Infinite Boundary Friction Limit for Weak Solutions of the Stochastic Navier-Stokes Equations
Daniel Goodair
TL;DR
This work analyzes the large boundary friction limit for the 2D stochastic Navier–Stokes equations with Navier boundary conditions under transport-stretching noise. It proves that, starting from $u_0 o L^2_{ ext{σ}}$, the unique weak solutions $u^n$ with friction coefficients $oldsymbol{α}_n o ty$ converge in probability to the unique weak solution under no-slip in the topology $C([0,T];W^{- rac{ε}{2},2}_{σ})igcap L^2([0,T];L^2_{σ})$ for every $0<ε o 1$. The proof combines uniform interior estimates, boundary vanishing, tightness, limit identification, and a Gyongy–Krylov argument to upgrade subsequential convergence to full convergence. This result is novel in the stochastic setting and for weak solutions with $L^2$ initial data, expanding the understanding of boundary-friction and no-slip limits in fluid dynamics.
Abstract
We address convergence of the unique weak solutions of the 2D stochastic Navier-Stokes equations with Navier boundary conditions, as the boundary friction is taken uniformly to infinity, to the unique weak solution under the no-slip condition. Our result is that for initial velocity in $L^2_x$, the convergence holds in probability in $C_tW^{-\varepsilon,2}_x \cap L^2_tL^2_x$ for any $0 < \varepsilon$. The noise is of transport-stretching type, although the theorem holds with other transport, multiplicative and additive noise structures. This seems to be the first work concerning the large boundary friction limit with noise, and convergence for weak solutions, due to only $L^2_{x}$ initial data, appears new even deterministically.