Incompressible Limit of Three Dimensional Compressible Viscoelastic Systems with Vanishing Shear Viscosity

Abstract

This work concerns the incompressible limit of compressible viscoelastic systems when the shear viscosity converges to zero. The incompressible limit is characterised by the large value of the volume viscosity. In the limit, the dispersive effect of pressure waves disappears and the global convergence to the limit system around an equilibrium is justified with the help of vector fields.

Appendix: Local Energy Estimates

Let $\mathcal {V}$ and $\mathcal {W}$ be finite-dimensional inner product space over $\mathbb {R}$. Let $u:[0,T)\times \mathbb {R}^n\rightarrow \mathcal {V}$ be the solution to the systems

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u+\mathcal {A}(\nabla )u-\mu \Delta u-\lambda \nabla (\nabla \cdot u)=f,\\ \mathcal {B}(\nabla ) u=g. \end{array}\right. } \end{aligned}$$

(6.1)

Here,

$$\begin{aligned} \mathcal {A}(\nabla )= & {} \mathcal {A}_k\nabla _k, \quad \mathcal {A}_k\in \mathcal {L}(\mathcal {V},\mathcal {V}),\\ \mathcal {B}(\nabla )= & {} \mathcal {B}_k\nabla _k, \quad \mathcal {B}_k\in \mathcal {L}(\mathcal {V},\mathcal {W}), \end{aligned}$$

where $\mathcal {A}_k$ and $\mathcal {B}_k$ are constant matrices.

For the above systems (6.1), we make the following assumptions:

(1) $\mathcal {A}(\nabla )$ is symmetric, which means that

$$\begin{aligned} \mathcal {A}^k=(\mathcal {A}^k)^T, \quad \mathrm {for}\ 1\leqq k\leqq n. \end{aligned}$$

(6.2)

(2)

$$\begin{aligned} \mathrm {ker}\mathcal {A}(\xi )\cap \mathrm {ker}\mathcal {B}(\xi )=\{0\},\quad \forall \ 0\ne \xi \in \mathbb {R}^n. \end{aligned}$$

(6.3)

(3) There exist smooth maps such that

$$\begin{aligned} V:\mathcal {SO}(\mathbb {R}^n)\rightarrow \mathcal {SO}(\mathcal {V}), \quad \mathrm {and}\quad W:\mathcal {SO}(\mathbb {R}^n)\rightarrow \mathcal {L}(\mathcal {W},\mathcal {W}), \end{aligned}$$

such that, for every $\xi \in \mathbb {R}^n$ and $R\in \mathcal {SO}(\mathbb {R}^n)$, we have

$$\begin{aligned} A(R\xi )=V(R)A(\xi )V(R)^* \end{aligned}$$

(6.4)

and

$$\begin{aligned} B(R\xi )=W(R)B(\xi )V(R)^*. \end{aligned}$$

(6.5)

Lemma 6.1

Let $n\geqq 2$. Suppose the conditions (6.2) and (6.3) hold. All sufficiently regular solutions of (6.1) satisfy the estimate

$$\begin{aligned}&\Vert t\mathcal {A}(\nabla )u-r\partial _r u\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}^2 +\mu t\Vert \nabla u\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}^2+\lambda t \Vert \nabla \cdot u\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}^2\\&\qquad +(2\mu \lambda +\lambda ^2)t^2\Vert \nabla (\nabla \cdot u)\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}^2+\mu ^2t^2\Vert \Delta u\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}^2\\&\quad \lesssim \Vert Su -tf\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}^2. \end{aligned}$$

Proof

We follow the arguments in [8, 29]. According to the definition of S and the first equation in (6.1), we have

$$\begin{aligned} t\mathcal {A}(\nabla )u-r\partial _r u-\mu t\Delta u-\lambda t\nabla (\nabla \cdot u)= tf-S u. \end{aligned}$$

