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

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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. CarrilloS. LisiniG. 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. DolbeaultB. 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. GianazzaG. 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. JordanD. 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. LisiniD. 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. LoiblD. 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. MatthesR. 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. CarrilloS. LisiniG. 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. DolbeaultB. 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. GianazzaG. 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. JordanD. 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. LisiniD. 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. LoiblD. 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. MatthesR. 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

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