Strong well-posedness of the two-dimensional stochastic Navier-Stokes equation on moving domains
Ping Chen, Tianyi Pan, Tusheng Zhang
TL;DR
This work proves the strong $H^1$ well-posedness of the two-dimensional stochastic Navier–Stokes equation with multiplicative noise on moving domains by reducing to a piecewise variational framework on short time intervals via the Piola transform. It then patches these local solutions to obtain a global result on $[0,T]$, using a uniform moment bound with the function $\Theta(x)=\log(1+\log(1+x))$ to pass to the limit in a truncated problem. The analysis relies on transforming the problem to a fixed domain, establishing equivalence with a variational SPDE, and verifying monotonicity-type conditions in a time-dependent Gelfand triple, all without smallness assumptions. Norm-equivalence results on time-dependent domains support the stability and consistency of the variational approach, making this a robust contribution to stochastic fluid-structure interaction and moving-boundary SPDE theory.
Abstract
In this paper, we establish the strong($H^1$) well-posedness of the two dimensional stochastic Navier-Stokes equation with multiplicative noise on moving domains. Due to the nonlocality effect, this equation exhibits a ``piecewise" variational setting. Namely the global well-posedness of this equation is decomposed into the well-posedness of a family of stochastic partial differential equations(SPDEs) in the variational setting on each small time-interval. We first examine the well-posedness on each time interval, which does not have (nonhomogeneous) coercivity. Subsequently, we give an estimate of lower bound of length of the time-interval, which enables us to achieve the global well-posedness.