Constructing a Relativistic Heat Flow by Transport Time Steps

Mccann, Robert J. ; Puel, Marjolaine

Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009), p. 2539-2580 / Harvested from Numdam

Détail

Publié le : 2009-01-01

MR 2569908 | Zbl pre05649886

DOI : https://doi.org/10.1016/j.anihpc.2009.06.006

Bibliographie

[1] Agueh Martial, Existence of Solutions to Degenerate Parabolic Equations Via the Monge-Kantorovich Theory, Adv. Differential Equations 10 (3) (2005) 309-360. | MR 2123134 | Zbl 1103.35051

[2] Luigi Ambrosio, Steepest descent flows and applications to spaces of probability measures, Lectures Notes, Santander, July 2004. | Zbl 1115.35012

[3] Ambrosio Luigi, Gigli Nicola, Savaré Giuseppe, Gradient Flows in Metric Spaces and in the Spaces of Probability Measures, Lectures Math. ETH Zurich, Birkhäuser Verlag, Basel, 2005. | MR 2129498 | Zbl 1090.35002

[4] Luigi Ambrosio, Aldo Pratelli, Existence and Stability Results in the ${L}^{1}$ Theory of Optimal Transportation, Lecture Notes in Math. | Zbl 1065.49026

[5] Ambrosio Luigi, Tilli Paolo, Topics on Analysis in Metric Spaces, Oxford Lecture Ser. Math. Appl., vol. 25, Oxford University Press, Oxford, 2004. | MR 2039660 | Zbl 1080.28001

[6] Anzellotti Gianni, Pairing Between Measures and Bounded Functions and Compensated Compactness, Ann. Mat. Pura Appl. (4) 135 (1983) 293-318. | MR 750538 | Zbl 0572.46023

[7] Andreu Fuensanta, Caselles Vicent, Mazón José M., Existence and Uniqueness of Solution for a Parabolic Quasilinear Problem for Linear Growth Functionals With ${L}^{1}$ Data, Math. Ann. 322 (2002) 139-206. | MR 1924690 | Zbl 1056.35085

[8] Andreu Fuensanta, Caselles Vicent, Mazón José M., Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, Progr. Math., vol. 223, Birkhäuser Verlag, 2004. | MR 2033382 | Zbl 1053.35002

[9] Andreu Fuensanta, Caselles Vicent, Mazón José M., A Strongly Degenerate Quasilinear Equation: the Elliptic Case, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (3) (2004) 555-587. | Numdam | MR 2099249 | Zbl 1117.35022

[10] Andreu Fuensanta, Caselles Vicent, Mazón José M., A Strong Degenerate Quasilinear Equation: the Parabolic Case, Arch. Ration. Mech. Anal. (2005). | MR 2185664 | Zbl 1112.35111

[11] Andreu Fuensanta, Caselles Vicent, Mazón José M., A Strongly Degenerate Quasilinear Elliptic Equation, Nonlinear Anal. 61 (2005) 637-669. | Zbl pre02167837

[12] Andreu Fuensanta, Caselles Vicent, Mazón José M., The Cauchy Problem for a Strong Degenerate Quasilinear Equation, J. Eur. Math. Soc. (JEMS) 7 (2005) 361-393. | MR 2156605 | Zbl 1082.35089

[13] Fuensanta Andreu, Vicent Caselles, José M. Mazón, Salvador Moll, The speed of propagation of the support of solutions of a tempered diffusion equation, preprint.

[14] Brenier Yann, Extended Monge-Kantorovich Theory, in: Optimal Transportation and Applications, Martina Franca, 2001, Lecture Notes in Math., vol. 1813, Springer, Berlin, 2003, pp. 91-121. | MR 2006306 | Zbl 1064.49036

[15] Brezis Haim, Analyse Fonctionnelle Et Ses Applications, Masson, 1983. | MR 697382 | Zbl 0511.46001

[16] Brezis Haim, Opérateurs Maximaux Monotones Et Semi-Groupes De Contractions Dans Les Espaces De Hilbert, North-Holland Math. Stud., vol. 5, North-Holland Publishing Co./American Elsevier Publishing Co., Inc., Amsterdam, London/New York, 1973, (in French). | MR 348562 | Zbl 0252.47055

