We discuss the analysis and stability of a family of cross-diffusion boundary value problems with nonlinear diffusion and drift terms. We assume that these systems are close, in a suitable sense, to a set of decoupled and linear problems. We focus on stability estimates, that is, continuous dependence of solutions with respect to the nonlinearities in the diffusion and in the drift terms. We establish well-posedness and stability estimates in an appropriate Banach space. Under additional assumptions we show that these estimates are time independent. These results apply to several problems from mathematical biology; they allow comparisons between the solutions of different models a priori. For specific cell motility models from the literature, we illustrate the limit of the stability estimates we have derived numerically, and we document the behaviour of the solutions for extremal values of the parameters.
Mots-clés : Cross diffusion, continuous dependence, quasilinear parabolic systems.
@article{M2AN_2018__52_3_1109_0,
author = {Alasio, Luca and Bruna, Maria and Capdeboscq, Yves},
title = {Stability estimates for systems with small cross-diffusion\protect\textsuperscript{,}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1109--1135},
publisher = {EDP-Sciences},
volume = {52},
number = {3},
year = {2018},
doi = {10.1051/m2an/2018036},
zbl = {1403.35126},
mrnumber = {3865560},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2018036/}
}
TY - JOUR AU - Alasio, Luca AU - Bruna, Maria AU - Capdeboscq, Yves TI - Stability estimates for systems with small cross-diffusion, JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1109 EP - 1135 VL - 52 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2018036/ DO - 10.1051/m2an/2018036 LA - en ID - M2AN_2018__52_3_1109_0 ER -
%0 Journal Article %A Alasio, Luca %A Bruna, Maria %A Capdeboscq, Yves %T Stability estimates for systems with small cross-diffusion, %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1109-1135 %V 52 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2018036/ %R 10.1051/m2an/2018036 %G en %F M2AN_2018__52_3_1109_0
Alasio, Luca; Bruna, Maria; Capdeboscq, Yves. Stability estimates for systems with small cross-diffusion,. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 1109-1135. doi : 10.1051/m2an/2018036. http://www.numdam.org/articles/10.1051/m2an/2018036/
[1] Analyse fonctionnelle: Théorie et applications. Collection Mathématiques Appliquées pour la Maî trise. [Collection of Applied Mathematics for the Master’s Degree]. Masson, Paris (1983). | MR | Zbl
[2] , , , and , Cross-diffusion systems with excluded-volume effects and asymptotic gradient flow structures. J. Nonlinear Sci. 27 (2017) 687–719. | DOI | MR | Zbl
[3] and , Diffusion of multiple species with excluded-volume effects. J. Chem. Phys. 137 (2012) 204116. | DOI
[4] , , , and , Nonlinear cross-diffusion with size exclusion. SIAM J. Math. Anal. 42 (2010) 2842. | DOI | MR | Zbl
[5] and , Continuous dependence estimates and homogenization of quasi-monotone systems of fully nonlinear second order parabolic equations. Nonlinear Anal. Theor. 75 (2012) 5103–5118. | DOI | MR | Zbl
[6] , , and , Quasilinear parabolic systems. J. Differ. Equ. 20 (1976) 441–472. | DOI | MR | Zbl
[7] and , Analysis of a parabolic cross-diffusion population model without self-diffusion. J. Differ. Equ. 224 (2006) 39–59. | DOI | MR | Zbl
[8] and , Mathematical Analysis and Numerical Methods for Science and Technology. Springer Verlag, Berlin (1993). | MR | Zbl
[9] , Partial Differential Equations. American Mathematical Society (1998). | MR | Zbl
[10] and , An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs. Vol. 2 of Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Normale, Pisa (2005). | MR | Zbl
[11] and , On the partial regularity of weak solutions of nonlinear parabolic systems. Math. Z. 179 (1982) 437–451. | DOI | MR | Zbl
[12] , Elliptic Problems in Nonsmooth Domains. Vol. 24 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, MA (1985). | MR | Zbl
[13] , The boundedness-by-entropy method for cross-diffusion systems. Nonlinearity 28 (2015) 1963. | DOI | MR | Zbl
[14] , First order quasilinear equations in several independent variables. Math. USSR SB+ 10 (1970) 217. | DOI | MR | Zbl
[15] , , and , Linear and Quasi-Linear Equations of Parabolic Type. Vol. 23. American Mathematical Society (1988). | Zbl
[16] and , Problèmes aux limites non homogènes. Dunod, Paris (1968) Vol. 1–3. | MR | Zbl
[17] , Parabolic Equations in Biology: Growth, Reaction, Movement and Diffusion. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, Cham (2015). | DOI | MR | Zbl
[18] and , Models of collective cell behaviour with crowding effects: comparing lattice-based and lattice-free approaches. J. R. Soc. Interface 9 (2012) 2983–2996. | DOI
[19] , Explicit h2-estimates and pointwise bounds for solutions of second-order elliptic boundary value problems. J. Math. Anal. Appl. 165 (1992) 36–61. | DOI | MR | Zbl
[20] , , and , Spatial segregation of interacting species. J. Theor. Biol. 79 (1979) 83–99. | DOI | MR
[21] , , and , Multi-species simple exclusion processes. Phys. A: Stat. Mech. Appl. 388 (2009) 399–406. | DOI
[22] and , Some (new) counterexamples of parabolic systems. Commentat. Math. Univ. Carol. 36 (1995) 503–510 | MR | Zbl
[23] , Elliptic Differential Equations and Obstacle Problems. The University Series in Mathematics. Plenum Press, New York (1987). | MR | Zbl
[24] and , Positivity-preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal. 37 (2000) 523–555. | DOI | MR | Zbl
Cité par Sources :
