[Le problème de Cauchy pour les systèmes paraboliques quasi-linéaires : une nouvelle approche]
We study a class of parabolic quasilinear systems, in which the diffusion matrix is not uniformly elliptic, but satisfies the Petrovskii condition (positivity of eigenvalues’ real part). 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 $\textnormal{L}^2$ and then explore the endpoint Besov case $B^{d/p}_{p,1}$. We also exemplify our method on the SKT system, showing the existence of local, non-negative, strong solutions.
Nous étudions une classe de systèmes quasi-linéaires paraboliques, dans lesquels la matrice de diffusion n’est pas uniformément elliptique mais satisfait la condition de Petrovskii (partie réelle de toutes les valeurs propres strictement positive). Le caractère bien posé localement pour de tels système a été établi par Amann dans les années 90, par une méthode de semi-groupe. Nous revisitons d’abord ces résultats dans le contexte des espaces de Sobolev modelés sur $L^2$ puis explorons le cas limite des espaces de Besov $B^{d/p}_{p,1}$. Nous illustrons également notre méthode sur le système SKT, en montrant localement l’existence de solutions fortes et positives.
Accepté le :
Publié le :
Keywords: Quasi-linear parabolic systems, Petrovskii’s condition, Littlewood-Paley, paraproduct
Mots-clés : Systèmes quasi-linéaires paraboliques, condition de Petrovskii, Littlewood-Paley, paraproduit
Isabelle Gallagher 1, 2 ; Ayman Moussa 1, 3
CC-BY 4.0
@article{JEP_2025__12__1633_0,
author = {Isabelle Gallagher and Ayman Moussa},
title = {The {Cauchy} problem for quasi-linear~parabolic systems revisited},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1633--1676},
publisher = {\'Ecole polytechnique},
volume = {12},
year = {2025},
doi = {10.5802/jep.319},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.319/}
}
TY - JOUR AU - Isabelle Gallagher AU - Ayman Moussa TI - The Cauchy problem for quasi-linear parabolic systems revisited JO - Journal de l’École polytechnique — Mathématiques PY - 2025 SP - 1633 EP - 1676 VL - 12 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.319/ DO - 10.5802/jep.319 LA - en ID - JEP_2025__12__1633_0 ER -
%0 Journal Article %A Isabelle Gallagher %A Ayman Moussa %T The Cauchy problem for quasi-linear parabolic systems revisited %J Journal de l’École polytechnique — Mathématiques %D 2025 %P 1633-1676 %V 12 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.319/ %R 10.5802/jep.319 %G en %F JEP_2025__12__1633_0
Isabelle Gallagher; Ayman Moussa. The Cauchy problem for quasi-linear parabolic systems revisited. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1633-1676. doi: 10.5802/jep.319
[1] - “Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems”, in Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992), Teubner-Texte Math., vol. 133, Teubner, Stuttgart, 1993, p. 9-126 | DOI | MR | Zbl
[2] - “On the initial value problem for parabolic systems of differential equations”, Bull. Amer. Math. Soc. 65 (1959) no. 5, p. 310 - 318 | DOI | MR | Zbl
[3] - Fourier analysis and nonlinear partial differential equations, Grundlehren Math. Wissen., vol. 343, Springer, Heidelberg, 2011 | DOI | Zbl | MR
[4] - “Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires”, Ann. Sci. École Norm. Sup. (4) 14 (1981) no. 2, p. 209-246 | MR | DOI | Numdam | Zbl
[5] - “On the Maxwell-Stefan approach to multicomponent diffusion”, in Parabolic problems, Progr. Nonlinear Differential Equations Appl., vol. 80, Birkhäuser/Springer Basel AG, Basel, 2011, p. 81-93 | DOI | Zbl
[6] - “Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations”, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003) no. 4, p. 645-668 | DOI | MR | Numdam | Zbl
