Abstract
In this paper we devote our attention to a class of weighted ultrafast diffusion equations arising from the problem of quantisation for probability measures. These equations have a natural gradient flow structure in the space of probability measures endowed with the quadratic Wasserstein distance. Exploiting this structure, in particular through the so-called JKO scheme, we introduce a notion of weak solutions, prove existence, uniqueness, BV and H1 estimates, L1 weighted contractivity, Harnack inequalities, and exponential convergence to a steady state.
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Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, 2nd edn. Birkhäuser Verlag, Basel (2008)
Aubin, J.-P.: Un théorème de compacité. C. R. Acad. Sci. Paris 256(24), 5042–5044 (1963)
Bénilan, P., Crandall, M.G.: The continuous dependence on \(\phi \) of solutions of \(u_t-\Delta \phi (u)=0\). Indiana Univ. Math. J. 30(2), 161–177 (1981)
Berryman, J.G., Holland, J.C.: Asymptotic behaviour of the nonlinear differential equation \(n_t=(n^{-1}n_x)_x\). J. Math. Phys. 23(6), 983–987 (1982)
Billinglsey, P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics, 2nd edn. Wiley, New York (1999)
Bonforte, M., Grillo, G., Vázquez, J.L.: Fast diffusion flow on manifolds of nonpositive curvature. J. Evol. Equ. 8(1), 99–128 (2008)
Bonforte, M., Grillo, G., Vázquez, J.L.: Behaviour near extinction for the fast diffusion equation on bounded domains. J. Math. Pures Appl. (9) 97(1), 1–38 (2012)
Bonforte, M., Vázquez, J.L.: Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations. Adv. Math. 223(2), 529–578 (2010)
Bonforte, M., Vázquez, J.L.: A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains. Arch. Ration. Mech. Anal. 218(1), 317–362 (2015)
Bonforte, M., Vázquez, J.L.: Fractional nonlinear degenerate diffusion equations on bounded domains part I. Existence, uniqueness and upper bounds. Nonlinear Anal. TMA 131, 363–398 (2016)
Bouchitté, G., Jimenez, C., Rajesh, M.: Asymptotique d'un problème de positionnement optimal. C. R. Math. Acad. Sci. Paris 335(10), 853–858 (2002)
Bourne, D.P., Roper, S.M.: Centroidal power diagrams, Lloyd's algorithm and applications to optimal location problems. SIAM J. Numer. Anal. 53(6), 2545–2569 (2015)
Caffarelli, L.: A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity. Ann. of Math. 131(1), 129–134 (1990)
Caffarelli, L.: Interior \(W^{2, p}\) estimates for solutions of the Monge-Ampère equation. Ann. Math. 131(1), 135–150 (1990)
Caffarelli, L.: Some regularity properties of solutions of Monge-Ampère equation. Commun. Pure Appl. Math. 44(8–9), 965–969 (1991)
Caglioti, E., Golse, F., Iacobelli, M.: A gradient flow approach to quantisation of measures. Math. Models Methods Appl. Sci. 25(10), 1845–1885 (2015)
Caglioti, E., Golse, F., Iacobelli, M.: Quantization of measures and gradient flows: a perturbative approach in the 2-dimensional case. Ann. Inst. H. Poincaré Anal. Non Linéaire 35(6), 1531–1555 (2018)
Chayes, J.T., Osher, S., Ralston, J.: On singular diffusion equations with applications to self-organized criticality. Commun. Pure Appl. Math. 46(10), 1363–1377 (1993)
Cordero-Erausquin, D.: Sur le transport de mesures périodiques. C. R. Math. Acad. Sci. Paris 329(3), 199–202 (1999)
Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence. Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser Verlag, Basel (1992)
Daskalopoulos, P., Del Pino, M.: On nonlinear parabolic equations of very fast diffusion. Arch. Ration. Mech. Anal. 137(4), 363–380 (1997)
Daskalopoulos, P., Kenig, C.E.: Degenerate Diffusions. Initial Value Problems and Local Regularity Theory. Tracts in Mathematics, European Mathematical Society, Zürich (2007)
De Gennes, P.G.: Wetting: statics and dynamics. Rev. Mod. Phys. 57(3), 827–863 (1985)
De Philippis, G., Mészáros, A.R., Santambrogio, F., Velichkov, B.: \(BV\) estimates in optimal transportation and applications. Arch. Ration. Mech. Anal. 219(2), 829–860 (2016)
De Philippis, G., Figalli, A.: The Monge-Ampère equation and its link to optimal transportation. Bull. Am. Math. Soc. 51(4), 527–580 (2014)
Di Francesco, M., Matthes, D.: Curves of steepest descent are entropy solutions for a class of degenerate convection-diffusion equations Calc. Var. Partial Differ. Equ. 50(1–2), 199–230 (2014)
