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The Initial Value Problem for Singular SPDEs via Rough Paths

Claudia Raithel, Jonas Sauer

Abstract

In this contribution we develop a solution theory for singular quasilinear stochastic partial differential equations subject to an initial condition. We obtain our solution theory as a perturbation of the rough path approach developed to handle the space-time periodic problem by Otto and Weber (2019). As in their work, we assume that the forcing is of class $C^{α-2}$ for $α\in (\frac{2}{3},1)$ and space-time periodic and, additionally, that the initial condition is of class $C^α$ and periodic. We contribute to the analytic aspects of the theory. Indeed, we show that we can enforce the initial condition via correcting the previously obtained space-time periodic solution with an initial boundary layer which may be handled in a completely deterministic manner. Uniqueness is obtained in the class of solutions which are corrected in this way by an initial boundary layer. Moreover, stability of the solutions with respect to perturbations of the data is established.

The Initial Value Problem for Singular SPDEs via Rough Paths

Abstract

In this contribution we develop a solution theory for singular quasilinear stochastic partial differential equations subject to an initial condition. We obtain our solution theory as a perturbation of the rough path approach developed to handle the space-time periodic problem by Otto and Weber (2019). As in their work, we assume that the forcing is of class for and space-time periodic and, additionally, that the initial condition is of class and periodic. We contribute to the analytic aspects of the theory. Indeed, we show that we can enforce the initial condition via correcting the previously obtained space-time periodic solution with an initial boundary layer which may be handled in a completely deterministic manner. Uniqueness is obtained in the class of solutions which are corrected in this way by an initial boundary layer. Moreover, stability of the solutions with respect to perturbations of the data is established.

Paper Structure

This paper contains 27 sections, 21 theorems, 374 equations.

Table of Contents

  1. Keywords:
  2. AMS subject classifications:
  3. Introduction
  4. Our perturbative approach:
  5. Rough path method of Otto and Weber (for $\alpha \in (\frac{2}{3},1)$):
  6. Notation
  7. Set-up and overview of our strategy
  8. Definitions and tools
  9. Usage of the periodic offline products
  10. New "offline" products
  11. Reconstruction lemmas
  12. Discussion and statement of our results
  13. Proof of Lemma \ref{['KrylovSafonov']} (Safonov Lemma): Main PDE ingredient
  14. Argument for Theorem \ref{['linear_theorem']}: Treatment of the linear problem
  15. Proof of Proposition \ref{['linear_forcing']}: Treatment of linear space-time periodic problem with irregular forcing
  16. ...and 12 more sections

Key Result

Lemma 1

Let $\alpha \in (0,1)$, then a distribution $f$ on $\mathbb{R}^2$ satisfies

Theorems & Definitions (55)

  • Definition 1: Modelledness
  • Remark 1: Modelledness notational conventions
  • Remark 2: Modelling for additive vs. multiplicative noise
  • Definition 2: Parameterized constant coefficient solutions
  • Definition 3: Negative Hölder norm
  • Definition 4: Local Hölder seminorm
  • Lemma 1: Equivalent $C^{\alpha -2}$-norm
  • Definition 5: Extensions to negative times
  • Remark 3: Usage of the extensions
  • Proposition 1
  • ...and 45 more