Critical and super-critical abstract parabolic equations

Tomasz Dlotko; Tongtong Liang; Yejuan Wang

doi:10.3934/dcdsb.2019238

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2020, Volume 25, Issue 4: 1517-1541. Doi: 10.3934/dcdsb.2019238

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Critical and super-critical abstract parabolic equations

1.
Institute of Mathematics, University of Silesia, Katowice, Poland
2.
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China

Received: December 2018

Early access: November 2019

Published: April 2020

This work was supported by NSF of China (Grants No. 41875084, 11571153), the Fundamental Research Funds for the Central Universities under Grant Nos. lzujbky-2018-ot03 and lzujbky-2018-it58

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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 quasi-geostrophic equation. Various technical estimates related to that problems and their fractional generalizations are also presented in the paper.

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Mathematics Subject Classification: Primary: 35A25, 35Q30, 26A33.

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