# American Institute of Mathematical Sciences

August  2017, 10(4): 919-933. doi: 10.3934/dcdss.2017047

## The gradient flow of a generalized Fisher information functional with respect to modified Wasserstein distances

 Zentrum für Mathematik, Technische Universität München, 85747 Garching, Germany

Received  March 2016 Revised  October 2016 Published  April 2017

This article is concerned with the existence of nonnegative weak solutions to a particular fourth-order partial differential equation: it is a formal gradient flow with respect to a generalized Wasserstein transportation distance with nonlinear mobility. The corresponding free energy functional is referred to as generalized Fisher information functional since it is obtained by autodissipation of another energy functional which generates the heat flow as its gradient flow with respect to the aforementioned distance. Our main results are twofold: For mobility functions satisfying a certain regularity condition, we show the existence of weak solutions by construction with the well-known minimizing movement scheme for gradient flows. Furthermore, we extend these results to a more general class of mobility functions: a weak solution can be obtained by approximation with weak solutions of the problem with regularized mobility.

Citation: Jonathan Zinsl. The gradient flow of a generalized Fisher information functional with respect to modified Wasserstein distances. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 919-933. doi: 10.3934/dcdss.2017047
##### References:
 [1] 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.Google Scholar [2] J. -D. Benamou and Y. Brenier, A computational fluid mechanics solution to the MongeKantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002. Google Scholar [3] A. Blanchet and P. Laurençot, The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in $\mathbb{R}^d$, d ≥ 3, Comm. Partial Differential Equations, 38 (2013), 658-686. Google Scholar [4] J. A. Carrillo, S. Lisini, G. Savaré and D. Slepčev, Nonlinear mobility continuity equations and generalized displacement convexity, J. Funct. Anal., 258 (2010), 1273-1309. doi: 10.1016/j.jfa.2009.10.016. Google Scholar [5] J. Dolbeault, B. Nazaret and G. Savaré, A new class of transport distances between measures, Calc. Var. Partial Differential Equations, 194 (2009), p133. doi: 10.1007/s00526-008-0182-5. Google Scholar [6] 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. Google Scholar [7] 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. Google Scholar [8] P. -L. Lions and C. Villani, Régularité optimale de racines carrées, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 1537-1541. Google Scholar [9] S. Lisini and A. Marigonda, On a class of modified Wasserstein distances induced by concave mobility functions defined on bounded intervals, Manuscripta Math., 133 (2010), 197-224. doi: 10.1007/s00229-010-0371-3. Google Scholar [10] 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. Google Scholar [11] D. Loibl, D. Matthes and J. Zinsl, Existence of weak solutions to a class of fourth order partial differential equations with Wasserstein gradient structure, Potential Analysis, 45 (2016), 755-776. doi: 10.1007/s11118-016-9565-y. Google Scholar [12] 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. Google Scholar [13] R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634. Google Scholar [14] 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. Google Scholar [15] R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2 (2003), 395-431. Google Scholar [16] C. Villani, Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 2003.Google Scholar [17] J. Zinsl and D. Matthes, Transport distances and geodesic convexity for systems of degenerate diffusion equations, Calc. Var. Partial Differential Equations, 54 (2015), 3397-3438. doi: 10.1007/s00526-015-0909-z. Google Scholar

