Advances in Differential Equations

A priori bounds for Gevrey--Sobolev norms of space-periodic three-dimensional solutions to equations of hydrodynamic type

Vladislav Zheligovsky

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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.

Article information

Source
Adv. Differential Equations Volume 16, Number 9/10 (2011), 955-976.

Dates
First available in Project Euclid: 17 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355703183

Mathematical Reviews number (MathSciNet)
MR2850760

Zentralblatt MATH identifier
1351.76007

Subjects
Primary: 76B03: Existence, uniqueness, and regularity theory [See also 35Q35] 76D03: Existence, uniqueness, and regularity theory [See also 35Q30] 35Q35: PDEs in connection with fluid mechanics 35B65: Smoothness and regularity of solutions

Citation

Zheligovsky, Vladislav. A priori bounds for Gevrey--Sobolev norms of space-periodic three-dimensional solutions to equations of hydrodynamic type. Adv. Differential Equations 16 (2011), no. 9/10, 955--976. https://projecteuclid.org/euclid.ade/1355703183.


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