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Asymptotic behavior of solutions of the linearized Euler equations near a shear layer

Dongfen Bian, Emmanuel Grenier

Abstract

In this article, thanks to a new and detailed study of the Green's function of Rayleigh equation near the extrema of the velocity of a shear layer, we obtain optimal bounds on the asymptotic behaviour of solutions to the linearized incompressible Euler equations both in the whole plane, the half plane and the periodic case, and improve the description of the so called "vorticity depletion property" discovered by F. Bouchet and H. Morita by putting into light a localization property of the solutions of Rayleigh equation near an extremal velocity.

Asymptotic behavior of solutions of the linearized Euler equations near a shear layer

Abstract

In this article, thanks to a new and detailed study of the Green's function of Rayleigh equation near the extrema of the velocity of a shear layer, we obtain optimal bounds on the asymptotic behaviour of solutions to the linearized incompressible Euler equations both in the whole plane, the half plane and the periodic case, and improve the description of the so called "vorticity depletion property" discovered by F. Bouchet and H. Morita by putting into light a localization property of the solutions of Rayleigh equation near an extremal velocity.
Paper Structure (20 sections, 12 theorems, 242 equations)

This paper contains 20 sections, 12 theorems, 242 equations.

Table of Contents

  1. Introduction
  2. Setup and main results
  3. Study of Rayleigh equation
  4. Regular point
  5. Near a critical point
  6. Near an extremal point
  7. The case $\alpha = 0$ and $U_s(y) = y^2$
  8. The general case
  9. At infinity
  10. Proof of Propositions \ref{['extension']} and \ref{['localization']}
  11. Asymptotic behavior of $\psi_\alpha(t,y)$
  12. Green function of Rayleigh equation
  13. Study of $G_\alpha^{int}$
  14. Away from extremal values
  15. Near an extremal value
  16. ...and 5 more sections

Key Result

Proposition 2.1

(Extension of the dispersion relation) Under the assumptions (A1) and (A2), $\psi_{-,\alpha,c}(0)$ may be extended by continuity to ${\Bbb R}^+ - \{ (c_{extr}^l)_{1 \le l \le L} \}$.

Theorems & Definitions (21)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 11 more