Di Marino, S., Maury, B., Santambrogio, F.: Measure sweeping processes. J. Convex Anal. 23(2), 567–601 (2016)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, Revised edn. Textbooks in Mathematics. CRC Press, Boca Raton (2015)
Esteban, J.R., Rodríguez, A., Vázquez, J.L.: A nonlinear heat equation with singular diffusivity. Commun. Partial Differ. Equ. 13(8), 985–1039 (1988)
Fejes Tóth, L.: Lagerungen in der Ebene auf der Kugel und im Raum. Grundlehren der mathematischen Wissenschaften. Springer Verlag, Berlin (1953)
Figalli, A.: The Monge-Ampère equation and its applications. Zürich Lectures in Advanced Mathematics. European Mathematical Society, Zürich (2017)
Galaktionov, V.A., Peletier, L.A., Vázquez, J.L.: Asymptotics of the fast-diffusion equation with critical exponent. SIAM J. Math. Anal. 31(5), 1157–1174 (2000)
Graf, S., Luschgy, H.: Foundations of Quantization for Probability Distributions. Lecture Notes in Mathematics. Springer Verlag, Berlin (2000)
Herrero, M.A., Pierre, M.: The Cauchy problem for \(u_t=\Delta u^m\) when \(0<m< 1\). Trans. Am. Math. Soc. 291, 145–158 (1985)
Iacobelli, M.: Asymptotic quantisation for probability measures on Riemannian manifolds. ESAIM Control Optim. Calc. Var. 22(3), 770–785 (2016)
Iacobelli, M.: Asymptotic analysis for a very fast diffusion equation arising from the 1D quantisation problem. Discrete Contin. Dyn. Syst. (to appear)
Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)
Kloeckner, B.: Approximation by finitely supported measures. ESAIM Control Optim. Calc. Var. 18(2), 343–359 (2012)
Lavenant, H., Santambrogio, F.: Optimal density evolution with congestion: \(L^\infty \) bounds via flow interchange techniques and applications to variational mean field games. Commun. Partial Differ. Equ. (to appear)
Lloyd, S.P.: Least squares quantization in PCM. IEEE Trans. Inf. Theory 28(2), 129–137 (1982)
Matthes, D., McCann, R.J., Savaré, G.: A family of nonlinear fourth order equations of gradient flow type. Comm. Partial Differential Equations 34(11), 1352–1397 (2009)
Mérigot, Q.: A multiscale approach to optimal transport. Comput. Graph. Forum 30(5), 1583–1592 (2011)
McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128(1), 153–159 (1997)
Morgan, F., Bolton, R.: Hexagonal economic regions solve the location problem. Am. Math. Mon. 109(2), 165–172 (2002)
Mosconi, S., Tilli, P.: \(\Gamma \)-convergence for the irrigation problem. J. Conv. Anal. 12(1), 145–158 (2005)
Moser, J.: A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations. Commun. Pure Appl. Math. 13(3), 457–468 (1980)
Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001)
Santambrogio, F.: Optimal Transport for Applied Mathematicians. Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Basel (2015)
Santambrogio, F.: Euclidean, metric, and Wasserstein gradient flows: an overview. Bull. Math. Sci. 7(1), 87–154 (2017)
Vázquez, J.L.: Nonexistence of solutions for nonlinear heat equations of fast-diffusion type. J. Math. Pures. Appl. 71(6), 503–526 (1992)
Vázquez, J.L.: Failure of the strong maximum principle in nonlinear diffusion. Existence of needles. Commun. Partial Differ. Equ. 30(9), 1263–1303 (2005)
Vázquez, J.L.: Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Mathematics and Its Applications. Oxford University Press, New York (2006)
Vázquez, J.L.: The Porous Medium Equation. Mathematical Theory. Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2007)
Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics. American Mathematical Society, Providence (2003)
Acknowledgements
MI thanks Matteo Bonforte for his comments to a preliminary version of this manuscript and José A. Carrillo for interesting discussions on this topic during a visit at Imperial College in 2015. FSP thanks the CNA at Carnegie Mellon University for their kind support. FS thanks Imperial College for their warm hospitality in 2017 when this work started, via a CNRS-Imperial fellowship.
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Communicated by D. Kinderlehrer
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Iacobelli, M., Patacchini, F.S. & Santambrogio, F. Weighted Ultrafast Diffusion Equations: From Well-Posedness to Long-Time Behaviour. Arch Rational Mech Anal 232, 1165–1206 (2019). https://doi.org/10.1007/s00205-018-01341-w
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DOI: https://doi.org/10.1007/s00205-018-01341-w