ENERGY CONSERVATION FOR SOLUTIONS OF INCOMPRESSIBLE VISCOELASTIC FLUIDS

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 μ=0,energy conservation is proved for u and F in certain Besovs paces.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∈B_(3,c(N))^(1/3),and F∈B_(3,∞)^(1/3).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^(T)(0,T;L^(n)(Ω))for any 1/r+1/s≤1/2,with s≥4,and F∈L^(m)(0,T;L^(n)(Ω))for any 1/m+1/n≤1/2,with n≥4.