The weak solutions of Hopf type to 2D Maxwell flows with infinite number of relaxation times

N. A. Karazeeva

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 47, Tome 461 (2017), pp. 140-147

Résumé

The system of equations, describing motion of fluids of Maxwell type is considered $$ \frac\partial{\partial t}v+v\cdot\nabla v-\int_0^t K(t-\tau)\Delta v(x,\tau)\,d\tau+\nabla p=f(x,t),\quad\operatorname{div}v=0. $$ Here $K(t)$ is exponential series $K(t)=\sum_{s=1}^\infty\beta_se ^{-\alpha_st}$. The existence of weak solution for initial boundary value problem $$ v(x,0)=v_0(x),\quad v\cdot n|_{\partial\Omega}=0,\quad\operatorname{rot}v|_{\partial\Omega}=0 $$ is proved.