Self-similar imploding solutions of the relativistic Euler equations

TL;DR

This paper constructs self-similar smooth imploding solutions for the relativistic Euler equations across various dimensions and equations of state, advancing understanding of finite time blow-up phenomena in nonlinear wave equations.

Contribution

It introduces a new class of self-similar solutions for relativistic Euler equations applicable in multiple dimensions and for a range of equations of state.

Findings

Existence of smooth imploding solutions in 2D and 3D.

Solutions applicable for all >1 in physical space.

Progress towards understanding blow-up in nonlinear wave equations.

Abstract

Motivated by recent breakthrough on smooth imploding solutions of compressible Euler, we construct self-similar smooth imploding solutions of isentropic relativistic Euler equations with isothermal equation of state $p=\frac1\ell\varrho$ for \textit{all} $\ell>1$ in physical space dimension $d=2,3$ and for $\ell>1$ close to 1 in higher dimensions. This work is a crucial step toward solving the long-standing problem: finite time blow-up of the supercritical defocusing nonlinear wave equation.