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Global Existence and Asymptotic Equivalence to Barenblatt-type Solutions for the Physical Vacuum Free Boundary Problem of Damped Compressible Euler Equations in M-D

Huihui Zeng

Abstract

For the physical vacuum free boundary problem of the damped compressible Euler equations in both 2D and 3D, we prove the global existence of smooth solutions and justify their time-asymptotic equivalence to the corresponding Barenblatt self-similar solutions derived from the porous media equation under Darcy's law approximation, provided the initial data are small perturbations of the Barenblatt solutions. Building on the 3D almost global existence result in [Zeng, Arch. Ration. Mech. Anal. 239, 553--597 (2021)], our key contribution lies in improving the decay rate of the time derivative of the perturbation from $-1$ (as previously established) to $-1-\varepsilon$ for a fixed constant $\varepsilon > 0$. This critical enhancement ensures time integrability and hence global existence. Together with the previous 1D result in [Luo--Zeng, Comm. Pure Appl. Math. 69, 1354--1396 (2016)], the results obtained in this paper provide a complete answer to the question raised in [Liu, T.-P.: Jpn. J. Appl. Math. 13, 25--32 (1996)]. Moreover, we also consider the problem with time-dependent damping of the form $(1+t)^{-λ}$ for $0 < λ< 1$. Notably, our framework unifies the treatment of both time-dependent ($0 < λ< 1$) and time-independent ($λ= 0$) damping cases across dimensions. We further quantify the decay rates of the density and velocity, as well as the expansion rate of the physical vacuum boundary.

Global Existence and Asymptotic Equivalence to Barenblatt-type Solutions for the Physical Vacuum Free Boundary Problem of Damped Compressible Euler Equations in M-D

Abstract

For the physical vacuum free boundary problem of the damped compressible Euler equations in both 2D and 3D, we prove the global existence of smooth solutions and justify their time-asymptotic equivalence to the corresponding Barenblatt self-similar solutions derived from the porous media equation under Darcy's law approximation, provided the initial data are small perturbations of the Barenblatt solutions. Building on the 3D almost global existence result in [Zeng, Arch. Ration. Mech. Anal. 239, 553--597 (2021)], our key contribution lies in improving the decay rate of the time derivative of the perturbation from (as previously established) to for a fixed constant . This critical enhancement ensures time integrability and hence global existence. Together with the previous 1D result in [Luo--Zeng, Comm. Pure Appl. Math. 69, 1354--1396 (2016)], the results obtained in this paper provide a complete answer to the question raised in [Liu, T.-P.: Jpn. J. Appl. Math. 13, 25--32 (1996)]. Moreover, we also consider the problem with time-dependent damping of the form for . Notably, our framework unifies the treatment of both time-dependent () and time-independent () damping cases across dimensions. We further quantify the decay rates of the density and velocity, as well as the expansion rate of the physical vacuum boundary.

Paper Structure

This paper contains 7 sections, 6 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Reformulation of the problem and main results
  3. Proof of Theorem \ref{['mainthm']}
  4. Preliminaries
  5. Curl estimates
  6. Energy estimates
  7. Proof of Theorem \ref{['orig']}

Key Result

Theorem 2.1

Let $n=2,3$ and $0\le \lambda<1$. There exists a positive constant $\bar{\varepsilon}$ depending only on $n,\lambda,\gamma$ and $M$ such that if $\mathscr{E}(0)\le \bar{\varepsilon}$, then the problem system admits a global smooth solution in $[0,\infty)\times \Omega$ satisfying where $C$ is a positive constant which only depend on $n,\lambda,\gamma$ and $M$ but do not depend on the time $t$.

Theorems & Definitions (7)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4