Local and global existence for the stochastic Prandtl equation driven by multiplicative noises in two and three dimensions

TL;DR

This paper proves local and global existence results for the stochastic Prandtl equation in two and three dimensions, showing that noise can have a regularizing effect leading to global solutions with high probability.

Contribution

It introduces new higher-order estimates in conormal Sobolev spaces and constructs approximate schemes to establish well-posedness of the stochastic Prandtl equation.

Findings

Existence of local solutions in tangentially analytic and Sobolev spaces.

Global Gevrey-2 solutions with high probability due to noise regularization.

Linear growth of the solution radius over time.

Abstract

In this paper, we are concerned with the local and global existence for the stochastic Prandtl equation in two and three dimensions, which governs the velocity field inside the boundary layer that appears in the inviscid limit of the stochastic Navier-Stokes equation with non-slip boundary condition. New problem arises when establishing the well-posedness in the stochastic regime: one can never derive a pathwise control of the energy functional which is used to describe the analytic radius of the solution in the deterministic setting. To this end, we establish higher-order estimates in the conormal Sobolev spaces in order to get rid of the dependence of the analytic radius on the unknown. Three approximate schemes are constructed for solving the stochastic Prandtl equation, and a local well-posedness is obtained in a tangentially analytic and normally Sobolev-type space. Furthermore, we study the stochastic Prandtl equation driven by a tangential random diffusion, and find that the noise regularizes the equation in the sense that in both two and three dimensions, there exists a global Gevrey-2 solution with high probability and the radius growing linearly in time.