2023/06/06 by Hui Li, Weiren Zhao, Li, Hui +1 · 1 citation
Mathematics · Engineering · #Navier-Stokes equation solutions #Fluid Dynamics and Turbulent Flows #Advanced Mathematical Physics Problems
paper · pdf · doi:10.48550/arxiv.2306.03555
In this paper, we study the long-time behavior of the solutions to the two-dimensional incompressible free Navier Stokes equation (without forcing) with small viscosity ν, when the initial data is close to stable monotone shear flows. We prove the asymptotic stability and obtain the sharp stability threshold ν(1)/(2) for perturbations in the critical space HlogxL2y. Specifically, if the initial velocity Vin and the corresponding vorticity Win are ν(1)/(2)-close to the shear flow (bin(y),0) in the critical space, i.e., ‖Vin-(bin(y),0)‖_Lx,y2+‖Win-(-∂ybin)‖HlogxL2y≤ εν(1)/(2), then the velocity V(t) stay ν(1)/(2)-close to a shear flow (b(t,y),0) that solves the free heat equation (∂t-ν∂yy)b(t,y)=0. We also prove the enhanced dissipation and inviscid damping, namely, the nonzero modes of vorticity and velocity decay in the following sense ‖W≠‖L2\lesssim εν(1)/(2)e^-cν(1)/(3)t and ‖V≠‖_L2tL2x,y\lesssim εν(1)/(2). In the proof, we construct a time-dependent wave operator corresponding to the Rayleigh operator b(t,y)∂x-∂yyb(t,y)∂xΔ-1, which could be useful in future studies.