Noise-Driven Free Boundaries In The Compressible Navier-Stokes Equations

TL;DR

This paper studies a stochastic free-boundary problem for the 3D compressible Navier-Stokes equations, incorporating noise into the boundary evolution and momentum, and establishes local well-posedness with positive density.

Contribution

It introduces a stochastic model with noise in the boundary and momentum equations, and proves local pathwise well-posedness using stochastic and deterministic analysis techniques.

Findings

Established local pathwise well-posedness of the stochastic free-boundary Navier-Stokes problem.

Demonstrated the existence of solutions with strictly positive density.

Used stochastic maximal regularity and contraction arguments for analysis.

Abstract

A stochastic free-boundary problem for the three-dimensional barotropic compressible Navier--Stokes equations is studied. The main feature of the model is that the free boundary is transported by a Stratonovich stochastic flow, so that the noise enters the kinematic boundary condition and hence the evolution of the moving domain. An additional It\^o forcing in the momentum equation is also allowed. The problem is transformed by a stochastic Lagrangian map generated by the velocity and the transport vector fields. In these coordinates the density is represented through the Jacobian of the flow, and the remaining system is solved by combining stochastic maximal regularity, deterministic %\rL^p%-%\rL^q$ estimates, and a localized contraction argument. Local pathwise well-posedness is obtained up to an a.s. positive stopping time, with strictly positive density and pathwise uniqueness.