# Formation of finite-time singularities for nonlinear elastodynamics with small initial disturbances

## TL;DR

This paper demonstrates that solutions to certain nonlinear hyperbolic systems, including elastodynamics and magnetohydrodynamics, can develop finite-time singularities even with small initial disturbances, extending previous results on fluid equations.

## Contribution

It provides a simplified proof technique for finite-time singularity formation and applies it to elastodynamics and magnetohydrodynamics, showing small data blow-up.

## Key findings

- Classical solutions blow up in finite time for small initial data
- A new test function simplifies the proof of singularity formation
- Extends singularity results to elastodynamics and magnetohydrodynamics

## Abstract

This article concerns the formation of finite-time singularities in solutions to quasilinear hyperbolic systems with small initial data. By constructing a special test function, we first present a simpler proof of the main result in Sideris' "Formation of singularities in three-dimensional compressible fluids": the global classical solution is non-existent for compressible Euler equation even for some small initial data. Then we apply this approach to nonlinear elastodynamics and magnetohydrodynamics, showing that the classical solutions to these equations can still blow up in finite time even if the initial data is small enough.