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On local well-posedness of the stochastic incompressible density-dependent Euler equations

Claudia Espitia, David A. C. Mollinedo, Christian Olivera

TL;DR

The paper addresses the local well-posedness of stochastic incompressible density-dependent Euler equations in R^3 under both multiplicative and additive noise. By reducing the stochastic system to a random PDE through a velocity transformation and deriving robust a priori estimates for transport, elliptic, and linear Euler subproblems, it constructs local-in-time, pathwise solutions via successive approximations. It proves existence, pathwise uniqueness, and the existence of maximal pathwise solutions for both noise types, highlighting a rigorous framework for stochastic density-dependent fluids. These results extend the deterministic theory to stochastic settings and provide a foundation for further investigations into regularization by noise and more general stochastic perturbations.

Abstract

In this paper we study the stochastic inhomogeneous incompressible Euler equations in the whole space $\RR^3$. We prove the existence and pathwise uniqueness of local solutions with both additive and multiplicative stochastic noise. Our approach is based on reducing our problem to a random problem and some estimations for type transport equations.

On local well-posedness of the stochastic incompressible density-dependent Euler equations

TL;DR

The paper addresses the local well-posedness of stochastic incompressible density-dependent Euler equations in R^3 under both multiplicative and additive noise. By reducing the stochastic system to a random PDE through a velocity transformation and deriving robust a priori estimates for transport, elliptic, and linear Euler subproblems, it constructs local-in-time, pathwise solutions via successive approximations. It proves existence, pathwise uniqueness, and the existence of maximal pathwise solutions for both noise types, highlighting a rigorous framework for stochastic density-dependent fluids. These results extend the deterministic theory to stochastic settings and provide a foundation for further investigations into regularization by noise and more general stochastic perturbations.

Abstract

In this paper we study the stochastic inhomogeneous incompressible Euler equations in the whole space . We prove the existence and pathwise uniqueness of local solutions with both additive and multiplicative stochastic noise. Our approach is based on reducing our problem to a random problem and some estimations for type transport equations.
Paper Structure (16 sections, 35 theorems, 241 equations)

This paper contains 16 sections, 35 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. Space and definition of solutions
  3. A priori estimates
  4. A priori estimates for transport equation.
  5. A priori estimates for elliptic equations.
  6. A priori estimates for linear Euler equation.
  7. Proof of the theorem \ref{['maintheorem']}
  8. Successive approximations
  9. Convergence results
  10. Uniqueness of solutions
  11. Maximal pathwise solutions
  12. Proof of the theorem \ref{['maintheorembi']}
  13. Successive approximations
  14. Convergence results
  15. Existence and uniqueness
  16. ...and 1 more sections

Key Result

Theorem 1.1

Suppose that for $3<p<\infty$, the function $\rho_0$ satisfies and the vector field $v_0$ verifies Then, there exists a unique local pathwise solution $(\rho,\nabla \pi, v, \tau)$ for Problem eq0 in the sense of Definitions deflocalsolution and localuniqueness. Moreover, there also exists a unique maximal pathwise solution in the sense of Definitions defmaximalsolution and maxuniqueness.

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Local pathwise solution
  • Definition 2.2: Local pathwise solution
  • Definition 2.3: Uniqueness
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6: Maximal pathwise solution
  • Definition 2.7: Maximal pathwise solution
  • Definition 2.8: Uniqueness
  • ...and 60 more