Taking the $L^2$ norm on both sides of the equation above, one obtains

$$\begin{aligned}&\Vert t \mathcal {A}(\nabla )u- r\partial _r u\Vert _{L^2(\mathbb {R}^n,\mathcal {V})}^2+2\big \langle r\partial _r u-t\mathcal {A}(\nabla ) u, \mu t \Delta u+\lambda t \nabla (\nabla \cdot u)\big \rangle _{L^2(\mathbb {R}^n,\mathcal {V})}\nonumber \\&\qquad +2\mu \lambda t^2\langle \Delta u, \nabla (\nabla \cdot u)\rangle _{L^2(\mathbb {R}^n,\mathcal {V})} +\mu ^2t^2\Vert \Delta u\Vert _{L^2(\mathbb {R}^n,\mathcal {V})}^2+\lambda ^2 t^2\Vert \nabla (\nabla \cdot u)\Vert _{L^2(\mathbb {R}^n,\mathcal {V})}^2\nonumber \\&\quad \lesssim \Vert Su-tf\Vert _{L^2(\mathbb {R}^n,\mathcal {V})}^2. \end{aligned}$$

(6.6)

Due to the symmetry of the coefficient matrices, by integration by parts, we have

$$\begin{aligned} \langle \mathcal {A}(\nabla ) u, \Delta u\rangle _{L^2(\mathbb {R}^n, \mathcal {V})}=\langle A_k\nabla _k u, \Delta u\rangle _{L^2(\mathbb {R}^n, \mathcal {V})}=0. \end{aligned}$$

Similarly, we also derive

$$\begin{aligned} \langle \mathcal {A}(\nabla )u, \nabla (\nabla \cdot u)\rangle _{L^2(\mathbb {R}^n, \mathcal {V})}&=\langle A_k \nabla _k u, \nabla (\nabla \cdot u)\rangle _{L^2(\mathbb {R}^n, \mathcal {V})} =0. \end{aligned}$$

Moreover we have

$$\begin{aligned}&2\langle r\partial _r u, \mu t\Delta u\rangle _{L^2(\mathbb {R}^n, \mathcal {V})}\\&\quad =2\mu t\langle x_j \nabla _j u, \Delta u\rangle _{L^2(\mathbb {R}^n, \mathcal {V})}\\&\quad =-2\mu t\langle x_j \nabla _j \nabla _k u, \nabla _k u\rangle _{L^2(\mathbb {R}^n, \mathcal {V})} -2\mu t\langle \nabla _k u, \nabla _k u\rangle _{L^2(\mathbb {R}^n, \mathcal {V})}\\&\quad =3\mu t\langle \nabla _k u, \nabla _k u\rangle _{L^2(\mathbb {R}^n, \mathcal {V})}-2\mu t \langle \nabla _k u, \nabla _k u\rangle _{L^2(\mathbb {R}^n, \mathcal {V})}\\&\quad =\mu t\Vert \nabla u\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}^2 \end{aligned}$$

and

$$\begin{aligned}&2\langle r\partial _r u, \lambda t\nabla (\nabla \cdot u)\rangle _{L^2(\mathbb {R}^n, \mathcal {V})} \\&\quad = 2\lambda t\langle x_k\nabla _k u_i, \partial _i \nabla _j u_j \rangle _{L^2(\mathbb {R}^n, \mathcal {V})}\\&\quad =-2\lambda t\langle x_k \nabla _k \nabla _i u_i, \nabla _j u_j\rangle _{L^2(\mathbb {R}^n, \mathcal {V})} -2\lambda t\Vert \nabla \cdot u\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}^2\\&\quad =\lambda t\Vert \nabla \cdot u\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}^2. \end{aligned}$$

For the third term on the left hand side of (6.6), we have

$$\begin{aligned} 2\mu \lambda t^2 \langle \Delta u, \nabla (\nabla \cdot u)\rangle _{L^2(\mathbb {R}^n, \mathcal {V})}&=-2\mu \lambda t^2\langle \Delta \nabla \cdot u, \nabla \cdot u\rangle _{L^2(\mathbb {R}^n, \mathcal {V})}\\&=2\mu \lambda t^2\Vert \nabla (\nabla \cdot u)\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}^2. \end{aligned}$$