[7] - “Analysis of a parabolic cross-diffusion population model without self-diffusion”, J. Differential Equations 224 (2006) no. 1, p. 39-59 | DOI | MR | Zbl
[8] - “When do cross-diffusion systems have an entropy structure?”, J. Differential Equations 278 (2021), p. 60-72 | DOI | MR | Zbl
[9] - “Symmetrization and entropy inequality for general diffusion equations”, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) no. 9, p. 963-968 | DOI | MR | Zbl
[10] - “On the entropic structure of reaction-cross diffusion systems”, Comm. Partial Differential Equations 40 (2015) no. 9, p. 1705-1747 | DOI | MR | Zbl
[11] - “On fundamental solutions of parabolic systems. II”, Mat. Sb. (N.S.) 53 (1961), p. 73-136 | MR | Zbl
[12] - Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, NJ, 1964 | MR | Zbl
[13] - Multicomponent flow modeling, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999 | DOI | MR | Zbl
[14] - “Asymptotic stability of equilibrium states for multicomponent reactive flows”, Math. Models Methods Appl. Sci. 8 (1998) no. 2, p. 251-297 | DOI | MR | Zbl
[15] - “Global smooth solutions for triangular reaction-cross diffusion systems”, Bull. Sci. Math. 189 (2023), article ID 103342, 27 pages | DOI | MR | Zbl
[16] - “Gradient estimates and global existence of smooth solutions to a cross-diffusion system”, SIAM J. Math. Anal. 47 (2015) no. 3, p. 2122-2177 | DOI | MR | Zbl
[17] - “Entropy methods for diffusive partial differential equations”, Lecture Notes, TU Wien, Summer Term 2025, available online | MR
[18] - Entropy methods for diffusive partial differential equations, SpringerBriefs in Math., Springer, Cham, 2016 | DOI | MR | Zbl
[19] - “Existence analysis of Maxwell–Stefan systems for multicomponent mixtures”, SIAM J. Math. Anal. 45 (2013) no. 4, p. 2421-2440 | Zbl | DOI | MR
[20] - Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Ph. D. Thesis, Kyoto University, 1984
[21] - “On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws”, Tohoku Math. J. (2) 40 (1988) no. 3, p. 449-464 | DOI | MR | Zbl
[22] - Lectures on elliptic and parabolic equations in Sobolev spaces, Graduate Studies in Math., vol. 96, American Mathematical Society, Providence, RI, 2008 | DOI | MR | Zbl
[23] - Linear and quasilinear equations of parabolic type, Transl. of Math. Monographs, vol. 23, American Mathematical Society, Providence, RI, 1968 | DOI | Zbl
[24] - “On the uniqueness of the solution of Cauchy’s problem for a linear parabolic equation”, Mat. Sb. (N.S.) 27 (1950) no. 2, p. 175-184 | MR
[25] - Introduction to smooth manifolds, Graduate Texts in Math., Springer, New York, 2012 | MR
[26] - “Équations et systèmes paraboliques: quelques questions nouvelles” (2012-2013), Cours au Collège de France, cours/equations-et-systemes-paraboliques-quelques-questions-nouvelles
[27] - “Le problème de Cauchy pour les équations paraboliques”, J. Math. Soc. Japan 8 (1956) no. 4, p. 269-299 | MR | Zbl
[28] - “The work of G. E. Shilov in the theory of generalized functions and differential equations”, Russian Math. Surveys 33 (1978) no. 4, p. 219 | DOI | Zbl
[29] - “Über das Cauchysche Problem für Systeme von partiellen Differentialgleichungen”, Rec. Math. Moscou, n. Ser. 2 (1937), p. 815-868 | Zbl
[30] - Matrices: Theory and applications, Graduate Texts in Math., vol. 216, Springer, New York, 2010 | DOI | MR | Zbl
[31] - “Spatial segregation of interacting species”, J. Theoret. Biol. 79 (1979) no. 1, p. 83-99 | DOI | MR
[32] - “A patient-specific anisotropic diffusion model for brain tumour spread”, Bull. Math. Biology 80 (2017) no. 5, p. 1259–1291 | DOI | MR | Zbl
Cité par Sources :