilde{n}}_0,{\tilde{{\varvec{u}}}}_0)$ with strictly positive ${\tilde{\varrho }}_0$ and ${\tilde{n}}_0$. Let $(\varrho ,n,{\varvec{u}})$ be a dissipative weak solution to (1.1)–(1.5) emanating from the same initial data. It follows from (A.9) that

$$\begin{aligned}&{\mathcal {E}}\Big ((\varrho ,n,{\varvec{u}})\,\Big |\,({\tilde{\varrho }},{\tilde{n}},{\tilde{{\varvec{u}}}})\Big )(\tau ) +\int _0^{\tau }\int _{\Omega } \Big ( {\mathbb {S}}(\nabla _x{\varvec{u}}-\nabla _x {\tilde{{\varvec{u}}}}):(\nabla _x {\varvec{u}}-\nabla _x {\tilde{{\varvec{u}}}}) \Big )\mathrm{d} x\mathrm{d} t +\int _{ \overline{\Omega }} \mathrm{d} {\mathcal {D}}(\tau ) \nonumber \\&\quad \le \int _0^{\tau }\int _{\Omega } ({\tilde{{\varvec{u}}}}-{\varvec{u}})\cdot \Big [{\mathfrak {r}}\Big (\partial _t {\tilde{{\varvec{u}}}} +{\varvec{u}}\cdot \nabla _x {\tilde{{\varvec{u}}}}\Big )-\mathrm{div}_x {\mathbb {S}} (\nabla _x {\tilde{{\varvec{u}}}}) \Big ] \mathrm{d} x\mathrm{d} t \nonumber \\&\quad + \int _0^{\tau }\int _{\Omega } ({\tilde{\varrho }}-\varrho )\partial _t \left( \frac{\gamma }{\gamma -1}{\tilde{\varrho }}^{\gamma -1} \right) \mathrm{d} x\mathrm{d} t + \int _0^{\tau }\int _{\Omega } ({\tilde{n}}-n)\partial _t \left( \frac{\alpha }{\alpha -1}{\tilde{n}}^{\alpha -1} \right) \mathrm{d} x\mathrm{d} t \nonumber \\&\quad +\int _0^{\tau }\int _{\Omega } ({\tilde{\varrho }}{\tilde{{\varvec{u}}}}-\varrho {\varvec{u}})\cdot \nabla _x \left( \frac{\gamma }{\gamma -1}{\tilde{\varrho }}^{\gamma -1} \right) \mathrm{d} x\mathrm{d} t +\int _0^{\tau }\int _{\Omega } ({\tilde{n}}{\tilde{{\varvec{u}}}}-n {\varvec{u}}) \cdot \nabla _x \left( \frac{\alpha }{\alpha -1}{\tilde{n}}^{\alpha -1} \right) \mathrm{d} x\mathrm{d} t .\nonumber \\&\quad -\int _0^{\tau }\int _{\Omega }\Big [ p(\varrho ,n)-p({\tilde{\varrho }},{\tilde{n}}) \Big ]\mathrm{div}_x {\tilde{{\varvec{u}}}} \mathrm{d} x\mathrm{d} t -\int _0^{\tau }\int _{ \overline{\Omega }}\nabla _x {\tilde{{\varvec{u}}}} : \mathrm{d} \mu _c(t) \mathrm{d} t . \end{aligned}$$

(A.10)

In light of the fact that $({\tilde{\varrho }},{\tilde{n}},{\tilde{{\varvec{u}}}})$ solves (1.1) in the classical sense, we furthermore rewrite the relative energy inequality as (see for instance [16] for similar calculations)

