The influence of viscous coefficients on the lifespan of 3-D anisotropic Navier-Stokes system

TL;DR

This paper establishes lower bound estimates on the lifespan of solutions to the 3-D anisotropic Navier-Stokes system, considering various configurations of viscous coefficients, including cases where some coefficients vanish, to better understand fluid behavior and nonlinear dynamics.

Contribution

It provides new lower bound estimates for the lifespan of anisotropic Navier-Stokes solutions, accounting for different viscous coefficient scenarios, including degenerate cases.

Findings

Derived lower bounds depend on the anisotropic viscous coefficients.

Analyzed cases with all positive, one zero, or two zero viscous coefficients.

Enhanced understanding of nonlinear structure in classical Navier-Stokes equations.

Abstract

The anisotropic Navier-Stokes system arises in geophysical fluid dynamics, which is derived by changing $-\nu\D$ in the classical Navier-Stokes system to $-(\nu_1\partial^2_1+\nu_2\partial^2_2+\nu_3\partial^2_3)$. Here $\nu_1,\,\nu_2\,,\nu_3$ are the viscous coefficients, which can be different from each other. This reflects that the fluid can behave differently in each direction. The purpose of this paper is to derive some lower bound estimates on the lifespan to such anisotropic Navier-Stokes system. We not only investigate the case when $\nu_1,\,\nu_2\,,\nu_3$ are all positive, but also the more sophisticated cases when one or two of them vanish. We find that in these lower bound estimates, the weights of $\nu_1,\,\nu_2\,,\nu_3$ are not equal. A detailed study of this problem can also help us to have a better understanding of the nonlinear structure in the classical Navier-Stokes system.