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Well-posedness and decay for a nonlinear propagation wave model in atmospheric flows

Diego Alonso-Orán, Rafael Granero-Belinchón

Abstract

In this note, we provide two results concerning the global well-posedness and decay of solutions to an asymptotic model describing the nonlinear wave propagation in the troposphere, namely, the morning glory phenomenon. The proof of the first result combines a pointwise estimate together with some interpolation inequalities to close the energy estimates in Sobolev spaces. The second proof relies on suitable Wiener-like functional spaces.

Well-posedness and decay for a nonlinear propagation wave model in atmospheric flows

Abstract

In this note, we provide two results concerning the global well-posedness and decay of solutions to an asymptotic model describing the nonlinear wave propagation in the troposphere, namely, the morning glory phenomenon. The proof of the first result combines a pointwise estimate together with some interpolation inequalities to close the energy estimates in Sobolev spaces. The second proof relies on suitable Wiener-like functional spaces.
Paper Structure (3 sections, 2 theorems, 53 equations)

This paper contains 3 sections, 2 theorems, 53 equations.

Table of Contents

  1. Introduction and main result
  2. Proof of Theorem \ref{['thm1']}
  3. Proof of Theorem \ref{['thm2']}

Key Result

Theorem 1

Let $u_0\in H^{1}(\Omega)$ a zero mean function. Then, the corresponding solution satisfies Furthermore, if $u_0$ is such that $\left\| u_{0} \right\|_{L^{\infty}(\Omega)}$ is sufficiently small. Then, there exists a unique global in time solution $u\in C([0,\infty),H^1(\Omega))\cap L^2(0,\infty;H^2(\Omega))$.

Theorems & Definitions (3)

  • Theorem 1
  • Remark 1
  • Theorem 2