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Waiting Time Solutions in gas dynamics

Juhi Jang, Jiaqi Liu, Nader Masmoudi

TL;DR

The paper addresses waiting-time phenomena for the 1D compressible Euler equations with a vacuum boundary under the physical vacuum condition, for $\gamma\in(1,3)$. It develops a self-similar reduction using a two-parameter scaling and analyzes an associated ODE system in the $(U,H)$ plane, including a key special solution $H^{\text{sp}}(U)$. It proves the existence of a two-parameter continuum of self-similar waiting-time solutions, parameterized by $\mu\in(0,1)$ and $\gamma\in(1,3)$, featuring a stationary left vacuum boundary for $t<0$ that transitions to a moving boundary with a physical vacuum for $t>0$, after a finite waiting time. The construction relies on a detailed phase-portrait analysis and barrier arguments to connect critical points $C\to A\to E\to D\to B$, yielding Hölder regularity near the singular point and a Lipschitz discontinuity along the emergent sonic curve.

Abstract

In this article, we construct a continuum family of self-similar waiting time solutions for the one-dimensional compressible Euler equations for the adiabatic exponent $\ga\in(1,3)$ in the half-line with the vacuum boundary. The solutions are confined by a stationary vacuum interface for a finite time with at least $C^1$ regularity of the velocity and the sound speed up to the boundary. Subsequently, the solutions undergo the change of the behavior, becoming only Hölder continuous near the singular point, and simultaneously transition to the solutions to the vacuum moving boundary Euler equations satisfying the physical vacuum condition. When the boundary starts moving, a weak discontinuity emanating from the singular point along the sonic curve emerges. The solutions are locally smooth in the interior region away from the vacuum boundary and the sonic curve.

Waiting Time Solutions in gas dynamics

TL;DR

The paper addresses waiting-time phenomena for the 1D compressible Euler equations with a vacuum boundary under the physical vacuum condition, for . It develops a self-similar reduction using a two-parameter scaling and analyzes an associated ODE system in the plane, including a key special solution . It proves the existence of a two-parameter continuum of self-similar waiting-time solutions, parameterized by and , featuring a stationary left vacuum boundary for that transitions to a moving boundary with a physical vacuum for , after a finite waiting time. The construction relies on a detailed phase-portrait analysis and barrier arguments to connect critical points , yielding Hölder regularity near the singular point and a Lipschitz discontinuity along the emergent sonic curve.

Abstract

In this article, we construct a continuum family of self-similar waiting time solutions for the one-dimensional compressible Euler equations for the adiabatic exponent in the half-line with the vacuum boundary. The solutions are confined by a stationary vacuum interface for a finite time with at least regularity of the velocity and the sound speed up to the boundary. Subsequently, the solutions undergo the change of the behavior, becoming only Hölder continuous near the singular point, and simultaneously transition to the solutions to the vacuum moving boundary Euler equations satisfying the physical vacuum condition. When the boundary starts moving, a weak discontinuity emanating from the singular point along the sonic curve emerges. The solutions are locally smooth in the interior region away from the vacuum boundary and the sonic curve.
Paper Structure (11 sections, 15 theorems, 101 equations, 2 figures)

This paper contains 11 sections, 15 theorems, 101 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Self-similar ODEs and the main result
  3. Riemann invariants and simple wave solutions
  4. Scaling invariance and self-similarity as $x\rightarrow 0$ and $t\rightarrow 0$
  5. Self-similar ODE
  6. Connection from $C$ to $A$, $E$ and $D$
  7. Basic phase portrait analysis
  8. Connection from $C$ to $A$
  9. Continuation through $A$, $E$ and $D$
  10. Connection from $B$ to $D$
  11. Proof of Theorem \ref{['main theorem']}

Key Result

Lemma 2.2

For any $\gamma\in(1,3)$ and $\mu\in(0,1)$, $H^{\text{sp}}(U) = \frac{(\gamma-1)^2}{4}U^2$ is always a solution to the ODE ODE.

Figures (2)

  • Figure 1: The structure of the waiting time solution. The red line shows the moving boundary $\{ y \equiv y_B \}=\{x=y_B t^\delta\}$. The blue line corresponds to the sonic curve $\{ y \equiv y_D \}=\{x=y_D t^\delta\}$ representing the weak discontinuity.
  • Figure 2: $\gamma = 1.816$, $\mu=0.716$, Dashed line: $H^\text{sp} = \frac{(\gamma-1)^2}{4}U^2$, Red line: $F=0$, Blue line: $G=0$, Green line: $\Delta=0$.

Theorems & Definitions (35)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Remark 2.4
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4: Special solution before the change of behavior
  • ...and 25 more