The Cauchy problem for an inviscid and non-diffusive Oldroyd-B model in two dimensions

TL;DR

This paper proves the global existence, uniqueness, and decay rates of solutions for a non-diffusive, inviscid Oldroyd-B model in two dimensions, addressing challenges due to lack of dissipation.

Contribution

It establishes the first global well-posedness and decay results for the non-diffusive Oldroyd-B model in 2D with small initial data.

Findings

Global existence and uniqueness of strong solutions.

Optimal decay rates for spatial derivatives.

Analysis of the slow decay due to lack of dissipation.

Abstract

A two-dimensional inviscid and diffusive Oldroyd-B model was investigated by [T. M. Elgindi, F. Rousset, Commun. Pure Appl. Math. 68 (2015), 2005--2021] where the global existence and uniqueness of the strong solution were established for arbitrarily large initial data. As pointed out by [A. V. Bhave, R. C. Armstrong, R. A. Brown, J. Chem. Phys. 95(1991), 2988--3000], the diffusion coefficient is significantly smaller than other effects, it is interesting to study the non-diffusive model. In the present work, we obtain the global-in-time existence and uniqueness of the strong solution to the non-diffusive model with small initial data via deriving some uniform regularity estimates and taking vanishing diffusion limits. In addition, the large time behavior of the solution is studied and the optimal time-decay rates for each order of spatial derivatives are obtained. The main challenges focus on the lack of dissipation and regularity effects of the system and on the slower decay in the two-dimensional settings. A combination of the spectral analysis and the Fourier splitting method is adopted.