Global unique solvability of the 1D stochastic Navier-Stokes-Korteweg equations

Abstract

We prove the global well-posedness of the one-dimensional Navier-Stokes-Korteweg equations driven by a stochastic multiplicative noise. The analysis is performed for the general case of capillarity and viscosity coefficients $k(\rho)= \rho^\beta, \, \beta \in \mathbb{R},\, \mu(\rho)=\rho^\alpha, \, \alpha \ge 0,$ which are not coupled through a BD relation. Global existence and uniqueness of solutions is obtained in the regularity class of strong pathwise solutions, which are strong solutions in PDEs and also in the sense of probability. We first make use of a multi-layer approximation scheme and a stochastic compactness argument to establish the local well-posedness result for any $\alpha$ and $\beta.$ Then, we apply a BD entropy method which provides control of the vacuum states of the density and allows to perform an extension argument. Global well-posedness is thus obtained in the range where there is no vacuum and the strong coercivity condition $2\alpha-4 \le \beta \le 2 \alpha-1,$ introduced in [49], holds. As a byproduct, we also cover the deterministic setting $\mathbb{F}=0,$ which to the best of our knowledge is likewise an open problem in the fluid dynamics literature.