Aggregation-confinement-diffusion evolutions with saturation: Regularity and long-time asymptotics

Yousef Alamri

doi:10.3934/dcds.2025157

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2026, Volume 49: 75-105. Doi: 10.3934/dcds.2025157

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Aggregation-confinement-diffusion evolutions with saturation: Regularity and long-time asymptotics

Yousef Alamri

Computer, Electrical and Mathematical Sciences, and Engineering (CEMSE) Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia

Received: February 2025

Revised: September 2025

Early access: October 13, 2025

Published online: October 13, 2025

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Abstract

We establish Hölder regularity for the weak solution to a degenerate diffusion equation in the presence of a local (drift) potential and nonlocal (interaction) term, posed in a bounded domain with no-flux boundary conditions. The degeneracy is due to saturation, i.e., it occurs when the solution reaches its maximal value. As a byproduct of the established regularity and the underlying dissipative structure of the evolution, we prove the uniform convergence of contractive solutions to a stationary state as $ t\to \infty $.

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Mathematics Subject Classification: Primary: 35B65, 35B40, 35K65; Secondary: 35B45.

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Figure 1. The standard parabolic cylinder $ Q(R^2, R) $ contained in the rescaled cylinder $ Q(\theta_1 R^2, R) $

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References

[1]	R. Bailo, J. A. Carrillo and D. Gómez-Castro, Aggregation-diffusion equations for collective behaviour in the sciences, preprint, arXiv: 2405.16679, 2024.
[2]	R. Bailo, J. A. Carrillo and J. Hu, Bound-preserving finite-volume schemes for systems of continuity equations with saturation, SIAM J. Appl. Math., 83 (2023), 1315-1339. doi: 10.1137/22M1488703.
[3]	M. Burger, M. Di Francesco and Y. Dolak-Struss, The Keller-Segel model for chemotaxis with prevention of overcrowding: Linear vs. nonlinear diffusion, SIAM Journal on Mathematical Analysis, 38 (2006), 1288-1315. doi: 10.1137/050637923.
[4]	J. A. Carrillo, K. Craig and Y. Yao, Aggregation-diffusion equations: Dynamics, asymptotics, and singular limits, Active Particles, Volume 2: Advances in Theory, Models, and Applications, 2 (2019), 65-108.
[5]	J. A. Carrillo, A. Fernández-Jiménez and D. Gómez-Castro, Aggregation-diffusion equations with saturation, arXiv preprint arXiv: 2410.10040 (2024).
[6]	J. A. Carrillo and R. S. Gvalani, Phase transitions for nonlinear nonlocal aggregation-diffusion equations, Communications in Mathematical Physics, 382 (2021), 485-545. doi: 10.1007/s00220-021-03977-4.
[7]	J. A. Carrillo, R. Gvalani, G. Pavliotis and A. Schlichting, Long-time behaviour and phase transitions for the Mckean-Vlasov equation on the torus, Archive for Rational Mechanics and Analysis, 235 (2020), 635-690. doi: 10.1007/s00205-019-01430-4.
[8]	J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani andA. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatshefte für Mathematik, 133 (2001), 1-82. doi: 10.1007/s006050170032.
[9]	J. A. Carrillo, S. Lisini, G. Savaré and D. Slepčev, Nonlinear mobility continuity equations and generalized displacement convexity, Journal of Functional Analysis, 258 (2010), 1273-1309. doi: 10.1016/j.jfa.2009.10.016.
[10]	J. A. Carrillo, L. Wang and C. Wei, Structure preserving primal dual methods for gradient flows with nonlinear mobility transport distances, SIAM Journal on Numerical Analysis, 62 (2024), 376-399. doi: 10.1137/23M1562068.
[11]	L. Chayes, I. Kim and Y. Yao, An aggregation equation with degenerate diffusion: Qualitative property of solutions, SIAM Journal on Mathematical Analysis, 45 (2013), 2995-3018. doi: 10.1137/120874965.
[12]	L. Chayes and V. Panferov, The Mckean-Vlasov equation in finite volume, Journal of Statistical Physics, 138 (2010), 351-380. doi: 10.1007/s10955-009-9913-z.
[13]	E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Memorie della Accademia delle Scienze di Torino. Classe di Scienze Matematiche Fisiche e Natural, 3 (1957), 25-43.
[14]	E. DiBenedetto, Regularity results for the porous media equation, Annali di Matematica Pura ed Applicata, 121 (1979), 249-262. doi: 10.1007/BF02412006.
[15]	E. DiBenedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 13 (1986), 487-535.
[16]	E. DiBenedetto, Degenerate Parabolic Equations, Universitext. Springer-Verlag, New York, 1993.
[17]	E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer, 2012.
[18]	M. Di Francesco, S. Fagioli and E. Radici, Deterministic particle approximation for nonlocal transport equations with nonlinear mobility, Journal of Differential Equations, 266 (2019), 2830-2868. doi: 10.1016/j.jde.2018.08.047.
[19]	J. Dolbeault, B. Nazaret and G. Savaré, A new class of transport distances between measures, Calculus of Variations and Partial Differential Equations, 34 (2009), 193-231. doi: 10.1007/s00526-008-0182-5.
[20]	S. Fagioli and E. Radici, Solutions to aggregation-diffusion equations with nonlinear mobility constructed via a deterministic particle approximation, Mathematical Models and Methods in Applied Sciences, 28 (2018), 1801-1829. doi: 10.1142/S0218202518400067.
[21]	S. Fagioli and O. Tse, On gradient flow and entropy solutions for nonlocal transport equations with nonlinear mobility, Nonlinear Analysis, 221 (2022), 112904. doi: 10.1016/j.na.2022.112904.
[22]	G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. Ⅰ. Macroscopic limits, Journal of Statistical Physics, 87 (1997), 37-61. doi: 10.1007/BF02181479.
[23]	D. Gómez-Castro, Convergence of a finite-volume scheme for aggregation-diffusion equations with saturation, arXiv preprint arXiv: 2507.11132 (2025).
[24]	T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Advances in Applied Mathematics, 26 (2001), 280-301. doi: 10.1006/aama.2001.0721.
[25]	R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359.
[26]	S. Lisini and A. Marigonda, On a class of modified Wasserstein distances induced by concave mobility functions defined on bounded intervals, Manuscripta Mathematica, 133 (2010), 197-224. doi: 10.1007/s00229-010-0371-3.
[27]	F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Communications in Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.
[28]	K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canadian Applied Mathematics Quarterly, 10 (2002), 501-543.
[29]	M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, Journal of Differential Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045.
[30]	J. M. Urbano, Hölder continuity of local weak solutions for parabolic equations exhibiting two degeneracies, Advances in Differential Equations, 6 (2001), 327-358. doi: 10.57262/ade/1357141214.
[31]	J. M. Urbano, The Method of Intrinsic Scaling, vol. 1930 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2008, A systematic approach to regularity for degenerate and singular PDEs.
[32]	Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations, World Scientific, 2001.
[33]	N. Zamponi and A. Jüngel, Analysis of degenerate cross-diffusion population models with volume filling, Annales de l'Institut Henri Poincaré C. Analyse Non Linéaire, 34(2017), 1-29. doi: 10.1016/j.anihpc.2015.08.003.