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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, (2005).
|
[2] |
L. Ambrosio and G. Savare, "Gradient Flows of Probability Measures,", Handbook of Differential Equations, 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.
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.
|
[5] |
E. De Giorgi, New problems on minimizing movements,, Boundary Value Problems for PDE and Applications (eds. C. Baiocchi and J. L. Lions), (1993), 81.
|
[6] |
Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity,, SIAM J. Appl. Math., 66 (2005), 286.
doi: 10.1137/040612841. |
[7] |
N. Gigli and F. Otto, Entropic Burgers' equation via a minimizing movement scheme based on the Wasserstein metric,, submitted., (). Google Scholar |
[8] |
R. Jordan and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1.
doi: 10.1137/S0036141096303359. |
[9] |
E. F Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225.
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, 669 (1996). Google Scholar |
[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.
doi: 10.1016/0022-0396(77)90085-7. |
[13] |
F. Otto and Weinan E., Thermodynamically driven incompressible fluid mixtures,, J. Chem. Phys., 107 (1997).
doi: 10.1063/1.474153. |
[14] |
B. Perthame, PDE models for chemotactic movements: parabolic, hyperbolic and kinetic,, Appl. Math., 49 (2004), 539.
doi: 10.1007/s10492-004-6431-9. |
[15] |
G. Peyre, Toolbox Fast Marching - A toolbox for Fast Marching and level sets computations,, software, (2008). Google Scholar |
[16] |
M. Renardy and R. C. Rogers, "An Introduction to Partial Differential Equations,", Texts in App. Math., 13 (2004).
|
[17] |
C. Villani, "Topics in Optimal Transportation,", Grad. Stud. Math., 58 (2003).
|
[18] |
C. Villani, Optimal transport, old and new,, Grundlehren der mathematischen Wissenschaften, 338 (2009). Google Scholar |
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, (2005).
|
[2] |
L. Ambrosio and G. Savare, "Gradient Flows of Probability Measures,", Handbook of Differential Equations, 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.
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.
|
[5] |
E. De Giorgi, New problems on minimizing movements,, Boundary Value Problems for PDE and Applications (eds. C. Baiocchi and J. L. Lions), (1993), 81.
|
[6] |
Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity,, SIAM J. Appl. Math., 66 (2005), 286.
doi: 10.1137/040612841. |
[7] |
N. Gigli and F. Otto, Entropic Burgers' equation via a minimizing movement scheme based on the Wasserstein metric,, submitted., (). Google Scholar |
[8] |
R. Jordan and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1.
doi: 10.1137/S0036141096303359. |
[9] |
E. F Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225.
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, 669 (1996). Google Scholar |
[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.
doi: 10.1016/0022-0396(77)90085-7. |
[13] |
F. Otto and Weinan E., Thermodynamically driven incompressible fluid mixtures,, J. Chem. Phys., 107 (1997).
doi: 10.1063/1.474153. |
[14] |
B. Perthame, PDE models for chemotactic movements: parabolic, hyperbolic and kinetic,, Appl. Math., 49 (2004), 539.
doi: 10.1007/s10492-004-6431-9. |
[15] |
G. Peyre, Toolbox Fast Marching - A toolbox for Fast Marching and level sets computations,, software, (2008). Google Scholar |
[16] |
M. Renardy and R. C. Rogers, "An Introduction to Partial Differential Equations,", Texts in App. Math., 13 (2004).
|
[17] |
C. Villani, "Topics in Optimal Transportation,", Grad. Stud. Math., 58 (2003).
|
[18] |
C. Villani, Optimal transport, old and new,, Grundlehren der mathematischen Wissenschaften, 338 (2009). Google Scholar |
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