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arXiv · 2407.08226

The Cauchy Problem For Quasi-Linear Parabolic Systems Revisited

Isabelle Gallagher·Ayman Moussa
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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.

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Isabelle Gallagher, Ayman Moussa. 2024-07-11. The Cauchy Problem For Quasi-Linear Parabolic Systems Revisited. https://arxiv.org/abs/2407.08226

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