Global regularity and large time behavior for some inviscid Oldroyd-B models in $\mathbb{R}^2$

TL;DR

This paper proves the global existence and analyzes the large time decay of solutions for inviscid Oldroyd-B models in two dimensions, using energy estimates, Besov space techniques, and Fourier splitting methods.

Contribution

It establishes global strong solutions in Sobolev and Besov spaces for inviscid Oldroyd-B models and studies their large time decay behavior, extending previous results.

Findings

Global existence of strong solutions in Sobolev space.

Global existence in critical Besov space.

Decay rate of solutions in $H^1$ norm.

Abstract

In this paper, we are concerned with global strong solutions and large time behavior for some inviscid Oldroyd-B models. We first establish the energy estimate and B-K-M criterion for the 2-D co-rotation inviscid Oldroyd-B model. Then, we obtain global strong solutions with large data in Sobolev space by proving the boundedness of vorticity. As a corollary, we prove global existence of the corresponding inviscid Hooke model near equilibrium. Furthermore, we present global existence for the 2-D co-rotation inviscid Oldroyd-B model in critical Besov space by a refined estimate in Besov spaces with index $0$. Finally, we study large time behaviour for the noncorotation inviscid Oldroyd-B model. Applying the Fourier splitting method, we prove the $H^1$ decay rate for global strong solutions constructed by T. M. Elgindi and F. Rousset.