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A priori bounds for quasi-linear SPDEs in the full sub-critical regime

Felix Otto, Jonas Sauer, Scott Smith, Hendrik Weber

Abstract

This paper is concerned with quasi-linear parabolic equations driven by an additive forcing $ξ\in C^{α-2}$, in the full sub-critical regime $α\in (0,1)$. We are inspired by Hairer's regularity structures, however we work with a more parsimonious model indexed by multi-indices rather than trees. This allows us to capture additional symmetries which play a crucial role in our analysis. Assuming bounds on this model, which is modified in agreement with the concept of algebraic renormalization, we prove local a priori estimates on solutions to the quasi-linear equations modified by the corresponding counter terms.

A priori bounds for quasi-linear SPDEs in the full sub-critical regime

Abstract

This paper is concerned with quasi-linear parabolic equations driven by an additive forcing , in the full sub-critical regime . We are inspired by Hairer's regularity structures, however we work with a more parsimonious model indexed by multi-indices rather than trees. This allows us to capture additional symmetries which play a crucial role in our analysis. Assuming bounds on this model, which is modified in agreement with the concept of algebraic renormalization, we prove local a priori estimates on solutions to the quasi-linear equations modified by the corresponding counter terms.

Paper Structure

This paper contains 20 sections, 11 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. A Formal Series Expansion for the Solution
  3. The Renormalized Model
  4. Model Space and Structure Group
  5. The Model Space
  6. Homogeneities
  7. The Structure Group
  8. Linear forms on $(u,\nu)$ space
  9. Projections
  10. Modelled Distributions
  11. Graded Continuity Lemma
  12. Interpolation Inequalities
  13. Proof of the Continuity Lemma
  14. Jets and Proof of the Main Theorem
  15. Approximation by Jets
  16. ...and 5 more sections

Key Result

Lemma 1

Let $\beta_{1},\beta_{2}$ be multi-indices.

Theorems & Definitions (31)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • Remark 4
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 21 more