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Well-posedness of a reaction-diffusion model with stochastic dynamical boundary conditions

Mario Maurelli, Daniela Morale, Stefania Ugolini

TL;DR

This work studies the well-posedness of a nonlinear reaction-diffusion system on the half-line with a stochastic dynamical boundary condition driven by a Jacobi-type SDE. A splitting strategy separates a heat equation with rough boundary from a nonlinear remainder, enabling pathwise estimates that overcome low boundary regularity. The authors prove global existence and pathwise uniqueness of mild solutions for Hölder boundary data, and establish stability with respect to boundary conditions, including adaptiveness for stochastic boundaries. The results extend deterministic boundary analyses to a stochastic, bounded-noise setting and provide a rigorous framework for the coupled s-c dynamics in a porosity/sulphation model.

Abstract

We study the well-posedness of a nonlinear reaction diffusion partial differential equation system on the half-line coupled with a stochastic dynamical boundary condition, a random system arising from the description of the chemical reaction of sulphur dioxide with calcium carbonate stones. The boundary condition is given by a Jacobi process, solution to a stochastic differential equation with a mean-reverting drift and a bounded diffusion coefficient. The main result is the global existence and the pathwise uniqueness of mild solutions. The proof relies on a splitting strategy, which allows to deal with the low regularity of the dynamical boundary condition.

Well-posedness of a reaction-diffusion model with stochastic dynamical boundary conditions

TL;DR

This work studies the well-posedness of a nonlinear reaction-diffusion system on the half-line with a stochastic dynamical boundary condition driven by a Jacobi-type SDE. A splitting strategy separates a heat equation with rough boundary from a nonlinear remainder, enabling pathwise estimates that overcome low boundary regularity. The authors prove global existence and pathwise uniqueness of mild solutions for Hölder boundary data, and establish stability with respect to boundary conditions, including adaptiveness for stochastic boundaries. The results extend deterministic boundary analyses to a stochastic, bounded-noise setting and provide a rigorous framework for the coupled s-c dynamics in a porosity/sulphation model.

Abstract

We study the well-posedness of a nonlinear reaction diffusion partial differential equation system on the half-line coupled with a stochastic dynamical boundary condition, a random system arising from the description of the chemical reaction of sulphur dioxide with calcium carbonate stones. The boundary condition is given by a Jacobi process, solution to a stochastic differential equation with a mean-reverting drift and a bounded diffusion coefficient. The main result is the global existence and the pathwise uniqueness of mild solutions. The proof relies on a splitting strategy, which allows to deal with the low regularity of the dynamical boundary condition.
Paper Structure (14 sections, 24 theorems, 151 equations)

This paper contains 14 sections, 24 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Notations.
  3. Acknowledgements.
  4. A PDE - ODE model with a nonsmooth boundary condition
  5. An example of stochastic dynamical boundary condition
  6. A model for the porosity
  7. A splitting strategy
  8. Heat equation with nonsmooth boundary condition: some bounds
  9. A linear parabolic equation: a priori estimates
  10. Existence, uniqueness and regularity for the nonlinear system
  11. The linear system.
  12. The nonlinear system
  13. Stability with respect to the boundary condition.
  14. Appendix

Key Result

Proposition 1

Let us consider equation eq:jacobi, with $\alpha,\sigma,\gamma,\eta\in \mathbb R_+$ and $\gamma\le \eta$. Assume that $\psi_0 \in [0,\eta]$. The solution of equation eq:jacobi exists (globally on $[0,T]$) and is pathwise unique. Furthermore, let us suppose that the following conditions upon the para Then for any $t\in (0,T],$ In particular, we have $\psi_t\in L^\infty(\mathbb R_+)$.

Theorems & Definitions (66)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 3
  • Proposition 4
  • proof
  • Definition 5: Heat kernels
  • Definition 6: Convolution $*_D$
  • Definition 7: Bounded positive mild solution
  • ...and 56 more