Critical and super-critical abstract parabolic equations

Abstract

Our purpose is to formulate an abstract result, motivated by the recent paper [8], allowing to treat the solutions of critical and super-critical equations as limits of solutions to their regularizations. In both cases we are improving the viscosity, making it stronger, solving the obtained regularizations with the use of Dan Henry's technique, then passing to the limit in the improved viscosity term to get a solution of the limit problem. While in case of the critical problems we will just consider a 'bit higher' fractional power of the viscosity term, for super-critical problems we need to use a version of the 'vanishing viscosity technique' that comes back to the considerations of E. Hopf, O.A. Oleinik, P.D. Lax and J.-L. Lions from 1950th. In both cases, the key to that method are the uniform with respect to the parameter estimates of the approximating solutions. The abstract result is illustrated with the Navier-Stokes equation in space dimensions 2 to 4, and with the 2-D quasigeostrophic equation. Various technical estimates related to that problems and their fractional generalizations are also presented in the paper.

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