Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local strong solutions to the stochastic third grade fluid equations with Navier boundary conditions

Yassine Tahraoui, Fernanda Cipriano

Abstract

This work is devoted to the study of non-Newtonian fluids of grade three on two-dimensional and three-dimensional bounded domains, driven by a nonlinear multiplicative Wiener noise. More precisely, we establish the existence and uniqueness of the local (in time) solution, which corresponds to an addapted stochastic process with sample paths defined up to a certain positive stopping time, with values in the Sobolev space H^3. Our approach combines a cut-off approximation scheme, a stochastic compactness arguments and a general version of Yamada-Watanabe theorem. This leads to the existence of a local strong pathwise solution.

Local strong solutions to the stochastic third grade fluid equations with Navier boundary conditions

Abstract

This work is devoted to the study of non-Newtonian fluids of grade three on two-dimensional and three-dimensional bounded domains, driven by a nonlinear multiplicative Wiener noise. More precisely, we establish the existence and uniqueness of the local (in time) solution, which corresponds to an addapted stochastic process with sample paths defined up to a certain positive stopping time, with values in the Sobolev space H^3. Our approach combines a cut-off approximation scheme, a stochastic compactness arguments and a general version of Yamada-Watanabe theorem. This leads to the existence of a local strong pathwise solution.
Paper Structure (21 sections, 20 theorems, 172 equations)

This paper contains 21 sections, 20 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Content of the study
  3. The functional setting
  4. The stochastic setting
  5. Multiplicative noise
  6. The main results
  7. Approximation (cut-off system)
  8. Approximation
  9. A priori estimates.
  10. Compactness
  11. Subsequence extractions
  12. Identification of the limit $\&$ Martingale solutions
  13. Proof of Theorem \ref{['exis-thm-mart']}
  14. The strong solution
  15. Local martingale solution of \ref{['I']}
  16. ...and 6 more sections

Key Result

Theorem 2.1

Suppose that $f \in (H^m(D))^d,\, m=0,1$. Then there exists a unique (up to a constant for $p$) solution $(h,p) \in (H^{m+2}(D))^d\times H^{m+1}(D)$ of the Stokes problem Stokes such that

Theorems & Definitions (40)

  • Theorem 2.1
  • Remark 2.1
  • Lemma 2.2
  • Definition 3.1
  • Definition 3.2
  • Remark 3.1
  • Theorem 3.1
  • Remark 3.2
  • Definition 4.1
  • Theorem 4.1
  • ...and 30 more