Theory of nonstationary flows of Kelvin--Voigt fluids

A. P. Oskolkov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 14, Tome 115 (1982), pp. 191-202

Résumé

One proves the global unique solvability in class $W_\infty^1(0,T;C^{2,\alpha}(\overline\Omega)\cap H(\Omega))$ of the initial-boundary-value problem for the quasilinear system $$ \frac{\partial\vec v}{\partial t}+v_k\frac{\partial\vec v}{\partial x_k}-\mu_1\frac{\partial\Delta\vec v}{\partial t}-\mu_0\Delta\vec v-\int_0^tK(t-\tau)\Delta\vec v(\tau)d\tau+\operatorname{grad}p=\vec f,\qquad\operatorname{div}\vec v=0,\quad\mu_1>0. $$ This system described the nonstationary flows of the elastic-viscous Kelvin-Voigt fluids with defining relation $$ \Bigl(1+\sum_{l=1}^L\lambda_l\frac{\partial^l}{\partial t^l}\Bigr)\sigma=2\Bigl(\nu+\sum_{m=1}^{L+1}\varkappa_m\frac{\partial^m}{\partial t^m}\Bigr)D,\qquad L=0,1,2,\dots;\quad\lambda_L,\varkappa_{L+1}>0. $$