Estimates of suitable weak solutions to the Navier-Stokes equations in critical Morrey spaces. (English) Zbl 1127.35042

Seregin, G. A.

Zap. Nauchn. Semin. POMI 336, 199-210 (2006) and J. Math. Sci., New York 143, No. 2, 2961-2968 (2007).

The author studies suitable weak solutions to the Navier-Stokes
\[ \frac{\partial v}{\partial t}+(v\cdot\nabla)v-\Delta v+\nabla p=0,\quad \operatorname{div} v=0, \quad x\in B\subset\mathbb R^3,\;t\in (-1,0). \] Here \(v=(v_1,v_2,v_3)\) is the velocity of a fluid, \(p\) is the pressure, \(B=\{x\in\mathbb R^3:| x| <1\}\). The regularity of solutions is defined in a frame of critical Morrey spaces by certain functionals which are invariant with respect to the natural scaling. The author proves some estimates for suitable weak solutions under the assumption that these functionals are bounded. The author hopes the results of the paper are the base for further investigations.

Reviewer: Il’ya Sh. Mogilevskij (Tver’)

MSC:

35Q30	Navier-Stokes equations
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B45	A priori estimates in context of PDEs
76D05	Navier-Stokes equations for incompressible viscous fluids
35D10	Regularity of generalized solutions of PDE (MSC2000)

Keywords:

weak solutions; Navier-Stokes equations; regularity