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Sharp lifespan estimate for the compressible Euler system with critical time-dependent damping in $\R^2$

Lv Cai, Ning-An Lai, Wen-Ze Su

Abstract

This paper concerns the long time existence to the smooth solutions of the compressible Euler system with critical time dependent damping in $\R^2$. We establish the sharp lifespan estimate from below, with respect to the small parameter of the initial perturbation. For this end, the vector fields $\widehat{Z}$ (defined below) are used instead of the usual one $Z$, to get better decay for the linear error terms. This idea may also apply to the long time behavior study of nonlinear wave equations with time-dependent damping.

Sharp lifespan estimate for the compressible Euler system with critical time-dependent damping in $\R^2$

Abstract

This paper concerns the long time existence to the smooth solutions of the compressible Euler system with critical time dependent damping in . We establish the sharp lifespan estimate from below, with respect to the small parameter of the initial perturbation. For this end, the vector fields (defined below) are used instead of the usual one , to get better decay for the linear error terms. This idea may also apply to the long time behavior study of nonlinear wave equations with time-dependent damping.
Paper Structure (24 sections, 42 theorems, 216 equations, 1 figure, 3 tables)

This paper contains 24 sections, 42 theorems, 216 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Review of the literature
  3. Euler system without damping
  4. Euler system with damping
  5. Preliminaries
  6. Reformulating as a hyperbolic system
  7. Local theory for the reformulated system
  8. Evolution of the support
  9. Reformulating as wave equations
  10. Vector fields and commutators
  11. Equations of derivatives
  12. Useful inequalities
  13. Reduction to a bootstrap argument
  14. Main result in the reformulated system
  15. Bootstrap argument
  16. ...and 9 more sections

Key Result

Theorem 1.1

Consider the system Euler eqnspressure law with $0 < \mu \leq 1$. If the perturbations $(\rho_0,u_0)$ satisfy then there exist constants $0<\epsilon_0\ll1$ and $C>0$, such that for all $\epsilon\in(0,\epsilon_0]$, Euler eqnspressure law admits a classical solution $(\rho,u) \in C^\infty ([ 0,T_{\epsilon} ]\times{\bf R}^2)$, where The quantities $\epsilon_0,C$ only depend on the perturbations $(u

Figures (1)

  • Figure 1: Flow chart 1

Theorems & Definitions (83)

  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.1
  • Lemma 2.1
  • ...and 73 more