Counterexamples of commutator estimates in the besov and the triebel - Lizorkin spaces related to the euler equations

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5 Citations (Scopus)

Abstract

This paper deals with the Kato.Ponce - type commutator estimates in the Besov space Bs p, q(ℝn) and the Triebel - Lizorkin space Fs p, q(ℝn) related to the Euler equations describing the motion of perfect incompressible fluids. We investigate the relation between the optimal bound of the commutator estimates and the solvability of the Euler equations. In particular, we show that these commutator estimates fail in Bs p, q(ℝn) and F s p, q(ℝn) with the critical differential order s = n/p + 1 and various exponents p and q.

Original languageEnglish
Pages (from-to)2473-2483
Number of pages11
JournalSIAM Journal on Mathematical Analysis
Volume42
Issue number6
DOIs
Publication statusPublished - Dec 1 2010
Externally publishedYes

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Commutator Estimate
Electric commutators
Triebel-Lizorkin Space
Euler equations
Euler Equations
Counterexample
Optimal Bound
Perfect Fluid
Besov Spaces
Incompressible Fluid
Solvability
Exponent
Fluids
Motion

All Science Journal Classification (ASJC) codes

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

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abstract = "This paper deals with the Kato.Ponce - type commutator estimates in the Besov space Bs p, q(ℝn) and the Triebel - Lizorkin space Fs p, q(ℝn) related to the Euler equations describing the motion of perfect incompressible fluids. We investigate the relation between the optimal bound of the commutator estimates and the solvability of the Euler equations. In particular, we show that these commutator estimates fail in Bs p, q(ℝn) and F s p, q(ℝn) with the critical differential order s = n/p + 1 and various exponents p and q.",
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