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On Vacuum Free Boundary Problem of the Spherically Symmetric Euler Equations with Damping and Solid Core

Yan-Lin Wang

Abstract

In this paper, the global existence of smooth solution and the long-time asymptotic stability of the equilibrium to vacuum free boundary problem of the spherically symmetric Euler equations with damping and solid core have been obtained for arbitrary finite positive gas constant $A$ in the state equation $p=A ρ^γ$ with $p$ being the pressure and $ρ$ the density, provided that $γ>4/3,$ initial perturbation is small and the radius of the equilibrium $R$ is suitably larger than the radius of the solid core $r_0$. Moreover, we obtain the pointwise convergence from the smooth solution to the equilibrium in a surprisingly exponential time-decay rate. The proof is mainly based on weighted energy method in Lagrangian coordinate.

On Vacuum Free Boundary Problem of the Spherically Symmetric Euler Equations with Damping and Solid Core

Abstract

In this paper, the global existence of smooth solution and the long-time asymptotic stability of the equilibrium to vacuum free boundary problem of the spherically symmetric Euler equations with damping and solid core have been obtained for arbitrary finite positive gas constant in the state equation with being the pressure and the density, provided that initial perturbation is small and the radius of the equilibrium is suitably larger than the radius of the solid core . Moreover, we obtain the pointwise convergence from the smooth solution to the equilibrium in a surprisingly exponential time-decay rate. The proof is mainly based on weighted energy method in Lagrangian coordinate.
Paper Structure (6 sections, 8 theorems, 131 equations)

This paper contains 6 sections, 8 theorems, 131 equations.

Table of Contents

  1. introduction
  2. Main results
  3. Proof of Theorem \ref{['thm1']} -- Theorem \ref{['thm3']}
  4. Elliptic Estimates
  5. Basic Energy Estimate
  6. Higher-Order Energy Estimates

Key Result

Theorem 2.1

Suppose $\gamma>\frac{4}{3}$ in the barotropic gas pressure pressure and $r_0< R\leq \frac{4}{3-\alpha}r_0<\infty$ with $\alpha=\frac{1}{\gamma-1},$ where $R$ is the finite radius of the equilibria and $r_0$ is the radius of the solid core. There exists a positive constant $\epsilon\in(0, \epsilon_0 for some suitably small positive constant $\delta$ independent of $t.$

Theorems & Definitions (14)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Proposition 3.1
  • Lemma 3.2: Lower-Order Elliptic Estimates
  • proof
  • Lemma 3.3: Higher-Order Elliptic Estimates
  • proof
  • ...and 4 more