Some model nonstationary systems in the theory of non-Newtonian fluids.~II

A. P. Oskolkov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 11, Tome 84 (1979), pp. 185-210

Résumé

For the non-stationary quasi-linear system \begin{gather*} \frac{\partial\bar{v}}{\partial{t}}+v_k\frac{\partial{v}}{\partial{x_k}}+\lambda\biggl[\frac{\partial^2{\bar{v}}}{\partial t^2}+v_{kt}\bar{v}_{x_k}+v_k\frac{\partial^2\bar{v}}{\partial t\partial x_k}\biggr]-\nu\Delta\bar{v}-\varkappa\frac{\partial\Delta\bar v}{\partial t}+\biggl(1+\lambda\frac{\partial}{\partial t}\biggr)\operatorname{grad}p=\bar{F}, \\ \operatorname{div}\bar{v}=0 \end{gather*} the local theorems of existence and uniqueness of generalized solutions with a finite energy integral $$ \max_{0\leq t\leq T}\int_\Omega(\bar{v}^2_x+\bar{v}^2_t)\,dx +\iint_{Q_T}\bar{v}^2_{xt}\,dx\,dt+\infty; $$ are proved. Different variants of regularized systems are constructed, for which the generalized solution exists “in the large”.