Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative estimates of $L^p$ maximal regularity for nonautonomous operators and global existence for quasilinear equations

Théo Belin, Pauline Lafitte

Abstract

In this work, we obtain quantitative estimates of the continuity constant for the $L^p$ maximal regularity of relatively continuous nonautonomous operators $\mathbb{A} : I \longrightarrow \mathcal{L}(D,X)$, where $D \subset X$ densely and compactly. They allow in particular to establish a new general growth condition for the global existence of strong solutions of Cauchy problems for nonlocal quasilinear equations for a certain class of nonlinearities $u \longrightarrow \mathbb{A}(u)$. The estimates obtained rely on the precise asymptotic analysis of the continuity constant with respect to perturbations of the operator of the form $\mathbb{A}(\cdot) + λI$ as $λ\longrightarrow \pm \infty$. A complementary work in preparation supplements this abstract inquiry with an application of these results to nonlocal parabolic equations in noncylindrical domains depending on the time variable.

Quantitative estimates of $L^p$ maximal regularity for nonautonomous operators and global existence for quasilinear equations

Abstract

In this work, we obtain quantitative estimates of the continuity constant for the maximal regularity of relatively continuous nonautonomous operators , where densely and compactly. They allow in particular to establish a new general growth condition for the global existence of strong solutions of Cauchy problems for nonlocal quasilinear equations for a certain class of nonlinearities . The estimates obtained rely on the precise asymptotic analysis of the continuity constant with respect to perturbations of the operator of the form as . A complementary work in preparation supplements this abstract inquiry with an application of these results to nonlocal parabolic equations in noncylindrical domains depending on the time variable.
Paper Structure (33 sections, 28 theorems, 97 equations)

This paper contains 33 sections, 28 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Operators, spaces and measurability
  4. Graph norms and domains of operators
  5. Functional spaces for $L^p$ maximal regularity
  6. Strong measurability of nonautonomous operators
  7. $L^p$ maximal regularity
  8. Autonomous operators: $m{}r_p(I)$
  9. Nonautonomous operators: $\mathcal{M{}R}_p(I)$
  10. Relatively continuous operators
  11. Main results and scheme of the proof
  12. Asymptotic behavior of diagonal perturbations
  13. Maximal regularity constant of relatively continuous operators
  14. Quantitative description of $K$
  15. Topologies for the persistence of maximal regularity in limits
  16. ...and 18 more sections

Key Result

Proposition 2.1

Let $A \in L^\infty(I;\mathcal{L}(D;X))$ be a nonautonomous operators satisfying df:NAi. Then $A \in \mathcal{M{}R}_p(I)$ if and only if for any $x \in \mathop{\mathrm{Tr}}\nolimits^p$, $f \in L^p(I;X)$ there exists a unique solution to eq:cp$^I_{x,f}$ and there exists $K \geq 0$, independent of $f$

Theorems & Definitions (60)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.3
  • Proposition 2.1
  • Proposition 2.2
  • ...and 50 more