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Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations
A congestion model for cell migration
1. | MAP5, UFR de Mathématiques et Informatique, Université Paris Descartes, 45 rue des Saints-Pères 75270 Paris cedex 06, France, France |
2. | Laboratoire de Mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay Cedex |
We propose a theoretical framework for the open-loop model (desired velocities are defined as gradients of given functions) based on a formulation in the form of a gradient flow in the Wasserstein space. We propose a numerical strategy to discretize the model, and illustrate its behaviour in the case of a prescribed velocity, and for the saturated Keller-Segel model.
References:
[1] |
L. Ambrosio, N. Gigli and G. Savare, Gradient flows in metric spaces in the space of probability measures, Lectures in Mathematics, ETH Zürich, (2005). |
[2] |
L. Ambrosio and G. Savare, "Gradient Flows of Probability Measures," Handbook of Differential Equations, Evolutionary Equations (ed. by C.M. Dafermos and E. Feireisl, Elsevier), 3, 2007. |
[3] |
J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2001), 375-393.
doi: 10.1007/s002110050002. |
[4] |
A. L. Dalibard and B. Perthame, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335. |
[5] |
E. De Giorgi, New problems on minimizing movements, Boundary Value Problems for PDE and Applications (eds. C. Baiocchi and J. L. Lions), Masson, (1993), 81-98. |
[6] |
Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM J. Appl. Math., 66 (2005), 286-308.
doi: 10.1137/040612841. |
[7] |
N. Gigli and F. Otto, Entropic Burgers' equation via a minimizing movement scheme based on the Wasserstein metric,, submitted., ().
|
[8] |
R. Jordan and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
[9] |
E. F Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[10] |
R. Kimmel and J. Sethian, Fast marching methods for computing distance maps and shortest paths, Technical Report, CPAM, Univ. of California, Berkeley, 669 (1996). |
[11] |
B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type,, Math. Mod. Meth. Appl. Sci., ().
|
[12] |
J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374.
doi: 10.1016/0022-0396(77)90085-7. |
[13] |
F. Otto and Weinan E., Thermodynamically driven incompressible fluid mixtures, J. Chem. Phys., 107 (1997), 10177.
doi: 10.1063/1.474153. |
[14] |
B. Perthame, PDE models for chemotactic movements: parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564.
doi: 10.1007/s10492-004-6431-9. |
[15] |
G. Peyre, Toolbox Fast Marching - A toolbox for Fast Marching and level sets computations, software, (2008). |
[16] |
M. Renardy and R. C. Rogers, "An Introduction to Partial Differential Equations," Texts in App. Math., 13, Springer-Verlag, New York, 2004. |
[17] |
C. Villani, "Topics in Optimal Transportation," Grad. Stud. Math., 58 AMS, Providence, 2003. |
[18] |
C. Villani, Optimal transport, old and new, Grundlehren der mathematischen Wissenschaften, 338 (2009). |
show all references
References:
[1] |
L. Ambrosio, N. Gigli and G. Savare, Gradient flows in metric spaces in the space of probability measures, Lectures in Mathematics, ETH Zürich, (2005). |
[2] |
L. Ambrosio and G. Savare, "Gradient Flows of Probability Measures," Handbook of Differential Equations, Evolutionary Equations (ed. by C.M. Dafermos and E. Feireisl, Elsevier), 3, 2007. |
[3] |
J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2001), 375-393.
doi: 10.1007/s002110050002. |
[4] |
A. L. Dalibard and B. Perthame, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335. |
[5] |
E. De Giorgi, New problems on minimizing movements, Boundary Value Problems for PDE and Applications (eds. C. Baiocchi and J. L. Lions), Masson, (1993), 81-98. |
[6] |
Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM J. Appl. Math., 66 (2005), 286-308.
doi: 10.1137/040612841. |
[7] |
N. Gigli and F. Otto, Entropic Burgers' equation via a minimizing movement scheme based on the Wasserstein metric,, submitted., ().
|
[8] |
R. Jordan and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
[9] |
E. F Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[10] |
R. Kimmel and J. Sethian, Fast marching methods for computing distance maps and shortest paths, Technical Report, CPAM, Univ. of California, Berkeley, 669 (1996). |
[11] |
B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type,, Math. Mod. Meth. Appl. Sci., ().
|
[12] |
J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374.
doi: 10.1016/0022-0396(77)90085-7. |
[13] |
F. Otto and Weinan E., Thermodynamically driven incompressible fluid mixtures, J. Chem. Phys., 107 (1997), 10177.
doi: 10.1063/1.474153. |
[14] |
B. Perthame, PDE models for chemotactic movements: parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564.
doi: 10.1007/s10492-004-6431-9. |
[15] |
G. Peyre, Toolbox Fast Marching - A toolbox for Fast Marching and level sets computations, software, (2008). |
[16] |
M. Renardy and R. C. Rogers, "An Introduction to Partial Differential Equations," Texts in App. Math., 13, Springer-Verlag, New York, 2004. |
[17] |
C. Villani, "Topics in Optimal Transportation," Grad. Stud. Math., 58 AMS, Providence, 2003. |
[18] |
C. Villani, Optimal transport, old and new, Grundlehren der mathematischen Wissenschaften, 338 (2009). |
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