Global existence and optimal decay rate of weak solutions to some inviscid Oldroyd-B models

TL;DR

This paper proves the global existence and optimal decay rates of weak solutions for certain inviscid Oldroyd-B models in 2D, extending classical results for Euler equations using advanced harmonic analysis techniques.

Contribution

It establishes the existence, energy conservation, and decay rates of weak solutions for both co-rotation and noncorotation inviscid Oldroyd-B models, including large and small data cases.

Findings

Global weak solutions exist under various conditions.

Energy conservation holds for co-rotation models.

Optimal decay rates are achieved for noncorotation models.

Abstract

This paper is devoted to global existence and optimal decay rate of weak solutions to some inviscid Oldroyd-B models with center diffusion. By virtue of the properties of Calderon-Zygmund operator and the Littlewood-Paley decomposition theory, we firstly prove that the 2-D co-rotation inviscid Oldroyd-B model admits global weak solutions with some large data under different integrability conditions. Furthermore, we prove the energy conservation of weak solutions for the co-rotation case. These obtained results generalize and cover the classical results for the Euler equation. Moreover, we establish global weak solutions with small data for the 2-D noncorotation inviscid Oldroyd-B model without damping. Finally, we prove optimal decay rate of global weak solutions for the noncorotation case by the improved Fourier splitting method.