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Amplitude–phase decompositions and the growth and decay of solutions of the incompressible Navier–Stokes and Euler equations

Thomas J. R. Hughes

Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, 201 East 24th Street, C0200, Austin TX 78712, USA

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https://doi.org/10.1142/S0218202513500759Cited by:0 (Source: Crossref)

Abstract

We introduce the concept of amplitude–phase decompositions and apply it to the study of growth and decay of solutions of the incompressible Navier–Stokes and Euler equations. Amplitudes coincide with functionals of physical and mathematical interest. We are able to explicitly solve for the amplitudes in terms of the phases. The results obtained provide insights into the growth and decay of enstrophy and viscous dissipation, and identify new criteria for solutions to remain smooth for all time or blow-up in finite time.

Keywords:

AMSC: 35Q30, 35Q31, 35Q35, 76B03, 76D03, 76D05, 76F02

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