$$\begin{aligned}&{\mathcal {E}}\Big ((\varrho ,n,{\varvec{u}})\,\Big |\,({\tilde{\varrho }},{\tilde{n}},{\tilde{{\varvec{u}}}})\Big )(\tau ) +\int _0^{\tau }\int _{\Omega } \Big ( {\mathbb {S}}(\nabla _x{\varvec{u}}-\nabla _x {\tilde{{\varvec{u}}}}):(\nabla _x {\varvec{u}}-\nabla _x {\tilde{{\varvec{u}}}}) \Big )\mathrm{d} x\mathrm{d} t +\int _{ \overline{\Omega }} \mathrm{d} {\mathcal {D}}(\tau )\nonumber \\&\quad \le \int _0^{\tau }\int _{\Omega } ({\tilde{{\varvec{u}}}}-{\varvec{u}})\cdot \Big [({\mathfrak {r}}-{\tilde{{\mathfrak {r}}}} )\partial _t {\tilde{{\varvec{u}}}} + ({\mathfrak {r}}{\varvec{u}}- {\tilde{{\mathfrak {r}}}} {\tilde{{\varvec{u}}}})\cdot \nabla _x {\tilde{{\varvec{u}}}} \Big ] \mathrm{d} x\mathrm{d} t \nonumber \\&\quad + \int _0^{\tau }\int _{\Omega }(\varrho -{\tilde{\varrho }}) ({\tilde{{\varvec{u}}}}-{\varvec{u}})\cdot \nabla _x \left( \frac{\gamma }{\gamma -1}{\tilde{\varrho }}^{\gamma -1} \right) \mathrm{d} x\mathrm{d} t \nonumber \\&\quad + \int _0^{\tau }\int _{\Omega }(n-{\tilde{n}}) ({\tilde{{\varvec{u}}}}-{\varvec{u}})\cdot \nabla _x \left( \frac{\alpha }{\alpha -1}{\tilde{n}}^{\alpha -1} \right) \mathrm{d} x\mathrm{d} t \nonumber \\&\quad -\int _0^{\tau }\int _{\Omega }\Big [ \Big ( \varrho ^{\gamma }-{\tilde{\varrho }}^{\gamma }-\gamma {\tilde{\varrho }}^{\gamma -1}(\varrho -{\tilde{\varrho }}) \Big ) + \Big ( n^{\alpha }-{\tilde{n}}^{\alpha }-\alpha {\tilde{n}}^{\alpha -1}(n-{\tilde{n}}) \Big ) \Big ] \mathrm{div}_x {\tilde{{\varvec{u}}}} \mathrm{d} x\mathrm{d} t \nonumber \\&\quad -\int _0^{\tau }\int _{ \overline{\Omega }}\nabla _x {\tilde{{\varvec{u}}}} : \mathrm{d} \mu _c(t) \mathrm{d} t =:\sum _{j=1}^5{\mathcal {R}}^{(j)}. \end{aligned}$$

(A.11)

Notice that we may rewrite ${\mathcal {R}}^{(1)}$ as

$$\begin{aligned} {\mathcal {R}}^{(1)}= & {} \int _0^{\tau }\int _{\Omega } ({\tilde{{\varvec{u}}}}-{\varvec{u}})\cdot ({\mathfrak {r}}-{\tilde{{\mathfrak {r}}}} ) \Big [\partial _t {\tilde{{\varvec{u}}}} + {\tilde{{\varvec{u}}}} \cdot \nabla _x {\tilde{{\varvec{u}}}} \Big ] \mathrm{d} x\mathrm{d} t \\&+\int _0^{\tau }\int _{\Omega } {\mathfrak {r}}({\varvec{u}}-{\tilde{{\varvec{u}}}}) \cdot \nabla _x {\tilde{{\varvec{u}}}} \cdot ({\tilde{{\varvec{u}}}}-{\varvec{u}}) \mathrm{d} x\mathrm{d} t =:{\mathcal {R}}^{(1)}_1+{\mathcal {R}}^{(1)}_2. \end{aligned}$$

Obviously, it holds

$$\begin{aligned} |{\mathcal {R}}^{(1)}_2|\lesssim \int _0^{\tau } {\mathcal {E}}\Big ((\varrho ,n,{\varvec{u}})\,\Big |\,({\tilde{\varrho }},{\tilde{n}},{\tilde{{\varvec{u}}}})\Big )(t) \mathrm{d} t . \end{aligned}$$

(A.12)

Next, to show the estimate of ${\mathcal {R}}^{(1)}_1$, we first recall [9, Lemma 14.3] for the following estimates.

$$\begin{aligned}&\Big [ \Big ( \varrho ^{\gamma }-{\tilde{\varrho }}^{\gamma }-\gamma {\tilde{\varrho }}^{\gamma -1}(\varrho -{\tilde{\varrho }}) \Big ) + \Big ( n^{\alpha }-{\tilde{n}}^{\alpha }-\alpha {\tilde{n}}^{\alpha -1}(n-{\tilde{n}}) \Big ) \Big ]\nonumber \\&\quad \gtrsim \left\{ \begin{aligned}&(\varrho -{\tilde{\varrho }})^2+(n-{\tilde{n}})^2 ,\,\, \text { if }\varrho \in [1/2\min _{\overline{Q_T}}{\tilde{\varrho }},2\max _{\overline{Q_T}}{\tilde{\varrho }}],n\in [1/2\min _{\overline{Q_T}}{\tilde{n}},2\max _{\overline{Q_T}}{\tilde{n}}], \\&1+\varrho ^{\gamma }+n^{\alpha } ,\,\, \text { otherwise} .\\ \end{aligned}\right. \end{aligned}$$

