ENERGY CONSERVATION FOR SOLUTIONS OF INCOMPRESSIBLE VISCOELASTIC FLUIDS

作者：He, Yiming; Zi, Ruizhao^*

来源：Acta Mathematica Scientia, 2021, 41(4): 1287-1301.

DOI：10.1007/s10473-021-0416-6

• Incompressible viscoelastic fluids
• weak solutions
• energy conservation

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 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 R-3, it is shown that the conditions on the velocity u and the deformation tensor F can be relaxed, that is, u is an element of B-3,c((N))1/3, and F is an element of N B-3,infinity(1/3). Finally, when mu > 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 is an element of L-T (0, T; L-s (Omega)) for any 1/r + 1/s <= 1/2, with s >= 4, and F is an element of L-m (0,T; L-n (Omega)) for any 1/m + 1/n <= 1/2, with n >= 4.