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Global regularity for the one-dimensional stochastic Quantum-Navier-Stokes equations

Donatella Donatelli, Lorenzo Pescatore, Stefano Spirito

Abstract

In this paper we prove the global in time well-posedness of strong solutions to the Quantum-Navier-Stokes equation driven by random initial data and stochastic external force. In particular, we first give a general local well-posedness result. Then, by means of the Bresch-Desjardins entropy, higher order energy estimates, and a continuation argument we prove that the density never vanishes, and thus that local strong solutions are indeed global.

Global regularity for the one-dimensional stochastic Quantum-Navier-Stokes equations

Abstract

In this paper we prove the global in time well-posedness of strong solutions to the Quantum-Navier-Stokes equation driven by random initial data and stochastic external force. In particular, we first give a general local well-posedness result. Then, by means of the Bresch-Desjardins entropy, higher order energy estimates, and a continuation argument we prove that the density never vanishes, and thus that local strong solutions are indeed global.
Paper Structure (21 sections, 23 theorems, 291 equations)

This paper contains 21 sections, 23 theorems, 291 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Definitions and main theorems
  5. Global regularity
  6. A priori estimates
  7. Proof of Theorem \ref{['Main Theorem global']}
  8. Proof of Theorem \ref{['Main Theorem local']}
  9. The approximating system
  10. Galerkin approximation
  11. Uniform estimates
  12. Compactness
  13. Identification of the limit
  14. Pathwise uniqueness
  15. Existence of a strong pathwise approximate solution
  16. ...and 6 more sections

Key Result

Theorem 2.3

(Local existence) Let $s \in \mathbb{N}$ satisfy $s > \frac{7}{2}, \; \gamma > 1.$ Let the coefficients $G_k$ satisfy G1-G2 and $(\rho_0,u_0)$ be an $\mathfrak{F}_0$-measurable, $H^{s+1}(\mathbb{T}) \times H^s(\mathbb{T})$- valued random variable satisfying C.I STRONG. Then there exists a unique max

Theorems & Definitions (36)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 26 more