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Estimates for solutions of the nonstationary Stokes problem in anisotropic Sobolev spaces and estimates for the resolvent of the Stokes operator. (English. Russian original) Zbl 1059.35101

Russ. Math. Surv. 58, No. 2, 331-365 (2003); translation from Usp. Mat. Nauk 58, No. 2, 123-156 (2003).
Summary: In this paper, which is mainly of a survey nature, a coercive estimate is proved in Sobolev spaces with a mixed norm to solve the non-stationary Stokes problem (with non-zero divergence) in bounded and exterior domains, and from the first estimate an estimate is proved for the resolvent of the Stokes operator. The latter proof uses the explicit representation of the solution of the problem in a half-space in terms of the Green’s matrix; pointwise estimates are derived for the elements of this matrix.

MSC:

35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D07 Stokes and related (Oseen, etc.) flows
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47A10 Spectrum, resolvent
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