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

L. Pescatore

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 $k(ρ)= ρ^β, \, β\in \mathbb{R},\, μ(ρ)=ρ^α, \, α\ge 0,$ 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 $α$ and $β.$ 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 $2α-4 \le β\le 2 α-1,$ 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.

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 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 and 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 introduced in [49], holds. As a byproduct, we also cover the deterministic setting which to the best of our knowledge is likewise an open problem in the fluid dynamics literature.

Paper Structure

This paper contains 14 sections, 12 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Definitions and main results
  3. Assumptions on the stochastic elements
  4. Definitions and main theorems
  5. A priori estimates
  6. High-order estimates
  7. Local well-posedness
  8. First level of approximation
  9. Second level of approximation
  10. Stochastic compactness
  11. Pathwise uniqueness
  12. Solvability of the first level of approximation
  13. Proof of Theorem \ref{['Main Theorem local']}
  14. Proof of the main result

Key Result

Theorem 2.3

Let $s \in \mathbb{N}$ satisfy $s > \frac{7}{2}.$ Assume the coefficients $F_k$ satisfy F1-F2. Then for any $\alpha \ge 0, \; \beta \in \mathbb{R},$ there exists a unique maximal local strong pathwise solution $(\rho,u,(\tau_R)_{R \in \mathbb{N}},\tau)$ to system main system-initial conditions in th

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Remark 3.3
  • proof
  • ...and 16 more