Combining all the above calculations, we arrive at the lemma. $\square$

Lemma 6.2

Let $n\geqq 2$. Assume the conditions (6.2) and (6.3) hold. Then there exist positive constants $\alpha$ and C depend on the coefficent of $\mathcal {A}_k$ and $\mathcal {B}_k$ such that

$$\begin{aligned}&\alpha t\Vert \nabla u\Vert _{L^2(\{r\leqq \alpha t\},\mathcal {V})}+ (\mu t)^\frac{1}{2}\Vert \nabla u\Vert _{L^2(\mathbb {R}^n, \mathcal {V})} +(\lambda t)^\frac{1}{2}\Vert \nabla \cdot u\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}\\&\qquad +\mu t\Vert \Delta u\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}+(2\mu \lambda +\lambda ^2)^\frac{1}{2} t\Vert \nabla (\nabla \cdot u)\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}\\&\quad \lesssim C\Vert u\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}+\Vert Su\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}+t\Vert f\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}+t\Vert g\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}. \end{aligned}$$

In addition, if the conditions (6.4) and (6.5) hold, then

$$\begin{aligned}&\Vert \langle \lambda _i t-r\rangle P_i \nabla u\Vert _{L^2(\{r\geqq \alpha t\}; \mathcal {V})}\nonumber \\&\quad \leqq C\Big [\Vert {{\tilde{\Omega }}} u\Vert _{L^2(\mathbb {R}^n; \mathcal {V})}+\Vert u\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}+\Vert Su\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}+t\Vert f\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}\Big ] \end{aligned}$$

(6.7)

and

$$\begin{aligned} \Vert r\mathcal {B}(\omega ) \nabla u\Vert _{L^2(\{r\geqq \alpha t\}; \mathcal {V})}\leqq C\Big [\Vert {{\tilde{\Omega }}} u\Vert _{^2}+\Vert u\Vert _{L^2}+\Vert rg\Vert _{L^2}\Big ]. \end{aligned}$$

(6.8)

Proof

Let $n\geqq 2$, due to the condition (6.3), we know that $|\mathcal {A}(\omega ) u|^2+|\mathcal {B}(\omega )u|^2$ vanishes if and only if $u=0$. In other words, the map $\mathcal {A}(\omega )^2+\mathcal {B}(\omega )^*\mathcal {B}(\omega )$ is positive definite and symmetric. If we let

$$\begin{aligned} (3\alpha )^2=\min \{ \lambda : \lambda \in \sigma (\mathcal {A}(\omega )^2+\mathcal {B}(\omega )^*\mathcal {B}(\omega )),\ \mathrm {for \ some} \ \omega \in S^{n-1} \}, \end{aligned}$$

then we have

$$\begin{aligned} (3\alpha )^2|u|^2_{\mathcal {V}}\leqq |\mathcal {A}(\omega ) u|^2_{\mathcal {V}}+|\mathcal {B}(\omega ) u|^2_{\mathcal {W}} \end{aligned}$$

for all $u\in \mathcal {V}$ and $\omega \in S^{n-1}$. Hence, by the Fourier transform, one obtains

$$\begin{aligned} 3\alpha \Vert \nabla u\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}\leqq \Vert \mathcal {A}(\nabla ) u\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}+\Vert \mathcal {B}(\nabla )u\Vert _{L^2(\mathbb {R}^n, \mathcal {W})}. \end{aligned}$$

(6.9)

We introduce the cut-off function $\xi \in C^\infty (\mathbb {R})$ with $0\leqq \xi \leqq 1$ and

$$\begin{aligned} \xi (s)={\left\{ \begin{array}{ll} 1, \quad \mathrm {if}\ s\leqq 1\\ 0, \quad \mathrm {if}\ s\geqq 2 \end{array}\right. } \end{aligned}$$

