Optimal decay rate for the 2-D compressible Oldroyd-B and Hall-MHD model

TL;DR

This paper establishes the optimal decay rates for strong solutions to the 2-D compressible Oldroyd-B and Hall-MHD models using advanced Fourier and Besov space techniques, without smallness assumptions on initial data.

Contribution

It introduces a novel combination of Fourier splitting, energy estimates, and Littlewood-Paley theory to determine optimal decay rates for these complex fluid models.

Findings

Decay rate of (1+t)^(-1/4) in L^2 norm.

Solutions belong to critical Besov spaces with negative index.

Achieves optimal decay in H^2 framework without smallness restrictions.

Abstract

In this paper, we are concerned with long time behavior of the strong solutions to the 2-D compressible Oldroyd-B and Hall-MHD model. By virtue of the improved Fourier splitting method and the time weighted energy estimate, we obtain the $L^2$ decay rate $(1+t)^{-\frac{1}{4}}$. According to the Littlewood-Paley theory, we prove that the solutions belong to the critical Besov space with negative index. Finally, we show optimal decay rate in $H^2$-framework without the smallness restriction of low frequencies.