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Riemann boundary value problems for the Chaplygin gas outside a convex cornered wedge

Bingsong Long

Abstract

We consider two-dimensional Riemann boundary value problems of Euler equations for the Chaplygin gas with two piecewise constant initial data outside a convex cornered wedge. In self-similar coordinates, when the flow at the wedge corner is subsonic, this problem can be reformulated as a boundary value problem for nonlinear degenerate elliptic equations in concave domains containing a corner larger than $π$. It is shown that there does not exist a global Lipschitz solution for this case. We analyze the sign of the flow velocity along a certain direction, and then obtain this result by deriving a contradiction. Besides, the unique existence of the solution to the problem is established when the flow at the wedge corner is supersonic. The results obtained here are also valid for the problem of shock diffraction by a convex cornered wedge.

Riemann boundary value problems for the Chaplygin gas outside a convex cornered wedge

Abstract

We consider two-dimensional Riemann boundary value problems of Euler equations for the Chaplygin gas with two piecewise constant initial data outside a convex cornered wedge. In self-similar coordinates, when the flow at the wedge corner is subsonic, this problem can be reformulated as a boundary value problem for nonlinear degenerate elliptic equations in concave domains containing a corner larger than . It is shown that there does not exist a global Lipschitz solution for this case. We analyze the sign of the flow velocity along a certain direction, and then obtain this result by deriving a contradiction. Besides, the unique existence of the solution to the problem is established when the flow at the wedge corner is supersonic. The results obtained here are also valid for the problem of shock diffraction by a convex cornered wedge.

Paper Structure

This paper contains 12 sections, 8 theorems, 58 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic facts
  4. Pressure wave structure
  5. Main theorem
  6. The case $u_1<c_1$
  7. Some useful estimates
  8. The sign of $u$
  9. Proof of the case $(i)$ of \ref{['thm: Main2']}
  10. The case $c_1<u_1<c_0+c_1$
  11. shock diffraction by a convex cornered wedge
  12. The critical case $u_1=c_1$

Key Result

Lemma 2.1

CQu12 If then the problem eq:1-D Euler system--1-D initial condition admits a self-similar solution. The relation between different state on the $(u,c)$ plane is shown in fg-cqu12. Moreover, the intermediate state is

Figures (8)

  • Figure 1: Riemann boundary value problems.
  • Figure 2: Four regions associated with a point $(u_l,c_l)$.
  • Figure 3: The case $u_1<c_1$.
  • Figure 4: The case $u_1<c_1$.
  • Figure 5: The case $c_1<u_1<c_0+c_1$.
  • ...and 3 more figures

Theorems & Definitions (12)

  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Remark 2.5
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.1
  • ...and 2 more