Energy Conservation for Solutions of Incompressible Viscoelastic Fluids

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 ℝ³, and the coefficient of viscosity μ = 0, energy conservation is proved for u and F in certain Besov spaces. Furthermore, in the whole space ℝ³, it is shown that the conditions on the velocity u and the deformation tensor F can be relaxed, that is, $u \in B_{3,c(\mathbb{N})}^{{1 \over 3}}$, and $F \in B_{3,\infty }^{{1 \over 3}}$. Finally, when μ > 0, in a periodic domain in ℝ^d again, a result independent of the spacial dimension is established. More precisely, it is shown that the energy is conserved for u ∈ L^r (0, T; L^s (Ω)) for any ${1 \over r} + {1 \over s} \leqslant {1 \over 2}$, with s ⩾ 4, and F ∈ L^m(0, T; Lⁿ(Ω)) for any ${1 \over m} + {1 \over n} \leqslant {1 \over 2}$, with n ⩾ 4.