Global solutions to the isentropic compressible Navier-Stokes equations with a class of large initial data

TL;DR

This paper proves the existence of global solutions to the isentropic compressible Navier-Stokes equations in multiple dimensions for large, highly oscillating initial data close to equilibrium, using dispersive estimates and hybrid Besov norms.

Contribution

It establishes global well-posedness for large initial data with arbitrary potential part norms, extending previous results by leveraging dispersive estimates and hybrid Besov spaces.

Findings

Global solutions exist for large, oscillating initial data near equilibrium.

Dispersive estimates are crucial for controlling nonlinear terms.

The results apply to initial data with large potential part in Besov spaces.

Abstract

In this paper, we consider the global well-posedness problem of the isentropic compressible Navier-Stokes equations in the whole space $\R^N$ with $N\ge2$. In order to better reflect the characteristics of the dispersion equation, we make full use of the role of the frequency on the integrability and regularity of the solution, and prove that the isentropic compressible Navier-Stokes equations admit global solutions when the initial data are close to a stable equilibrium in the sense of suitable hybrid Besov norm. As a consequence, the initial velocity with arbitrary $\dot{B}^{\fr{N}{2}-1}_{2,1}$ norm of potential part $\Pe^\bot u_0$ and large highly oscillating are allowed in our results. The proof relies heavily on the dispersive estimates for the system of acoustics, and a careful study of the nonlinear terms.