On the energy equality for the 3D incompressible viscoelastic flows

Abstract

In this paper, we study the problem of energy conservation for the solutions to the incompressible viscoelastic flows. First, we consider Leray-Hopf weak solutions in the bounded Lipschitz domain $\Omega$ in $\mathbb{R}^d\,\, (d\geq 2)$. We prove that under the Shinbrot type conditions $ u \in L^{q}_{loc}\left(0, T ; L^{p}(\Omega)\right) \text { for any } \frac{1}{q}+\frac{1}{p} \leq \frac{1}{2}, \text { with } p \geq 4,\text{ and } {\bf F} \in L^{r}_{loc}\left(0, T ; L^{s}(\Omega)\right) \text { for any } \frac{1}{r}+\frac{1}{s} \leq \frac{1}{2}, \text { with } s \geq 4 $, the boundary conditions $u|_{\partial\Omega}=0,\,\,{\bf F}\cdot n|_{\partial\Omega}=0$ can inhibit the boundary effect and guarantee the validity of energy equality. Next, we apply this idea to deal with the case $\Omega= \mathbb{R}^d\,\,(d=2, 3, 4)$, and showed that the energy is conserved for $u\in L_{loc}^{q}\left(0,T;L_{loc}^{p}\left(\mathbb{R}^{d}\right)\right)$ with $ \frac{2}{q}+\frac{2}{p}\leq1, p\geq 4 $ and $ {\bf F}\in L_{loc}^{r}\left(0,T;L_{loc}^{s}\left(\mathbb{R}^{d}\right)\right)\cap L^{\frac{4d+8}{d+4}}\left(0,T;L^{\frac{4d+8}{d+4}}\left(\mathbb{R}^{d}\right)\right)$ with $\frac{2}{r}+\frac{2}{s}\leq1, s\geq 4 $. This result shows that the behavior of solutions in the finite regions and the behavior at infinite play different roles in the energy conservation. Finally, we consider the problem of energy conservation for distributional solutions and show energy equality for the distributional solutions belonging to the so-called Lions class $L^4L^4$.