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September/October 2011 A priori bounds for Gevrey--Sobolev norms of space-periodic three-dimensional solutions to equations of hydrodynamic type
Vladislav Zheligovsky
Adv. Differential Equations 16(9/10): 955-976 (September/October 2011).

Abstract

We present a technique for derivation of a priori bounds for Gevrey--Sobolev norms of space-periodic three-dimensional solutions to evolutionary partial differential equations of hydrodynamic type. It involves a transformation of the flow velocity in the Fourier space, which introduces a feedback between the index of the norm and the norm of the transformed solution, and results in emergence of a mildly dissipative term. We illustrate the technique, using it to derive finite-time bounds for Gevrey--Sobolev norms of solutions to the Euler and inviscid Burgers equations, and global-in-time bounds for the Voigt-type regularizations of the Euler and Navier--Stokes equation (assuming that the respective norm of the initial condition is bounded). The boundedness of the norms implies analyticity of the solutions in space.

Citation

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Vladislav Zheligovsky. "A priori bounds for Gevrey--Sobolev norms of space-periodic three-dimensional solutions to equations of hydrodynamic type." Adv. Differential Equations 16 (9/10) 955 - 976, September/October 2011.

Information

Published: September/October 2011
First available in Project Euclid: 17 December 2012

zbMATH: 1351.76007
MathSciNet: MR2850760

Subjects:
Primary: 35B65, 35Q35, 76B03, 76D03

Rights: Copyright © 2011 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.16 • No. 9/10 • September/October 2011
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