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Maximal regularity for evolution equations with critical singular perturbations

Esmée Theewis, Mark Veraar

TL;DR

The paper develops two strands of perturbation theory for maximal $L^p$-regularity of evolution equations with critical, time-dependent perturbations. First, it proves that if $A$ has maximal $L^p$-regularity and the perturbation $B$ satisfies a sharp $L^q$-integrability bound in time with $q\in(p,\infty)$, then $A+B$ retains maximal $L^p$-regularity (endpoint cases handled separately). Second, it introduces a weighted, mixed-scale framework to treat perturbations that act across multiple spatial-temporal scales, yielding well-posedness for skeleton-like equations in large deviations by combining MRI with additional $L^r$-type components; it also provides embeddings and fractional time-regularity results. A dedicated r=1 case further extends the theory to Hilbert-space settings with transference arguments and two distinct proof strategies. Together, these results expand the applicability of maximal regularity to non-autonomous, singular perturbations and mixed-scale perturbations relevant to stochastic PDEs and large deviations.

Abstract

Assuming $A$ has maximal $L^p$-regularity, this paper investigates perturbations of $A$ by time-dependent operators $B$ that are unbounded and satisfy a critical $L^q$-integrability condition in time. We establish two main results. The first proves maximal $L^p$-regularity for the critical endpoint case, generalizing previous work by Prüss and Schnaubelt (2001). The second develops a weighted maximal regularity theory for mixed-scale perturbations, motivated by the linearized skeleton equations appearing in large deviations theory for stochastic PDEs.

Maximal regularity for evolution equations with critical singular perturbations

TL;DR

The paper develops two strands of perturbation theory for maximal -regularity of evolution equations with critical, time-dependent perturbations. First, it proves that if has maximal -regularity and the perturbation satisfies a sharp -integrability bound in time with , then retains maximal -regularity (endpoint cases handled separately). Second, it introduces a weighted, mixed-scale framework to treat perturbations that act across multiple spatial-temporal scales, yielding well-posedness for skeleton-like equations in large deviations by combining MRI with additional -type components; it also provides embeddings and fractional time-regularity results. A dedicated r=1 case further extends the theory to Hilbert-space settings with transference arguments and two distinct proof strategies. Together, these results expand the applicability of maximal regularity to non-autonomous, singular perturbations and mixed-scale perturbations relevant to stochastic PDEs and large deviations.

Abstract

Assuming has maximal -regularity, this paper investigates perturbations of by time-dependent operators that are unbounded and satisfy a critical -integrability condition in time. We establish two main results. The first proves maximal -regularity for the critical endpoint case, generalizing previous work by Prüss and Schnaubelt (2001). The second develops a weighted maximal regularity theory for mixed-scale perturbations, motivated by the linearized skeleton equations appearing in large deviations theory for stochastic PDEs.
Paper Structure (16 sections, 19 theorems, 152 equations, 1 table)

This paper contains 16 sections, 19 theorems, 152 equations, 1 table.

Table of Contents

  1. Introduction
  2. Setting with full maximal $L^p$-regularity
  3. Setting beyond maximal $L^p$-regularity
  4. Perturbations of maximal $L^p$-regularity
  5. Maximal $L^p$-regularity
  6. Main result
  7. Limiting case $p=q$
  8. Perturbations beyond maximal $L^p$-regularity
  9. A mixed-scale problem
  10. Embeddings of the mixed-scale maximal regularity space
  11. Main result
  12. Proof of Theorem \ref{['th: well-posed linear']}
  13. Further embeddings of the maximal regularity space
  14. Perturbations beyond maximal $L^p$-regularity: the case $r=1$
  15. Main result
  16. ...and 1 more sections

Key Result

Theorem 1.2

Suppose that Assumption assump:XA holds. Let $p\in (1, \infty)$ and $q\in (p, \infty)$. Suppose that $A$ has maximal $L^p$-regularity. Let $B\colon[0,T]\to \mathcal{L}(X_{1-1/q,1},X_0)$ be strongly measurable in the strong operator topology, and suppose that where $b\in L^q(0,T)$. Then $A+B$ has maximal $L^p$-regularity.

Theorems & Definitions (52)

  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2: Change of interval
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5
  • proof : Proof of Theorem \ref{['thm:perturbLp']}
  • ...and 42 more