Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Cauchy Problem For Quasi-Linear Parabolic Systems Revisited

Isabelle Gallagher, Ayman Moussa

TL;DR

This work develops a constructive local well-posedness theory for quasi-linear parabolic systems with diffusion matrices satisfying Petrovskii’s condition rather than uniform ellipticity. It replaces abstract semigroup methods with a Fourier-paraproduct framework, proving a priori estimates in Sobolev spaces $H^s$ with $s>d/2$ and in endpoint Besov spaces $B^{d/p}_{p,1}$, and giving a nonlinear fixed-point construction for short-time existence. A key feature is the sign-preserving structure that ensures nonnegativity of solutions in SKT-type cross-diffusion models, with concrete results for the Shigesada–Kawasaki–Teramoto system. The approach yields insights on the lifetime of solutions, stability, and regularity propagation, offering an alternative to Amann’s theory that is explicit and adaptable to critical function spaces.

Abstract

We study a class of parabolic quasilinear systems, in which the diffusion matrix is not uniformly elliptic, but satisfies a Petrovskii condition of positivity of the real part of the eigenvalues. Local well-posedness is known since the work of Amann in the 90s, by a semi-group method. We first revisit these results in the context of Sobolev spaces modelled on $L^2$ and then explore the endpoint Besov case $B_{p,1}^{d/p}$. We also exemplify our method on the SKT system, showing the existence of local, non-negative, strong solutions.

The Cauchy Problem For Quasi-Linear Parabolic Systems Revisited

TL;DR

This work develops a constructive local well-posedness theory for quasi-linear parabolic systems with diffusion matrices satisfying Petrovskii’s condition rather than uniform ellipticity. It replaces abstract semigroup methods with a Fourier-paraproduct framework, proving a priori estimates in Sobolev spaces with and in endpoint Besov spaces , and giving a nonlinear fixed-point construction for short-time existence. A key feature is the sign-preserving structure that ensures nonnegativity of solutions in SKT-type cross-diffusion models, with concrete results for the Shigesada–Kawasaki–Teramoto system. The approach yields insights on the lifetime of solutions, stability, and regularity propagation, offering an alternative to Amann’s theory that is explicit and adaptable to critical function spaces.

Abstract

We study a class of parabolic quasilinear systems, in which the diffusion matrix is not uniformly elliptic, but satisfies a Petrovskii condition of positivity of the real part of the eigenvalues. Local well-posedness is known since the work of Amann in the 90s, by a semi-group method. We first revisit these results in the context of Sobolev spaces modelled on and then explore the endpoint Besov case . We also exemplify our method on the SKT system, showing the existence of local, non-negative, strong solutions.
Paper Structure (39 sections, 30 theorems, 228 equations)

This paper contains 39 sections, 30 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. State of the art
  4. Sign-preserving property and application to the SKT model
  5. Notations
  6. Main results in the linear setting
  7. Plan of the paper
  8. Estimates in the linear case
  9. The case of a constant matrix field
  10. The case of a homogeneous in space matrix field
  11. Reduction of Theorem \ref{['thm:PetrovskiiHs']} to a single lemma
  12. Proof of Lemma \ref{['lem:goalL2:weak']}
  13. A useful corollary
  14. Existence theory and parabolic regularization in the linear case
  15. Proof of Theorem \ref{['thm:exlin']}
  16. ...and 24 more sections

Key Result

Theorem 1

Consider a smooth $A:\mathbb{R}^N\rightarrow \mathscr{P}$, and $s>d/2$. For any $(U^0,F)$ belonging to $\mathcal{D}_\infty^s$ there exists $T>0$ and a unique element $U$ of $E_T^s$ which solves the parabolic Cauchy problem eq:PtNL on $Q_T$. Moreover, if $(U^0_1,F_1),(U^0_2,F_2)$ lie in the ball of s

Theorems & Definitions (53)

  • Definition 1.1: Petrovskii condition
  • Theorem 1: Local well-posedness
  • Theorem 2: Lifetime and blow-up
  • Corollary 1.2
  • Definition 1.3
  • Proposition 1.4
  • Theorem 3
  • Remark 1.5
  • Theorem 4
  • Remark 1.6
  • ...and 43 more