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Perturbative global solutions of a large class of cross diffusion systems in any dimension

L Desvillettes, A Moussa

Abstract

This article focuses on a large family of cross-diffusion systems of the form $\partial$ t U-$Δ$A(U) = 0, in dimension d $\in$ N * , and where U $\in$ R 2. We show that under natural conditions on the nonlinearity A, those systems have a unique smooth (nonnegative for all components) solution when the initial data are small enough in a suitable norm.

Perturbative global solutions of a large class of cross diffusion systems in any dimension

Abstract

This article focuses on a large family of cross-diffusion systems of the form t U-A(U) = 0, in dimension d N * , and where U R 2. We show that under natural conditions on the nonlinearity A, those systems have a unique smooth (nonnegative for all components) solution when the initial data are small enough in a suitable norm.
Paper Structure (9 sections, 18 theorems, 98 equations)

This paper contains 9 sections, 18 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Global uniform a priori estimates for small initial data
  3. Proof of the existence theorem
  4. Proof of the stability and uniqueness theorem
  5. Some considerations on the case when the initial data are not small
  6. Duality estimates
  7. Bootstrap lemma
  8. Estimates for the heat kernel
  9. Classical SKT system: another method yielding global uniform a priori estimates

Key Result

Lemma 1

Consider $\mu\in\textnormal{L}_{\textnormal{loc}}^\infty(\mathbf{R}_+;\textnormal{L}^\infty(\mathbf{T}^d))$ such that $\inf \mu>0$, $z_{\textnormal{in}}\in\textnormal{H}^{-1}(\mathbf{T}^d)$, and $f\in\textnormal{L}_{\textnormal{loc}}^2(\mathbf{R}_+;\textnormal{L}^2(\mathbf{T}^d))$. Then, there exist Furthermore, this solution $z$ belongs to $\mathscr{C}([0,T];\textnormal{H}^{-1}(\mathbf{T}^d))$ an

Theorems & Definitions (36)

  • Lemma 1
  • proof
  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Corollary 1
  • Theorem 3
  • ...and 26 more