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Improved regularity for the stochastic fast diffusion equation

Ioana Ciotir, Dan Goreac, Jonas M. Tölle

Abstract

We prove that the solution to the singular-degenerate stochastic fast-diffusion equation with parameter $m\in (0,1)$, with zero Dirichlet boundary conditions on a bounded domain in any spatial dimension, and driven by linear multiplicative Wiener noise, exhibits improved regularity in the Sobolev space $W^{1,m+1}_0$ for initial data in $L^{2}$.

Improved regularity for the stochastic fast diffusion equation

Abstract

We prove that the solution to the singular-degenerate stochastic fast-diffusion equation with parameter , with zero Dirichlet boundary conditions on a bounded domain in any spatial dimension, and driven by linear multiplicative Wiener noise, exhibits improved regularity in the Sobolev space for initial data in .
Paper Structure (2 sections, 1 theorem, 22 equations)

This paper contains 2 sections, 1 theorem, 22 equations.

Table of Contents

  1. Introduction
  2. Proof of the main result

Key Result

Theorem 1.1

Assume that eq:coeff holds. Then the unique strong solution $u$ to equation eq:main with initial datum $u_0\in L^{2}(\Omega,{\mathcal{F}}_0,\mathbb{P};L^{2}({\mathcal{O}}))$ satisfies

Theorems & Definitions (2)

  • Theorem 1.1
  • proof : Proof of Theorem \ref{['thm:main']}