The Existence Theorems and the Liouville Theorem for the Steady-State Navier-Stokes Problems
Abstract
In the talk we present a survey of recent results (see [4]-[6]) on the existence theorems for the steady-state Navier-Stokes boundary value problems in the plane and axially symmetric 3D cases for bounded and exterior domains (the so called Leray problem, inspired by the classical paper [8]). One of the main tools is the Morse-Sard Theorem for the Sobolev functions $f\in W^2_1(\mathbb R^2)$ [1] (see also [2]-[3] for the multidimensional case). This theorem guaranties that almost all level lines of such functions are $C^1$-curves besides the function $f$ itself could be not $C^1$-regular.
Also we discuss the recent Liouville type theorem for the steady-state Navier-Stokes equations for axially symmetric 3D solutions in the absence of swirl (see [1]).
References
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