On attractors for equations describing the flow of generalized Newtonian fluids

G. A. Seregin

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 29, Tome 249 (1997), pp. 256-293

Résumé

We consider initial-boundary value problems for equations \begin{gather*} \partial_t v+(\nabla v)v-\operatorname{div}\sigma=g-\nabla p, \quad \operatorname{div}v=0, \\ \sigma=\frac{\partial D}{\partial\varepsilon}(\varepsilon (v)), \quad v\big|_{t=0}=a, \end{gather*} describing the $2D$ flow of generalized Newtonian fluids under periodical boundary conditions. It is supposed that $D(\varepsilon)\sim|\varepsilon|^p$ for $|\varepsilon|\gg 1$ and $1<p<2$. Under some additional restrictions imposed on the vector-valued field $g$ and the dissipative potential $D$ existence of a global solution for initial data having the finite $L_2$-norm $(\|a\|_2<+\infty$) is proved. If $\|\nabla a\|_2<+\infty$ and $\frac32\le p<2$, this solution is strong and unique. Strong solution exists and is unique for all $1<p<2$. The last result allows to define a semigroup of solution operators and to prove that it is of class I and possesses of a compact minimal global $\mathscr B$-attractor.