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On the Damped Euler--Monge--Ampère equations with Radial Symmetry: Critical Thresholds and Large-Time Behavior

Kunhui Luan

Abstract

We investigate the global well-posedness and large-time dynamics of the pressureless Euler--Monge--Ampère (EMA) system with velocity damping in multidimensions, subject to radially symmetric initial data. We first establish the phenomenon of critical thresholds, where subcritical initial data maintain global regularity, and supercritical initial data lead to finite time singularity formation. We provide two methods for constructing these thresholds: a refined spectral dynamics approach based on \cite{liu2002spectral} and a comparison principle based on Lyapunov functions introduced in \cite{bhatnagar2020critical2}. A key finding of this work is that the inclusion of linear damping effectively removes the initial density lower bound previously required in the undamped case \cite{tadmor2022critical} in certain regimes, allowing for global regularity even in the presence of vacuum or arbitrarily low density. Furthermore, for subcritical initial data, we prove an exponential decay rate to the equilibrium state. Our results unify and extend existing theories for 1D Euler--Poisson system and undamped multidimensional EMA system with radial symmetry.

On the Damped Euler--Monge--Ampère equations with Radial Symmetry: Critical Thresholds and Large-Time Behavior

Abstract

We investigate the global well-posedness and large-time dynamics of the pressureless Euler--Monge--Ampère (EMA) system with velocity damping in multidimensions, subject to radially symmetric initial data. We first establish the phenomenon of critical thresholds, where subcritical initial data maintain global regularity, and supercritical initial data lead to finite time singularity formation. We provide two methods for constructing these thresholds: a refined spectral dynamics approach based on \cite{liu2002spectral} and a comparison principle based on Lyapunov functions introduced in \cite{bhatnagar2020critical2}. A key finding of this work is that the inclusion of linear damping effectively removes the initial density lower bound previously required in the undamped case \cite{tadmor2022critical} in certain regimes, allowing for global regularity even in the presence of vacuum or arbitrarily low density. Furthermore, for subcritical initial data, we prove an exponential decay rate to the equilibrium state. Our results unify and extend existing theories for 1D Euler--Poisson system and undamped multidimensional EMA system with radial symmetry.
Paper Structure (20 sections, 16 theorems, 131 equations, 2 figures)

This paper contains 20 sections, 16 theorems, 131 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Comparison with the Euler--Poisson system.
  3. Outline of the paper.
  4. Background and statement of main results.
  5. Eulerian dynamics and the spectral dynamics
  6. 1D motivation and Riccati structure
  7. Spectral dynamics in Multidimensions
  8. The divergence approach
  9. Local well-posedness
  10. Global regularity versus finite time blowup
  11. Large-time behavior
  12. The damped Euler--Monge--Ampère system with radial symmetry.
  13. Spectral dynamics for the radial damped Euler--Monge--Ampére system.
  14. Proof of the main results.
  15. A vacuous critical threshold
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $s > \frac{n}{2}$. Consider the damped EMA system eqs:EMA with smooth radial initial data of the form Assume Then there exists a time $T>0$ such that the solution $\mathbf{\Omega}(\mathbf{x},t)$ satisfies Moreover, the life span $T$ can be extended as long as

Figures (2)

  • Figure 1: Phase portraits of the critical thresholds for the three damping regimes.
  • Figure 2: Illustration of the vector field.

Theorems & Definitions (39)

  • Theorem 2.1: Local well-posedness
  • Theorem 2.2: Critical thresholds
  • Theorem 2.3: Large Time Behavior
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4: Spectral form of the Monge--Ampère equation
  • ...and 29 more