Bounded weak solutions to a class of degenerate cross-diffusion systems

Laurençot, Philippe; Matioc, Bogdan-Vasile

Annales Henri Lebesgue, Tome 6 (2023), pp. 847-874

Bounded weak solutions are constructed for a degenerate parabolic system with a full diffusion matrix, which is a generalized version of the thin film Muskat system. Boundedness is achieved with the help of a sequence ${\left({ℰ}_n\right)}_{n\ge 2}$ of Liapunov functionals such that ${ℰ}_n$ is equivalent to the $L_n$-norm for each $n\ge 2$ and ${ℰ}_n^{1/n}$ controls the $L_{\infty }$-norm in the limit $n\to \infty$. Weak solutions are built by a compactness approach, special care being needed in the construction of the approximation in order to preserve the availability of the above-mentioned Liapunov functionals.

Des solutions faibles bornées sont construites pour un système parabolique dégénéré avec une matrice de diffusion pleine, qui est une version généralisée d’une approximation de type «  film mince » du système de Muskat. Le caractère borné des solutions est obtenu à l’aide d’une suite ${\left({ℰ}_n\right)}_{n\ge 2}$ de fonctionnelles de Liapunov avec les propriétés suivantes  : ${ℰ}_n$ est équivalente à la norme $L_n$ pour chaque $n\ge 2$ et ${ℰ}_n^{1/n}$ contrôle la norme $L_{\infty }$ dans la limite $n\to \infty$. Les solutions faibles sont construites par une méthode de compacité, la construction des approximations requérant une attention particulière afin d’être compatibles avec les fonctionnelles de Liapunov mentionnées ci-dessus.

Classification : 35K65, 35K51, 37L45, 35B65
Keywords: Degenerate parabolic system, cross-diffusion, boundedness, Liapunov functionals, global existence

[ACCL19] Aït Hammou Oulhaj, Ahmed; Cancès, Clément; Chainais-Hillairet, Claire; Laurençot, Philippe Large time behavior of a two phase extension of the porous medium equation, Interfaces Free Bound., Volume 21 (2019) no. 2, pp. 199-229 | MR | DOI | Zbl

[AIJM18] Alkhayal, Jana; Issa, Samar; Jazar, Mustapha; Monneau, Régis Existence result for degenerate cross-diffusion system with application to seawater intrusion, ESAIM, Control Optim. Calc. Var., Volume 24 (2018) no. 4, pp. 1735-1758 | Zbl | MR | DOI

[BDFPS10] Burger, Martin; Di Francesco, Marco; Pietschmann, Jan-Frederik; Schlake, Bärbel Nonlinear cross-diffusion with size exclusion, SIAM J. Math. Anal., Volume 42 (2010) no. 6, pp. 2842-2871 | MR | Zbl | DOI

[BGB19] Bruell, Gabriele; Granero-Belinchón, Rafael On the thin film Muskat and the thin film Stokes equations, J. Math. Fluid Mech., Volume 21 (2019) no. 2, 33 | MR | Zbl

[BGHP85] Bertsch, Michiel; Gurtin, Morton E.; Hilhorst, Danielle; Peletier, Lambertus A. On interacting populations that disperse to avoid crowding: preservation of segregation, J. Math. Biol., Volume 23 (1985) no. 1, pp. 1-13 | Zbl | MR | DOI

[DGJ97] Degond, Pierre; Génieys, Stéphane; Jüngel, Ansgar Symmetrization and entropy inequality for general diffusion equations, C. R. Math. Acad. Sci. Paris, Volume 325 (1997) no. 9, pp. 963-968 | DOI | Zbl | MR

[DJ12] Dreher, Michael; Jüngel, Ansgar Compact families of piecewise constant functions in $L^p(0,T;B)$, Nonlinear Anal., Theory Methods Appl., Volume 75 (2012) no. 6, pp. 3072-3077 | MR | Zbl | DOI

[ELM11] Escher, Joachim; Laurençot, Philippe; Matioc, Bogdan-Vasile Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 28 (2011) no. 4, pp. 583-598 | MR | Numdam | DOI | Zbl

[EMM12] Escher, Joachim; Matioc, Anca-Voichita; Matioc, Bogdan-Vasile Modelling and analysis of the Muskat problem for thin fluid layers, J. Math. Fluid Mech., Volume 14 (2012) no. 2, pp. 267-277 | MR | DOI | Zbl

[GS14] Galiano, Gonzalo; Selgas, Virginia On a cross-diffusion segregation problem arising from a model of interacting particles, Nonlinear Anal., Real World Appl., Volume 18 (2014), pp. 34-49 | Zbl | MR | DOI

[GT01] Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order, Classics in Mathematics, Springer, 2001 | DOI | Zbl

[JM06] Jüngel, Ansgar; Matthes, Daniel An algorithmic construction of entropies in higher-order nonlinear PDEs, Nonlinearity, Volume 19 (2006) no. 3, pp. 633-659 | Zbl | MR | DOI

[Jün16] Jüngel, Ansgar Entropy methods for diffusive partial differential equations, SpringerBriefs in Mathematics, Springer, 2016 | Zbl | MR | DOI

[LM13] Laurençot, Philippe; Matioc, Bogdan-Vasile A gradient flow approach to a thin film approximation of the Muskat problem, Calc. Var. Partial Differ. Equ., Volume 47 (2013) no. 1-2, pp. 319-341 | Zbl | MR | DOI

[LM17] Laurençot, Philippe; Matioc, Bogdan-Vasile Finite speed of propagation and waiting time for a thin-film Muskat problem, Proc. R. Soc. Edinb., Sect. A, Math., Volume 147 (2017) no. 4, pp. 813-830 | MR | Zbl | DOI

[LM22] Laurençot, Philippe; Matioc, Bogdan-Vasile Bounded weak solutions to the thin film Muskat problem via an infinite family of Liapunov functionals, Trans. Am. Math. Soc., Volume 375 (2022) no. 8, pp. 5963-5986 | Zbl | MR | DOI

[Mie23] Mielke, Alexander On two coupled degenerate parabolic equations motivated by thermodynamics, J. Nonlinear Sci., Volume 33 (2023) no. 3, 42 | Zbl | MR | DOI