(A.13)

Moreover, following [12], we may decompose any measurable function f(t, x) as the sum of “essential part” and “residual part”:

$$\begin{aligned} {[}f]_{\text {ess}}(t,x)= \left\{ \begin{aligned}&f(t,x) ,\,\, \text { if }\varrho \in [1/2\min _{\overline{Q_T}}{\tilde{\varrho }},2\max _{\overline{Q_T}}{\tilde{\varrho }}],n\in [1/2\min _{\overline{Q_T}}{\tilde{n}},2\max _{\overline{Q_T}}{\tilde{n}}], \\&0,\,\, \text { otherwise} ; \\ \end{aligned}\right. \\ {[}f]_{\text {res}}(t,x)=f(t,x)-[f]_{\text {ess}}(t,x). \end{aligned}$$

Thus,

$$\begin{aligned}&{\mathcal {R}}^{(1)}_1= \int _0^{\tau }\int _{\Omega } ({\tilde{{\varvec{u}}}}-{\varvec{u}})\cdot [{\mathfrak {r}}-{\tilde{{\mathfrak {r}}}}]_{\text {ess}} \Big [\partial _t {\tilde{{\varvec{u}}}} + {\tilde{{\varvec{u}}}} \cdot \nabla _x {\tilde{{\varvec{u}}}} \Big ] \mathrm{d} x\mathrm{d} t \nonumber \\&\quad + \int _0^{\tau }\int _{\Omega } ({\tilde{{\varvec{u}}}}-{\varvec{u}})\cdot [{\mathfrak {r}}-{\tilde{{\mathfrak {r}}}}]_{\text {res}} \Big [\partial _t {\tilde{{\varvec{u}}}} + {\tilde{{\varvec{u}}}} \cdot \nabla _x {\tilde{{\varvec{u}}}} \Big ] \mathrm{d} x\mathrm{d} t . \end{aligned}$$

(A.14)

Observe first that

$$\begin{aligned}&\left| \int _0^{\tau }\int _{\Omega } ({\tilde{{\varvec{u}}}}-{\varvec{u}})\cdot [{\mathfrak {r}}-{\tilde{{\mathfrak {r}}}}]_{\text {ess}} \Big [\partial _t {\tilde{{\varvec{u}}}} + {\tilde{{\varvec{u}}}} \cdot \nabla _x {\tilde{{\varvec{u}}}} \Big ] \mathrm{d} x\mathrm{d} t \right| \nonumber \\&\quad \le \int _0^{\tau } \Vert \partial _t {\tilde{{\varvec{u}}}} + {\tilde{{\varvec{u}}}} \cdot \nabla _x {\tilde{{\varvec{u}}}} \Vert _{L^{\infty }(\Omega )} \Vert {\varvec{u}}- {\tilde{{\varvec{u}}}}\Vert _{L^2(\Omega )}\Vert [{\mathfrak {r}}-{\tilde{{\mathfrak {r}}}}]_{\text {ess}} \Vert _{L^2(\Omega )} \mathrm{d} t \nonumber \\&\quad \lesssim \varepsilon \int _0^{\tau }\int _{\Omega } \Big ( {\mathbb {S}}(\nabla _x{\varvec{u}}-\nabla _x {\tilde{{\varvec{u}}}}):(\nabla _x {\varvec{u}}-\nabla _x {\tilde{{\varvec{u}}}}) \Big )\mathrm{d} x\mathrm{d} t + C(\varepsilon ) \left( \Vert [\varrho -{\tilde{\varrho }}]_{\text {ess}} \Vert _{L^2(\Omega )}^2+\Vert [n-{\tilde{n}}]_{\text {ess}} \Vert _{L^2(\Omega )}^2 \right) \nonumber \\&\quad \lesssim \varepsilon \int _0^{\tau }\int _{\Omega } \Big ( {\mathbb {S}}(\nabla _x{\varvec{u}}-\nabla _x {\tilde{{\varvec{u}}}}):(\nabla _x {\varvec{u}}-\nabla _x {\tilde{{\varvec{u}}}}) \Big )\mathrm{d} x\mathrm{d} t +C(\varepsilon ) \int _0^{\tau } {\mathcal {E}}\Big ((\varrho ,n,{\varvec{u}})\,\Big |\,({\tilde{\varrho }},{\tilde{n}},{\tilde{{\varvec{u}}}})\Big )(t) \mathrm{d} t , \end{aligned}$$

