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A Meta Theorem for nonlinear stochastic coupled systems: Application to stochastic chemotaxis-Stokes porous media

Erika Hausenblas, Boris Jidjou Moghomye

Abstract

The purpose of the paper is twofold. Firstly, we want to present a Meta Theorem to show the existence of a martingale solution for coupled systems of non-linear stochastic differential equations. The idea is first to split the system by rewriting the non-linear part in a linear part acting on a given process $ξ$. This is done in such a way that the fixpoint with respect to $ξ$ would be the solution. However, to show the well posedness of the {\sl linearized} system, one needs a cut-off argument. Under which conditions one can handle the limits of the cut-off parameter to get in the end a martingale solution of the original system is given in the Meta-Theorem. Secondly, we want to verify the full applicability of the Meta Theorem by showing the existence of a martingale solution of a highly nonlinear chemotaxis system with underlying fluid dynamic.

A Meta Theorem for nonlinear stochastic coupled systems: Application to stochastic chemotaxis-Stokes porous media

Abstract

The purpose of the paper is twofold. Firstly, we want to present a Meta Theorem to show the existence of a martingale solution for coupled systems of non-linear stochastic differential equations. The idea is first to split the system by rewriting the non-linear part in a linear part acting on a given process . This is done in such a way that the fixpoint with respect to would be the solution. However, to show the well posedness of the {\sl linearized} system, one needs a cut-off argument. Under which conditions one can handle the limits of the cut-off parameter to get in the end a martingale solution of the original system is given in the Meta-Theorem. Secondly, we want to verify the full applicability of the Meta Theorem by showing the existence of a martingale solution of a highly nonlinear chemotaxis system with underlying fluid dynamic.
Paper Structure (5 sections, 5 theorems, 55 equations)

This paper contains 5 sections, 5 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. The main Abstract result
  3. Application to stochastic chemotaxis on fluid model
  4. Technical Preliminaries of the noise and the stochastic integral
  5. Some technical Proposition

Key Result

Theorem 2.2

Let $m>1$ be given. Let us assume that for any $R>0$ and $\kappa\in \mathbb{N}$, there exists a bounded and convex set in ${{ \mathcal{M} }}^m_{ \mathfrak{A}}(\mathbb{X})$, such that for all $\xi \in {{ \mathcal{M} }}^m_{ \mathfrak{A}}(\mathbb{X})$ the system (spdeskappa) has a unique probabilistic strong solution where $\mathbb{X}_j\subset\{ \eta: [0,T]\to U_j \}$, $j=1,...,J$, are given Banach

Theorems & Definitions (15)

  • Definition 2.1
  • Theorem 2.2: Meta-Theorem
  • Remark 2.3
  • proof
  • Definition 3.1
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Proposition B.1
  • proof
  • ...and 5 more