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Strong existence and uniqueness for a class of quasilinear stochastic evolution equations

Sebastian Bechtel, Esmée Theewis

Abstract

We establish existence of probabilistically strong solutions and pathwise uniqueness for a class of quasilinear stochastic evolution equations on bounded domains. Our results combine recent weak existence results for quasilinear stochastic evolution equations in an $L^p$-setting (with $p > 2$) with Yamada--Watanabe theory. To establish pathwise uniqueness, we rely on an $L^1$-contraction argument.

Strong existence and uniqueness for a class of quasilinear stochastic evolution equations

Abstract

We establish existence of probabilistically strong solutions and pathwise uniqueness for a class of quasilinear stochastic evolution equations on bounded domains. Our results combine recent weak existence results for quasilinear stochastic evolution equations in an -setting (with ) with Yamada--Watanabe theory. To establish pathwise uniqueness, we rely on an -contraction argument.
Paper Structure (4 sections, 5 theorems, 21 equations)

This paper contains 4 sections, 5 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Probabilistically weak solutions
  3. Pathwise uniqueness
  4. Conclusion of the main result

Key Result

Theorem 1.1

Let $h>1$ and suppose Assumption ass:ql. Then, there exists $p_0\in(2,\infty)$ such that, for all $p\in(2,p_0]$, for all $q\in(ph,\infty)$: if $u_0\in L^p(\Omega;B_{2,p,0}^{1-2/p}(D))\cap L^q(\Omega\times D)$, then there exists a unique global solution $u$ to eq:ql in the sense of Definition def: st

Theorems & Definitions (13)

  • Theorem 1.1: Strong existence and uniqueness
  • Remark 1.2: Comparison of the assumptions
  • Remark 1.3: Global solutions
  • Definition 1.4: Strong solutions
  • Example 1.5
  • Definition 2.1: Weak solution
  • Theorem 2.2: Weak existence
  • Lemma 3.1: Approximation of the absolute value
  • Lemma 3.2
  • proof
  • ...and 3 more