(A.15)

where we have employed the generalized Korn’s inequality in the second step and the first item of (A.17) in the third step. Next, it follows that

$$\begin{aligned}&\left| \int _0^{\tau }\int _{\Omega } ({\tilde{{\varvec{u}}}}-{\varvec{u}})\cdot [{\mathfrak {r}}-{\tilde{{\mathfrak {r}}}}]_{\text {res}} \Big [\partial _t {\tilde{{\varvec{u}}}} + {\tilde{{\varvec{u}}}} \cdot \nabla _x {\tilde{{\varvec{u}}}} \Big ] \mathrm{d} x\mathrm{d} t \right| \nonumber \\&\quad \lesssim \int _0^{\tau } \int _{\Omega } |{\varvec{u}}- {\tilde{{\varvec{u}}}}| \Big (|[\varrho -{\tilde{\varrho }}]_{\text {res}}|+|[n-{\tilde{n}}]_{\text {res}}| \Big )\mathrm{d} x\mathrm{d} t . \end{aligned}$$

(A.16)

We make a decomposition as follows.

$$\begin{aligned}&\int _0^{\tau } \int _{\Omega } |{\varvec{u}}- {\tilde{{\varvec{u}}}}| |[\varrho -{\tilde{\varrho }}]_{\text {res}}| \mathrm{d} x\mathrm{d} t \\&\quad =\int _0^{\tau } \int _{\varrho \le 1/2\min _{\overline{Q_T}}{\tilde{\varrho }}} |{\varvec{u}}- {\tilde{{\varvec{u}}}}| |[\varrho -{\tilde{\varrho }}]_{\text {res}}| \mathrm{d} x\mathrm{d} t +\int _0^{\tau } \int _{\varrho \ge 2\max _{\overline{Q_T}}{\tilde{\varrho }}} |{\varvec{u}}- {\tilde{{\varvec{u}}}}| |[\varrho -{\tilde{\varrho }}]_{\text {res}}| \mathrm{d} x\mathrm{d} t ; \end{aligned}$$

Making use of the generalized Korn’s inequality and the second item of (A.17),

$$\begin{aligned}&\int _0^{\tau } \int _{\varrho \le 1/2\min _{\overline{Q_T}}{\tilde{\varrho }}} |{\varvec{u}}- {\tilde{{\varvec{u}}}}| |[\varrho -{\tilde{\varrho }}]_{\text {res}}| \mathrm{d} x\mathrm{d} t \lesssim \int _0^{\tau } \Vert [1]_{\text {res}}\Vert _{L^2(\Omega )} \Vert {\varvec{u}}- {\tilde{{\varvec{u}}}} \Vert _{L^2(\Omega )} \mathrm{d} t \nonumber \\&\quad \lesssim \varepsilon \int _0^{\tau }\int _{\Omega } \Big ( {\mathbb {S}}(\nabla _x{\varvec{u}}-\nabla _x {\tilde{{\varvec{u}}}}):(\nabla _x {\varvec{u}}-\nabla _x {\tilde{{\varvec{u}}}}) \Big )\mathrm{d} x\mathrm{d} t +C(\varepsilon ) \int _0^{\tau } {\mathcal {E}}\Big ((\varrho ,n,{\varvec{u}})\,\Big |\,({\tilde{\varrho }},{\tilde{n}},{\tilde{{\varvec{u}}}})\Big )(t) \mathrm{d} t .\qquad \quad \end{aligned}$$

(A.17)

In the same spirit,

$$\begin{aligned}&\int _0^{\tau } \int _{\varrho \ge 2\max _{\overline{Q_T}}{\tilde{\varrho }}} |{\varvec{u}}- {\tilde{{\varvec{u}}}}| |[\varrho -{\tilde{\varrho }}]_{\text {res}}| \mathrm{d} x\mathrm{d} t \lesssim \int _0^{\tau }\int _{\Omega } |[1]_{\text {res}}| \sqrt{\varrho }|{\varvec{u}}- {\tilde{{\varvec{u}}}}| \sqrt{\varrho } \mathrm{d} x\mathrm{d} t \nonumber \\&\quad \lesssim \int _0^{\tau }\int _{\Omega } |[1]_{\text {res}}| \sqrt{{\mathfrak {r}}}|{\varvec{u}}- {\tilde{{\varvec{u}}}}| \sqrt{\varrho } \mathrm{d} x\mathrm{d} t \lesssim \int _0^{\tau } {\mathcal {E}}\Big ((\varrho ,n,{\varvec{u}})\,\Big |\,({\tilde{\varrho }},{\tilde{n}},{\tilde{{\varvec{u}}}})\Big )(t) \mathrm{d} t . \end{aligned}$$

