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On global smooth solutions to the 2D isentropic and irrotational Chaplygin gases with short pulse data

Bingbing Ding, Zhouping Xin, Huicheng Yin

Abstract

This paper establishes the global existence of smooth solutions to the 2D isentropic and irrotational Euler equations for Chaplygin gases with a general class of short pulse initial data, which, in particular, resolves in this special case, the Majda's conjecture on the non-formation of shock waves of solutions from smooth initial data for multi-dimensional nonlinear symmetric systems which are totally linearly degenerate. Comparing to the 4D case, the major difficulties in this paper are caused by the slower time decay and the largeness of the solutions to the 2D quasilinear wave equation, some new auxiliary energies and multipliers are introduced to overcome these difficulties.

On global smooth solutions to the 2D isentropic and irrotational Chaplygin gases with short pulse data

Abstract

This paper establishes the global existence of smooth solutions to the 2D isentropic and irrotational Euler equations for Chaplygin gases with a general class of short pulse initial data, which, in particular, resolves in this special case, the Majda's conjecture on the non-formation of shock waves of solutions from smooth initial data for multi-dimensional nonlinear symmetric systems which are totally linearly degenerate. Comparing to the 4D case, the major difficulties in this paper are caused by the slower time decay and the largeness of the solutions to the 2D quasilinear wave equation, some new auxiliary energies and multipliers are introduced to overcome these difficulties.
Paper Structure (21 sections, 38 theorems, 341 equations, 5 figures)

This paper contains 21 sections, 38 theorems, 341 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The formulation of the problem and main results
  3. Sketch for the proof of Theorem \ref{['main']}
  4. Global estimates of the solution in $A_{2\delta}$
  5. Global estimates of the solution in $B_{2\delta}$
  6. Notations
  7. Some preliminaries
  8. Local existence of the smooth solution $\phi$
  9. The related geometry and definitions
  10. Basic equalities in the null frames
  11. Bootstrap assumptions on $\partial\phi$ near $C_0$ and some related estimates
  12. Energy estimates for the linearized equation and some higher order $L^2$ estimates
  13. $L^2$ estimates for the highest order derivatives of tr$\chi$ and $\slashed\triangle\mu$
  14. Estimates for the derivatives of tr$\chi$
  15. Estimates for the derivatives of $\slashed\triangle\mu$
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\varepsilon_0\in(0,\frac{1}{2})$ hold and $(\phi_0, \phi_1)(s,\omega)\in C^{\infty}(\mathbb R\times\mathbb S^1)$ be any fixed smooth functions with compact supports in $(-1,0)$ for the variable $s$. Under the assumption i1, there exists a suitably small positive constant $\delta_0$ such that fo where $C>0$ is a constant independent of $\delta$ and $\varepsilon_0$.

Figures (5)

  • Figure 1: Space-time domain $D^1=\{(t,r): 1\le t\le t_0, 2-t\le r\le t\}$
  • Figure 2: Space domain for $1-3\delta\le r\le 1+\delta$ on $\Sigma_{1+2\delta}$
  • Figure 3: The indications of some domains
  • Figure 4: The domain $D_t$ inside $B_{2\delta}$
  • Figure 5: The Domains I, II, III, IV

Theorems & Definitions (76)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • ...and 66 more