Let $\xi _\alpha =\xi (r/(\alpha t)),$ then $\xi _\alpha u$ is supported in $\{ r\leqq 2\alpha t\}$. By (6.9), it holds

$$\begin{aligned} 3\alpha t \Vert \nabla (\xi _\alpha u)\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}\leqq&t\Vert \mathcal {A}(\nabla ) (\xi _\alpha u)\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}+t\Vert \mathcal {B}(\nabla ) (\xi _\alpha u)\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}\\ \leqq&\Vert (t \mathcal {A}(\nabla )-r\partial _r )(\xi _\alpha u)\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}+2\alpha t\Vert \nabla (\xi _\alpha u)\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}\\&+t\Vert \mathcal {B}(\nabla ) (\xi _\alpha u)\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}, \end{aligned}$$

which means that

$$\begin{aligned} \alpha t\Vert \nabla (\xi _\alpha u)\Vert _{L^2(\mathbb {R}^n,\mathcal {V})}\leqq \Vert (t\mathcal {A}(\nabla )-r\partial _r) (\xi _\alpha u)\Vert _{L^2(\mathbb {R}^n,\mathcal {V})}+t\Vert \mathcal {B}(\nabla ) (\xi _\alpha u)\Vert _{L^2(\mathbb {R}^n,\mathcal {V})}. \end{aligned}$$

Since $(\alpha t+r)|\partial _j \xi _\alpha |\leqq C$, then we have

$$\begin{aligned}&\alpha t\Vert \nabla _j u\Vert _{L^2(\{r\leqq \alpha t\},\mathcal {V})}\\&\quad \leqq \alpha t\Vert \xi _\alpha \nabla _j u\Vert _{L^2(\mathbb {R}^n, \mathcal {V})}\\&\quad \leqq \alpha t\Vert \nabla _j (\xi _\alpha u)\Vert _{L^2(\mathbb {R}^n,\mathcal {V})} +C\Vert u\Vert _{L^2(\mathbb {R}^n,\mathcal {V})}\\&\quad \leqq \Vert (t\mathcal {A}(\nabla )-r\partial _r)(\xi _\alpha u)\Vert _{L^2(\mathbb {R}^n,\mathcal {V})}+t\Vert \mathcal {B}(\nabla ) (\xi _\alpha u)\Vert _{L^2(\mathbb {R}^n,\mathcal {V})}+C\Vert u\Vert _{L^2(\mathbb {R}^n,\mathcal {V})}\\&\quad \leqq \Vert (t\mathcal {A}(\nabla )-r\partial _r)u\Vert _{L^2(\mathbb {R}^n,\mathcal {V})}+t\Vert \mathcal {B}(\nabla ) u\Vert _{L^2(\mathbb {R}^n,\mathcal {V})}+C\Vert u\Vert _{L^2(\mathbb {R}^n,\mathcal {V})} \end{aligned}$$

and

$$\begin{aligned}&|(\lambda _i t-r)P_i(\omega )\nabla _j u(t,x)|_{\mathcal {V}}\\&\quad =|P_i(\omega )(tA(\omega )-rI)\nabla _j u(t,x)|_{ \mathcal {V}}\\&\quad \leqq |(tA(\omega )-rI)\nabla _j u(t,x)|_{ \mathcal {V}}\\&\quad \leqq |(tA_k-r\omega _kI)\omega _k\nabla _j u(t,x)|_{\mathcal {V}}\\&\quad \leqq |(tA_k-r\omega _k I)(\omega _j\nabla _k+r^{-1}\Omega _{kj})u(t,x)|_{\mathcal {V}}\\&\quad \leqq |\omega _j(tA(\nabla )-r\partial _r)u+r^{-1}(tA_k-r\omega _k I)\Omega _{kj}u|_{\mathcal {V}}\\&\quad \leqq |(t\mathcal {A}(\nabla )-r\partial _r)u(t,x)|_\mathcal {V}+C|tr^{-1}+1||\Omega u(t,x)|_{\mathcal {V}}. \end{aligned}$$

which yields (6.7) by an integration over $\{r\geqq \alpha t\}$.

The inequality (6.8) can be justified in a similar fashion, and the proof of the lemma is completed. $\square$