(A.18)

The estimate of the second integral in (A.16) can be carried in exactly the same manner. Therefore, combining (A.12), (A.14)–(A.18),

$$\begin{aligned} |{\mathcal {R}}^{(1)}| \lesssim \varepsilon \int _0^{\tau }\int _{\Omega } \Big ( {\mathbb {S}}(\nabla _x{\varvec{u}}-\nabla _x {\tilde{{\varvec{u}}}}):(\nabla _x {\varvec{u}}-\nabla _x {\tilde{{\varvec{u}}}}) \Big )\mathrm{d} x\mathrm{d} t +C(\varepsilon ) \int _0^{\tau } {\mathcal {E}}\Big ((\varrho ,n,{\varvec{u}})\,\Big |\,({\tilde{\varrho }},{\tilde{n}},{\tilde{{\varvec{u}}}})\Big )(t) \mathrm{d} t . \end{aligned}$$

(A.19)

It is easy to see that ${\mathcal {R}}^{(2)}$ and ${\mathcal {R}}^{(3)}$ can be estimated analogously as above. Next, we observe that

$$\begin{aligned}&\Big | \Big ( \varrho ^{\gamma }-{\tilde{\varrho }}^{\gamma }-\gamma {\tilde{\varrho }}^{\gamma -1}(\varrho -{\tilde{\varrho }}) \Big ) + \Big ( n^{\alpha }-{\tilde{n}}^{\alpha }-\alpha {\tilde{n}}^{\alpha -1}(n-{\tilde{n}}) \Big ) \Big | \\&\quad \lesssim H(\varrho )-H({\tilde{\varrho }})-H'({\tilde{\varrho }})(\varrho -{\tilde{\varrho }})+G(n)-G({\tilde{n}})-G'({\tilde{n}})(n-{\tilde{n}}), \end{aligned}$$

whence

$$\begin{aligned} |{\mathcal {R}}^{(4)} |\lesssim \int _0^{\tau } {\mathcal {E}}\Big ((\varrho ,n,{\varvec{u}})\,\Big |\,({\tilde{\varrho }},{\tilde{n}},{\tilde{{\varvec{u}}}})\Big )(t) \mathrm{d} t . \end{aligned}$$

(A.20)

Finally,

$$\begin{aligned} |{\mathcal {R}}^{(5)}|=\Big |\int _0^{\tau }\int _{ \overline{\Omega }}\nabla _x {\tilde{{\varvec{u}}}} : \mathrm{d} \mu _c(t) \mathrm{d} t \Big | \le \Vert \nabla _x {\tilde{{\varvec{u}}}}\Vert _{L^{\infty }(\overline{Q_T})}\int _0^{\tau } \int _{ \overline{\Omega }} \mathrm{d}{\mathcal {D}}(t)\mathrm{d} t . \end{aligned}$$

(A.21)

Consequently, collecting the estimates above, we conclude from (A.11) that, upon choosing $\varepsilon >0$ sufficiently small,

$$\begin{aligned}&{\mathcal {E}}\Big ((\varrho ,n,{\varvec{u}})\,\Big |\,({\tilde{\varrho }},{\tilde{n}},{\tilde{{\varvec{u}}}})\Big )(\tau ) +\int _0^{\tau }\int _{\Omega } \Big ( {\mathbb {S}}(\nabla _x{\varvec{u}}-\nabla _x {\tilde{{\varvec{u}}}}):(\nabla _x {\varvec{u}}-\nabla _x {\tilde{{\varvec{u}}}}) \Big )\mathrm{d} x\mathrm{d} t +\int _{ \overline{\Omega }} \mathrm{d} {\mathcal {D}}(\tau ) \\&\quad \lesssim \int _0^{\tau } {\mathcal {E}}\Big ((\varrho ,n,{\varvec{u}})\,\Big |\,({\tilde{\varrho }},{\tilde{n}},{\tilde{{\varvec{u}}}})\Big )(t) \mathrm{d} t +\int _0^{\tau } \int _{ \overline{\Omega }} \mathrm{d}{\mathcal {D}}(t)\mathrm{d} t , \end{aligned}$$

which immediately finishes the proof of Proposition 1.1 by Gronwall’s inequality. $\square$