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Uniqueness results of a nonlinear stochastic diffusion-convection equation with reflection

Niklas Sapountzoglou

TL;DR

This work analyzes the uniqueness of solutions to a nonlinear stochastic diffusion-convection obstacle problem with reflection on a bounded Lipschitz domain under additive noise. The authors develop contraction-type results for regular solutions, obtaining a pathwise $L^2$-contraction when the operator $a$ is independent of the second argument and an $L^1$-contraction in the general case via a smoothing argument using $n_{\delta}$, while highlighting the limitations imposed by the lack of a general It\ô formula in higher dimensions. They further discuss regularization-based strategies and the need for extra space-time regularity to extend these results, showing that full uniqueness in the rough regime remains open. The appendix establishes cap$_p$-based quasi-continuity tools to handle the reflection measure and the obstacle constraint rigorously, providing a solid analytic foundation for the stochastic obstacle problem framework.

Abstract

We are interested in the uniqueness of solutions of a nonlinear, pseudomonotone, stochastic diffusion evolution problem with homogeneous Dirichlet boundary conditions with reflection, where the noise term is additive and given by a stochastic Itô integral with respect to a Hilbert space valued cylindrical Wiener process. In fact, since there is no Itô formula available for a solution in general, a general uniqueness result seems not to be available. Nevertheless, assuming more regularity for the solutions or the reflection, we may show some comparison principles.

Uniqueness results of a nonlinear stochastic diffusion-convection equation with reflection

TL;DR

This work analyzes the uniqueness of solutions to a nonlinear stochastic diffusion-convection obstacle problem with reflection on a bounded Lipschitz domain under additive noise. The authors develop contraction-type results for regular solutions, obtaining a pathwise -contraction when the operator is independent of the second argument and an -contraction in the general case via a smoothing argument using , while highlighting the limitations imposed by the lack of a general It\ô formula in higher dimensions. They further discuss regularization-based strategies and the need for extra space-time regularity to extend these results, showing that full uniqueness in the rough regime remains open. The appendix establishes cap-based quasi-continuity tools to handle the reflection measure and the obstacle constraint rigorously, providing a solid analytic foundation for the stochastic obstacle problem framework.

Abstract

We are interested in the uniqueness of solutions of a nonlinear, pseudomonotone, stochastic diffusion evolution problem with homogeneous Dirichlet boundary conditions with reflection, where the noise term is additive and given by a stochastic Itô integral with respect to a Hilbert space valued cylindrical Wiener process. In fact, since there is no Itô formula available for a solution in general, a general uniqueness result seems not to be available. Nevertheless, assuming more regularity for the solutions or the reflection, we may show some comparison principles.

Paper Structure

This paper contains 7 sections, 5 theorems, 31 equations.

Table of Contents

  1. The first section of your paper
  2. Introduction
  3. Notations and assumptions
  4. Notion of a solution
  5. Comparison principles for regular solutions
  6. Discussion of more general uniqueness results
  7. Appendix

Key Result

Theorem 2.1

Let the function $a$ be independent of the second variable and $(u, \eta)$ and $(v, \nu)$ be two solutions of Obstacle problem with initial values $u_0 \in L^2(\Omega \times D)$ and $v_0 \in L^2(\Omega \times D)$$\mathcal{F}_0$-measurable and non-negative a.e. in $\Omega \times D$ respectively. More

Theorems & Definitions (14)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 2.1
  • Remark 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 4 more