Note On the blowup criterion of smooth solution to the incompressible viscoelastic flow

TL;DR

This paper investigates conditions preventing blowup in smooth solutions of the Oldroyd model for incompressible viscoelastic flow, showing that boundedness of the velocity gradient in a specific integrable function space ensures global regularity.

Contribution

It establishes a new blowup criterion based on the integrability of the velocity gradient, extending understanding of solution regularity in Oldroyd models.

Findings

Smooth solutions do not blow up if the velocity gradient is integrable in time and bounded in space.

The criterion applies to both 2D and 3D cases.

Provides a mathematical condition for the global existence of solutions.

Abstract

We study the blowup criterion of smooth solution to the Oldroyd models. Let $(u(t,x), F(t,x)$ be a smooth solution in $[0,T)$, it is shown that the solution $(u(t,x), F(t,x)$ does not appear breakdown until $t=T$ provided $\nabla u(t,x)\in L^1([0,T]; L^\infty(\mr^n))$, $n=2,3$.