show all references

##### References:
 [1] 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.Google Scholar [2] J. -D. Benamou and Y. Brenier, A computational fluid mechanics solution to the MongeKantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002. Google Scholar [3] A. Blanchet and P. Laurençot, The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in $\mathbb{R}^d$, d ≥ 3, Comm. Partial Differential Equations, 38 (2013), 658-686. Google Scholar [4] J. A. Carrillo, S. Lisini, G. Savaré and D. Slepčev, Nonlinear mobility continuity equations and generalized displacement convexity, J. Funct. Anal., 258 (2010), 1273-1309. doi: 10.1016/j.jfa.2009.10.016. Google Scholar [5] J. Dolbeault, B. Nazaret and G. Savaré, A new class of transport distances between measures, Calc. Var. Partial Differential Equations, 194 (2009), p133. doi: 10.1007/s00526-008-0182-5. Google Scholar [6] 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. Google Scholar [7] 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. Google Scholar [8] P. -L. Lions and C. Villani, Régularité optimale de racines carrées, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 1537-1541. Google Scholar [9] S. Lisini and A. Marigonda, On a class of modified Wasserstein distances induced by concave mobility functions defined on bounded intervals, Manuscripta Math., 133 (2010), 197-224. doi: 10.1007/s00229-010-0371-3. Google Scholar [10] 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. Google Scholar [11] D. Loibl, D. Matthes and J. Zinsl, Existence of weak solutions to a class of fourth order partial differential equations with Wasserstein gradient structure, Potential Analysis, 45 (2016), 755-776. doi: 10.1007/s11118-016-9565-y. Google Scholar [12] 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. Google Scholar [13] R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634. Google Scholar [14] 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. Google Scholar [15] R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2 (2003), 395-431. Google Scholar [16] C. Villani, Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 2003.Google Scholar [17] J. Zinsl and D. Matthes, Transport distances and geodesic convexity for systems of degenerate diffusion equations, Calc. Var. Partial Differential Equations, 54 (2015), 3397-3438. doi: 10.1007/s00526-015-0909-z. Google Scholar
 [1] Bertram Düring, Daniel Matthes, Josipa Pina Milišić. A gradient flow scheme for nonlinear fourth order equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 935-959. doi: 10.3934/dcdsb.2010.14.935 [2] Jaime Angulo Pava, Carlos Banquet, Márcia Scialom. Stability for the modified and fourth-order Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 851-871. doi: 10.3934/dcds.2011.30.851 [3] Feliz Minhós, João Fialho. On the solvability of some fourth-order equations with functional boundary conditions. Conference Publications, 2009, 2009 (Special) : 564-573. doi: 10.3934/proc.2009.2009.564 [4] José A. Carrillo, Ansgar Jüngel, Shaoqiang Tang. Positive entropic schemes for a nonlinear fourth-order parabolic equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 1-20. doi: 10.3934/dcdsb.2003.3.1 [5] Chunhua Jin, Jingxue Yin, Zejia Wang. Positive periodic solutions to a nonlinear fourth-order differential equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1225-1235. doi: 10.3934/cpaa.2008.7.1225 [6] Gabriele Bonanno, Beatrice Di Bella. Fourth-order hemivariational inequalities. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 729-739. doi: 10.3934/dcdss.2012.5.729 [7] Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov. Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3913-3938. doi: 10.3934/dcds.2018170 [8] Andrea L. Bertozzi, Ning Ju, Hsiang-Wei Lu. A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1367-1391. doi: 10.3934/dcds.2011.29.1367 [9] Luca Calatroni, Bertram Düring, Carola-Bibiane Schönlieb. ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 931-957. doi: 10.3934/dcds.2014.34.931 [10] Lili Ju, Xinfeng Liu, Wei Leng. Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1667-1687. doi: 10.3934/dcdsb.2014.19.1667 [11] Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227 [12] Haitao Che, Haibin Chen, Yiju Wang. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018153 [13] Horst Osberger. Long-time behavior of a fully discrete Lagrangian scheme for a family of fourth order equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 405-434. doi: 10.3934/dcds.2017017 [14] Feliz Minhós. Periodic solutions for some fully nonlinear fourth order differential equations. Conference Publications, 2011, 2011 (Special) : 1068-1077. doi: 10.3934/proc.2011.2011.1068 [15] John B. Greer, Andrea L. Bertozzi. $H^1$ Solutions of a class of fourth order nonlinear equations for image processing. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 349-366. doi: 10.3934/dcds.2004.10.349 [16] Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677 [17] Takahiro Hashimoto. Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations. Conference Publications, 2003, 2003 (Special) : 393-402. doi: 10.3934/proc.2003.2003.393 [18] A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044 [19] Zhilin Yang, Jingxian Sun. Positive solutions of a fourth-order boundary value problem involving derivatives of all orders. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1615-1628. doi: 10.3934/cpaa.2012.11.1615 [20] Maria-Magdalena Boureanu. Fourth-order problems with Leray-Lions type operators in variable exponent spaces. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 231-243. doi: 10.3934/dcdss.2019016

2018 Impact Factor: 0.545