Stability Analysis of Bounded Solutions to 2-D Incompressible Navier-Stokes Equations in L^2_uloc Spaces

Abstract

This paper examines the stability analysis of two-dimensional incompressible fluids in $L_{uloc}^2$ spaces. Extending the classical stability framework for Euler equations in Yudovich-type settings, we establish the convergence of Navier-Stokes solutions toward Euler solutions as viscosity $\nu \to 0$, under the weakened assumptions $u_0^{\nu }\in L^{\infty }$ and ${\omega }_0^{\nu }\in L^{\infty }$ without imposing decay conditions at infinity. We first derive a priori estimates for the Navier-Stokes equations in $L^{\infty }$ spaces. Then, using harmonic analysis tools including Littlewood-Paley decomposition and Bony's paraproduct calculus, we prove the $L^{\infty }$-convergence of the velocity field and $L_{uloc}^2$-convergence of the vorticity as $\nu \to 0$.

arXiv comment25 pages, 1 figure