On the constants in a Kato inequality for the Euler and Navier-Stokes equations

2010/09/10 by Carlo Morosi, Livio Pizzocchero · 6 citations
Mathematics · Physics and Astronomy · #math.AP #math-ph #math.FA #math.MP #msc:76D05 #msc:26D10 #msc:46E35

paper · pdf · doi:10.3934/cpaa.2012.11.557

published as Commun. Pure Appl. Anal. 11(2)(2012), 557-586 · LaTeX, 39 pages. arXiv admin note: text overlap with arXiv:1007.4412 by the same authors, not concerning the main results

arxiv created 2010/09/10 · arxiv updated 2012/09/08

Abstract

We continue an analysis, started in [10], of some issues related to the incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus T^d. More specifically, we consider the quadratic term in these equations; this arises from the bilinear map (v, w) -> v . D w, where v, w : T^d -> R^d are two velocity fields. We derive upper and lower bounds for the constants in some inequalities related to the above bilinear map; these bounds hold, in particular, for the sharp constants G_{n d} = G_n in the Kato inequality | < v . D w | w >_n | <= G_n || v ||_n || w ||²_n, where n in (d/2 + 1, + infinity) and v, w are in the Sobolev spaces Hⁿ, H^(n+1) of zero mean, divergence free vector fields of orders n and n+1, respectively. As examples, the numerical values of our upper and lower bounds are reported for d=3 and some values of n. When combined with the results of [10] on another inequality, the results of the present paper can be employed to set up fully quantitative error estimates for the approximate solutions of the Euler/NS equations, or to derive quantitative bounds on the time of existence of the exact solutions with specified initial data; a sketch of this program is given.

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• A posteriori estimates for Euler and Navier-Stokes equations
• Smooth solutions of the Euler and Navier-Stokes equations from the a posteriori analysis of approximate solutions
• Large order Reynolds expansions for the Navier-Stokes equations
• New results on the constants in some inequalities for the Navier-Stokes quadratic nonlinearity

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