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Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations
A BDF2-approach for the non-linear Fokker-Planck equation
Boltzmannstra. 3, D-85747 Garching, Germany |
We prove convergence of a variational formulation of the BDF2 method applied to the non-linear Fokker-Planck equation. Our approach is inspired by the JKO-method and exploits the differential structure of the underlying $ L^2 $-Wasserstein space. The technique presented here extends and strengthens the results of our own recent work [
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, vol. 254, Clarendon Press Oxford, 2000. |
[2] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. |
[3] |
J.-D. Benamou, G. Carlier, Q. Mérigot and E. Oudet,
Discretization of functionals involving the monge–ampère operator, Numerische Mathematik, 134 (2016), 611-636.
doi: 10.1007/s00211-015-0781-y. |
[4] |
A. Blanchet and P. Laurençot,
The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in $\mathbb{R}^d,\ d\ge3$, Comm. Partial Differential Equations, 38 (2013), 658-686.
doi: 10.1080/03605302.2012.757705. |
[5] |
A. Blanchet, V. Calvez and J. A. Carrillo,
Convergence of the mass-transport steepest descent scheme for the subcritical Patlak–Keller–Segel model, SIAM Journal on Numerical Analysis, 46 (2008), 691-721.
doi: 10.1137/070683337. |
[6] |
V. Calvez and T. O. Gallouët, Blow-up phenomena for gradient flows of discrete homogeneous functionals, Applied Mathematics & Optimization, (2017), 1–29.
doi: 10.1007/s00245-017-9443-z. |
[7] |
J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev,
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.
doi: 10.1215/00127094-2010-211. |
[8] |
J. Carrillo, F. Patacchini, P. Sternberg and G. Wolansky,
Convergence of a particle method for diffusive gradient flows in one dimension, SIAM Journal on Mathematical Analysis, 48 (2016), 3708-3741.
doi: 10.1137/16M1077210. |
[9] |
J. A. Carrillo, H. Ranetbauer and M.-T. Wolfram, Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms, Journal of Computational Physics, 327 (2016), 186-202, URL http://www.sciencedirect.com/science/article/pii/S0021999116304612.
doi: 10.1016/j.jcp.2016.09.040. |
[10] |
J. A. Carrillo and J. S. Moll,
Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM Journal on Scientific Computing, 31 (2009), 4305-4329.
doi: 10.1137/080739574. |
[11] |
M. Di Francesco, A. Esposito and S. Fagioli,
Nonlinear degenerate cross-diffusion systems with nonlocal interaction, Nonlinear Analysis, 169 (2018), 94-117.
doi: 10.1016/j.na.2017.12.003. |
[12] |
B. Düring, D. Matthes and J. P. Milišic,
A gradient flow scheme for nonlinear fourth order equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 935-959.
doi: 10.3934/dcdsb.2010.14.935. |
[13] |
M. Erbar et al., The heat equation on manifolds as a gradient flow in the Wasserstein space, in Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, vol. 46, Institut Henri Poincaré, 2010, 1–23.
doi: 10.1214/08-AIHP306. |
[14] |
T. O. Gallouët and Q. Mérigot,
A Lagrangian scheme à la Brenier for the incompressible Euler Equations, Foundations of Computational Mathematics, 18 (2018), 835-865.
doi: 10.1007/s10208-017-9355-y. |
[15] |
L. Giacomelli and F. Otto,
Variatonal formulation for the lubrication approximation of the Hele-Shaw flow, Calculus of Variations and Partial Differential Equations, 13 (2001), 377-403.
doi: 10.1007/s005260000077. |
[16] |
U. Gianazza, G. Savaré and G. Toscani,
The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal., 194 (2009), 133-220.
doi: 10.1007/s00205-008-0186-5. |
[17] |
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, vol. 80, Birkhauser Verlag, 1984.
doi: 10.1007/978-1-4684-9486-0. |
[18] |
R. Jordan, D. Kinderlehrer and F. Otto,
The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
[19] |
O. Junge, D. Matthes and H. Osberger,
A fully discrete variational scheme for solving nonlinear fokker–planck equations in multiple space dimensions, SIAM Journal on Numerical Analysis, 55 (2017), 419-443.
