Global regularity and decay behavior for Leray equations with critical-dissipation and its application to self-similar solutions

Miao, Changxing; Zheng, Xiaoxin

doi:10.1090/tran/9148

Global regularity and decay behavior for Leray equations with critical-dissipation and its application to self-similar solutions
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by Changxing Miao and Xiaoxin Zheng;

Trans. Amer. Math. Soc. 377 (2024), 4365-4433

DOI: https://doi.org/10.1090/tran/9148

Published electronically: April 11, 2024

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Abstract:

In this paper, we show the global regularity and the optimal decay of weak solutions to the generalized Leray problem with critical dissipation. Our approach hinges on the maximal smoothing effect, $L^{p}$-type elliptic regularity of linearization, and the action of the heat semigroup generated by the fractional powers of Laplace operator on distributions with Fourier transforms supported in an annulus. As a by-product, we construct a self-similar solution to the three-dimensional incompressible Navier-Stokes equations. Most notably, we prove the global regularity and the optimal decay without the need for additional requirements found in existing literatures.

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