Weak solutions of the three-dimensional hypoviscous elastodynamics with finite kinetic energy

TL;DR

This paper constructs weak solutions with finite kinetic energy for 3D hypoviscous elastodynamics equations, extending the understanding of such solutions for fractional hypoviscosity levels below 1.

Contribution

It introduces a convex integration method with novel 2D intermittency blocks and temporal correctors tailored to the geometric structure of viscoelastic equations, addressing an open problem.

Findings

Existence of weak solutions with finite energy for fractional hypoviscosity $0 \\leq \theta < 1$

Development of a convex integration scheme with new building blocks

Extension of solutions to previously unexplored fractional viscosity regimes

Abstract

We construct weak solutions to the 3D hypoviscous incompressible elastodynamics with finite kinetic energy which was unknown in literatures. Our result holds for fractional hypoviscosity $(-\Delta)^\theta$, where $0\leq\theta<1$. The proof {consists of a convex integration scheme with new building blocks of 2D intermittency and suitable temporal correctors, which are motivated by} the inherent geometric structure of the viscoelastic equations.