doi: 10.1137/16M1056560. |
[20] |
D. Kinderlehrer, L. Monsaingeon and X. Xu,
A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2017), 137-164.
doi: 10.1051/cocv/2015043. |
[21] |
L. Laguzet,
High order variational numerical schemes with application to Nash -MFG vaccination games, Ric. Mat., 67 (2018), 247-269.
doi: 10.1007/s11587-018-0366-z. |
[22] |
P. Laurençot and B.-V. Matioc,
A gradient flow approach to a thin film approximation of the Muskat problem, Calc. Var. Partial Differential Equations, 47 (2013), 319-341.
doi: 10.1007/s00526-012-0520-5. |
[23] |
G. Legendre and G. Turinici, Second-order in time schemes for gradient flows in Wasserstein and geodesic metric spaces, Comptes Rendus Mathematique, 355 (2017), 345-353, URL http://www.sciencedirect.com/science/article/pii/S1631073X17300365.
doi: 10.1016/j.crma.2017.02.001. |
[24] |
S. Lisini, D. Matthes and G. Savaré,
Cahn-Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics, J. Differential Equations, 253 (2012), 814-850.
doi: 10.1016/j.jde.2012.04.004. |
[25] |
D. Matthes, R. J. McCann and G. Savaré,
A family of nonlinear fourth order equations of gradient flow type, Comm. Partial Differential Equations, 34 (2009), 1352-1397.
doi: 10.1080/03605300903296256. |
[26] |
D. Matthes and H. Osberger,
Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 697-726.
doi: 10.1051/m2an/2013126. |
[27] |
D. Matthes and S. Plazotta, A variational formulation of the BDF2 method for metric gradient flows, to appear in ESAIM: Mathematical Modelling and Numerical Analysis.
doi: 10.1051/m2an/2018045. |
[28] |
D. Matthes and B. Söllner, Convergent Lagrangian discretization for drift-diffusion with nonlocal aggregation, in Innovative Algorithms and Analysis, Springer, 16 (2017), 313–351.
doi: 10.1007/978-3-319-49262-9_12. |
[29] |
D. Matthes and J. Zinsl,
Existence of solutions for a class of fourth order cross-diffusion systems of gradient flow type, Nonlinear Analysis, 159 (2017), 316-338.
doi: 10.1016/j.na.2016.12.002. |
[30] |
F. Otto,
The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.
doi: 10.1081/PDE-100002243. |
[31] |
G. Peyré,
Entropic approximation of Wasserstein gradient flows, SIAM Journal on Imaging Sciences, 8 (2015), 2323-2351.
doi: 10.1137/15M1010087. |
[32] |
R. Rossi and G. Savaré,
Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze-Serie V, 2 (2003), 395-431.
|
[33] |
F. Santambrogio, Optimal transport for Applied Mathematicians, Springer, 2015.
doi: 10.1007/978-3-319-20828-2. |
[34] |
K.-T. Sturm, Convex functionals of probability measures and nonlinear diffusions on manifolds, Journal de mathématiques pures et appliquées, 84 (2005), 149–168.
doi: 10.1016/j.matpur.2004.11.002. |
[35] |
C. M. Topaz, A. L. Bertozzi and M. A. Lewis,
A nonlocal continuum model for biological aggregation, Bulletin of Mathematical Biology, 68 (2006), 1601-1623.
doi: 10.1007/s11538-006-9088-6. |
[36] |
C. Villani, Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 2003.
doi: 10.1007/b12016. |
[37] |
C. Villani, Optimal Transport: Old and New, vol. 338, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9. |
[38] |
M. Westdickenberg and J. Wilkening,
Variational particle schemes for the porous medium equation and for the system of isentropic euler equations, ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2010), 133-166.
doi: 10.1051/m2an/2009043. |
[39] |
J. Zinsl and D. Matthes,
Exponential convergence to equilibrium in a coupled gradient flow system modeling chemotaxis, Analysis & PDE, 8 (2015), 425-466.
