关键词: Incompressible viscoelastic fluids, weak solutions, energy conservation

Abstract: Some sufficient conditions of the energy conservation for weak solutions of incompressible viscoelastic flows are given in this paper. First, for a periodic domain in $\mathbb R^3$, and the coefficient of viscosity $\mu=0$, energy conservation is proved for $u$ and $F$ in certain Besov spaces. Furthermore, in the whole space $\mathbb R^3$, it is shown that the conditions on the velocity $u$ and the deformation tensor $F$ can be relaxed, that is, $u\in B^{\frac 13}_{3,c(\mathbb N)}$, and $F\in B^{\frac 13}_{3,\infty}$. Finally, when $\mu>0$, in a periodic domain in $\mathbb R^d$ again, a result independent of the spacial dimension is established. More precisely, it is shown that the energy is conserved for $u\in L^r(0,T;L^s(\Omega))$ for any $\frac 1r+\frac 1s\leqslant\frac 12$, with $s\geqslant4$, and $F\in L^m(0,T;L^n(\Omega))$ for any $\frac 1m+\frac 1n\leqslant\frac 12$, with $n\geqslant4$.

Key words: Incompressible viscoelastic fluids, weak solutions, energy conservation