Global Regularity to Incompressible Viscoelastic System With a Class of Large Initial Data

TL;DR

This paper establishes the global existence of smooth solutions for 3D incompressible viscoelastic systems with large initial data by introducing a novel cone-condition structure, advancing understanding of complex fluid dynamics.

Contribution

It introduces a new cone-condition structure that enables proving global regularity for large initial data in incompressible viscoelastic flows.

Findings

Global smooth solutions are obtained for large initial data.

The cone-condition restricts the solution to control nonlinear terms.

The approach extends previous results limited to small initial data.

Abstract

The global existence of solutions to incompressible viscoelastic flows has been a longstanding open problem, even for the global weak solution. Under some special structure ("div-curl" condition) the global small smooth solution was obtained in \cite{llzhou, cz}. However, the result with large initial data remains unknown up to now, and it is studied in this paper. We shall put forward a new structure: the cone-condition and then derive the global smooth solution to 3D incompressible viscoelastic system with a class of large initial data. The key is to gain an angle quantity in the estimate of nonlinear terms by restricting the solution to a cone.