doi: 10.2140/apde.2015.8.425. |
[40] |
J. Zinsl and D. Matthes,
Transport distances and geodesic convexity for systems of degenerate diffusion equations, Calculus of Variations and Partial Differential Equations, 54 (2015), 3397-3438.
doi: 10.1007/s00526-015-0909-z. |
show all references
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, vol. 254, Clarendon Press Oxford, 2000. |
[2] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. |
[3] |
J.-D. Benamou, G. Carlier, Q. Mérigot and E. Oudet,
Discretization of functionals involving the monge–ampère operator, Numerische Mathematik, 134 (2016), 611-636.
doi: 10.1007/s00211-015-0781-y. |
[4] |
A. Blanchet and P. Laurençot,
The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in $\mathbb{R}^d,\ d\ge3$, Comm. Partial Differential Equations, 38 (2013), 658-686.
doi: 10.1080/03605302.2012.757705. |
[5] |
A. Blanchet, V. Calvez and J. A. Carrillo,
Convergence of the mass-transport steepest descent scheme for the subcritical Patlak–Keller–Segel model, SIAM Journal on Numerical Analysis, 46 (2008), 691-721.
doi: 10.1137/070683337. |
[6] |
V. Calvez and T. O. Gallouët, Blow-up phenomena for gradient flows of discrete homogeneous functionals, Applied Mathematics & Optimization, (2017), 1–29.
doi: 10.1007/s00245-017-9443-z. |
[7] |
J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev,
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.
doi: 10.1215/00127094-2010-211. |
[8] |
J. Carrillo, F. Patacchini, P. Sternberg and G. Wolansky,
Convergence of a particle method for diffusive gradient flows in one dimension, SIAM Journal on Mathematical Analysis, 48 (2016), 3708-3741.
doi: 10.1137/16M1077210. |
[9] |
J. A. Carrillo, H. Ranetbauer and M.-T. Wolfram, Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms, Journal of Computational Physics, 327 (2016), 186-202, URL http://www.sciencedirect.com/science/article/pii/S0021999116304612.
doi: 10.1016/j.jcp.2016.09.040. |
[10] |
J. A. Carrillo and J. S. Moll,
Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM Journal on Scientific Computing, 31 (2009), 4305-4329.
doi: 10.1137/080739574. |
[11] |
M. Di Francesco, A. Esposito and S. Fagioli,
Nonlinear degenerate cross-diffusion systems with nonlocal interaction, Nonlinear Analysis, 169 (2018), 94-117.
doi: 10.1016/j.na.2017.12.003. |
[12] |
B. Düring, D. Matthes and J. P. Milišic,
A gradient flow scheme for nonlinear fourth order equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 935-959.
doi: 10.3934/dcdsb.2010.14.935. |
[13] |
M. Erbar et al., The heat equation on manifolds as a gradient flow in the Wasserstein space, in Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, vol. 46, Institut Henri Poincaré, 2010, 1–23.
doi: 10.1214/08-AIHP306. |
[14] |
T. O. Gallouët and Q. Mérigot,
A Lagrangian scheme à la Brenier for the incompressible Euler Equations, Foundations of Computational Mathematics, 18 (2018), 835-865.
doi: 10.1007/s10208-017-9355-y. |
[15] |
L. Giacomelli and F. Otto,
Variatonal formulation for the lubrication approximation of the Hele-Shaw flow, Calculus of Variations and Partial Differential Equations, 13 (2001), 377-403.
doi: 10.1007/s005260000077. |
[16] |
U. Gianazza, G. Savaré and G. Toscani,
The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal., 194 (2009), 133-220.
doi: 10.1007/s00205-008-0186-5. |
[17] |
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, vol. 80, Birkhauser Verlag, 1984.
doi: 10.1007/978-1-4684-9486-0. |
[18] |
R. Jordan, D. Kinderlehrer and F. Otto,
The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
[19] |
O. Junge, D. Matthes and H. Osberger,
A fully discrete variational scheme for solving nonlinear fokker–planck equations in multiple space dimensions, SIAM Journal on Numerical Analysis, 55 (2017), 419-443.
doi: 10.1137/16M1056560. |
[20] |
D. Kinderlehrer, L. Monsaingeon and X. Xu,
A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2017), 137-164.
