On the Damped Euler--Monge--Amp\`ere equations with Radial Symmetry: Critical Thresholds and Large-Time Behavior

TL;DR

This paper studies the damped Euler--Monge--Ampère system with radial symmetry, identifying critical thresholds for global regularity versus finite-time blow-up, and shows damping can ensure global solutions even with vacuum.

Contribution

It introduces new methods to determine critical thresholds and demonstrates that damping removes density lower bounds needed in undamped cases, extending existing theories.

Findings

Critical thresholds distinguish global regularity from blow-up.

Damping allows solutions with vacuum to remain regular.

Subcritical data decay exponentially to equilibrium.

Abstract

We investigate the global well-posedness and large-time dynamics of the pressureless Euler--Monge--Amp\`ere (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.