On limiting regime for modified Navier--Stokes equations in three-dimensional space

O. A. Ladyzhenskaya

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 11, Tome 84 (1979), pp. 131-146

Résumé

The description of the limit-set $\mathfrak{M}_R$ (when $t\to\infty$) for all solutions of the system $$ \frac{\partial\vec v}{\partial t}-\nu\Delta\vec{v}+\sum_{k=1}^3 v_k\frac{\partial\vec{v}}{\partial x_k}+\operatorname{grad}{p} =\vec{f}, \quad\operatorname{div}\vec{v}=0, $$ where $\nu=\mu_0+\mu_1\int_\Omega\vec{v}^{\,2}_x(x,t)\,dx$, $\mu_i=\operatorname{const}>0$ and $\Omega$ is bounded, which start at $t=0$ from the points of the ball $K_R=\{\vec{a}(x):\vec{a}(x)\in\overset\circ{J}(\Omega),\|\vec{a}\|_{2,\Omega}\leq{R}\}$ is given. Particullary, it is proved, that the semi-group $V_t$, $t\geq0$, corresponding to this problem, may be extended to the group $V_t$, $t\in\mathbb R^1$, which has some interesting properties.