doi: 10.1051/cocv/2015043. |
[21] |
L. Laguzet,
High order variational numerical schemes with application to Nash -MFG vaccination games, Ric. Mat., 67 (2018), 247-269.
doi: 10.1007/s11587-018-0366-z. |
[22] |
P. Laurençot and B.-V. Matioc,
A gradient flow approach to a thin film approximation of the Muskat problem, Calc. Var. Partial Differential Equations, 47 (2013), 319-341.
doi: 10.1007/s00526-012-0520-5. |
[23] |
G. Legendre and G. Turinici, Second-order in time schemes for gradient flows in Wasserstein and geodesic metric spaces, Comptes Rendus Mathematique, 355 (2017), 345-353, URL http://www.sciencedirect.com/science/article/pii/S1631073X17300365.
doi: 10.1016/j.crma.2017.02.001. |
[24] |
S. Lisini, D. Matthes and G. Savaré,
Cahn-Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics, J. Differential Equations, 253 (2012), 814-850.
doi: 10.1016/j.jde.2012.04.004. |
[25] |
D. Matthes, R. J. McCann and G. Savaré,
A family of nonlinear fourth order equations of gradient flow type, Comm. Partial Differential Equations, 34 (2009), 1352-1397.
doi: 10.1080/03605300903296256. |
[26] |
D. Matthes and H. Osberger,
Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 697-726.
doi: 10.1051/m2an/2013126. |
[27] |
D. Matthes and S. Plazotta, A variational formulation of the BDF2 method for metric gradient flows, to appear in ESAIM: Mathematical Modelling and Numerical Analysis.
doi: 10.1051/m2an/2018045. |
[28] |
D. Matthes and B. Söllner, Convergent Lagrangian discretization for drift-diffusion with nonlocal aggregation, in Innovative Algorithms and Analysis, Springer, 16 (2017), 313–351.
doi: 10.1007/978-3-319-49262-9_12. |
[29] |
D. Matthes and J. Zinsl,
Existence of solutions for a class of fourth order cross-diffusion systems of gradient flow type, Nonlinear Analysis, 159 (2017), 316-338.
doi: 10.1016/j.na.2016.12.002. |
[30] |
F. Otto,
The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.
doi: 10.1081/PDE-100002243. |
[31] |
G. Peyré,
Entropic approximation of Wasserstein gradient flows, SIAM Journal on Imaging Sciences, 8 (2015), 2323-2351.
doi: 10.1137/15M1010087. |
[32] |
R. Rossi and G. Savaré,
Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze-Serie V, 2 (2003), 395-431.
|
[33] |
F. Santambrogio, Optimal transport for Applied Mathematicians, Springer, 2015.
doi: 10.1007/978-3-319-20828-2. |
[34] |
K.-T. Sturm, Convex functionals of probability measures and nonlinear diffusions on manifolds, Journal de mathématiques pures et appliquées, 84 (2005), 149–168.
doi: 10.1016/j.matpur.2004.11.002. |
[35] |
C. M. Topaz, A. L. Bertozzi and M. A. Lewis,
A nonlocal continuum model for biological aggregation, Bulletin of Mathematical Biology, 68 (2006), 1601-1623.
doi: 10.1007/s11538-006-9088-6. |
[36] |
C. Villani, Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 2003.
doi: 10.1007/b12016. |
[37] |
C. Villani, Optimal Transport: Old and New, vol. 338, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9. |
[38] |
M. Westdickenberg and J. Wilkening,
Variational particle schemes for the porous medium equation and for the system of isentropic euler equations, ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2010), 133-166.
doi: 10.1051/m2an/2009043. |
[39] |
J. Zinsl and D. Matthes,
Exponential convergence to equilibrium in a coupled gradient flow system modeling chemotaxis, Analysis & PDE, 8 (2015), 425-466.
doi: 10.2140/apde.2015.8.425. |
[40] |
J. Zinsl and D. Matthes,
Transport distances and geodesic convexity for systems of degenerate diffusion equations, Calculus of Variations and Partial Differential Equations, 54 (2015), 3397-3438.
doi: 10.1007/s00526-015-0909-z. |
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