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

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 $u_0^{\nu }\in L^{\mathrm{\infty }}$ and ${\omega }_0^{\nu }\in L^{\mathrm{\infty }}$ without imposing decay conditions at infinity. We first derive a priori estimates for the Navier-Stokes equations in $L^{\mathrm{\infty }}$ spaces. Then, using harmonic analysis tools including Littlewood-Paley decomposition and Bony’s paraproduct calculus, we prove the $L^{\mathrm{\infty }}$-convergence of the velocity field and $L_{uloc}^2$-convergence of the vorticity as $\nu \to 0$.