[17] Caffarelli Luis A., Boundary Regularity of Maps With Convex Potentials. II, Ann. of Math. (2) 144 (3) (1996) 453-496. | MR 1426885 | Zbl 0916.35016

[18] Carrillo Jose Antonio, Toscani Giuseppe, Wasserstein Metric and Large-Time Asymptotics of Nonlinear Diffusion Equations, in: New Trends in Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2005, pp. 234-244, (in honor of the Salvatore Rionero 70th birthday). | MR 2163983 | Zbl 1089.76055

[19] Caselles Vicent, Convergence of the “relativistic” Heat Equation to the Heat Equation as $c\rightarrow \infty $, Publ. Mat. 51 (1) (2007) 121-142. | MR 2307149 | Zbl 1140.35466

[20] Chertock Alina, Kurganov Alexander, Rosenau Philip, Formation of Discontinuities in Flux-Saturated Degenerate Parabolic Equations, Nonlinearity 16 (2003) 1875-1898. | MR 2012846 | Zbl 1049.35111

[21] Cordero-Erausquin Dario, Non-Smooth Differential Properties of Optimal Transport, in: Recent Advances in the Theory and Applications of Mass Transport, Contemp. Math., vol. 353, Amer. Math. Soc., Providence, RI, 2004, pp. 61-71. | MR 2079070 | Zbl 1141.49040

[22] Dal Maso Gianni, Integral Representation on $\mathrm {BV}\left(\omega \right)$ of Γ-Limits of Variational Integrals, Manuscripta Math. 30 (1980) 387-416. | MR 567216 | Zbl 0435.49016

[23] De Cicco Virginia, Fusco Nicola, Verde Anna, On ${L}^{1}$-Lower Semicontinuity in BV, J. Convex Anal. 12 (1) (2005) 173-185. | MR 2135805 | Zbl 1115.49011

[24] Dunford Nelson, Schwartz Jaco, Linear Operators, Interscience Publishers, New York, 1958. | MR 117523 | Zbl 0084.10402

[25] Evans Lawrence C., Weak Convergence Methods for Nonlinear Partial Differential Equations, published for the Conference Board of the Mathematical Sciences, Washington, DC, CBMS Reg. Conf. Ser. Math., vol. 74, Amer. Math. Soc., Providence, RI, 1990. | MR 1034481 | Zbl 0698.35004

[26] Evans Lawrence C., Gangbo Wilfrid, Differential Equation Methods for the Monge Kantorovich Mass Transfer, Mem. Amer. Math. Soc. 653 (1999). | MR 1464149 | Zbl 0920.49004

[27] Evans Lawrence C., Gariepy Ronald F., Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, 1992. | MR 1158660 | Zbl 0804.28001

[28] Gangbo Wilfrid, Mccann Robert J., Optimal Maps in Monge's Mass Transport Problem, C. R. Acad. Sci. Paris Sér. I Math. 321 (12) (1995) 1653-1658. | MR 1367824 | Zbl 0858.49002

[29] Gangbo Wilfrid, Mccann Robert J., The Geometry of Optimal Transportation, Acta Math. 177 (2) (1996) 113-161. | MR 1440931 | Zbl 0887.49017

[30] Jordan Richard, Kinderlehrer David, Otto Felix, The Variational Formulation of the Fokker-Planck Equation, SIAM J. Math. Anal. 29 (1) (1998) 1-17. | MR 1617171 | Zbl 0915.35120

[31] Robert Kohn, Roger Temam, Dual Spaces of Stresses and Strains, With Applications to Hencky Plasticity, Appl. Math. Optim. 10 (1) (1983) 1-35. | MR 701898 | Zbl 0532.73039

[32] Lions Jacques-Louis, Quelques Methodes De Résolution Des Problèmes Aux Limites Non Linéaires, Dunod, 1969. | MR 259693 | Zbl 0189.40603

[33] Loeper Grégoire, On the Regularity of the Polar Factorization for Time Dependent Maps, Calc. Var. Partial Differential Equations 22 (3) (2005) 343-374. | MR 2118903 | Zbl 1116.35033

[34] Ma Xi-Nan, Trudinger Neil S., Wang Xu-Jia, Regularity of Potential Functions of the Optimal Transportation Problem, Arch. Ration. Mech. Anal. 177 (2) (2005) 151-183. | MR 2188047 | Zbl 1072.49035

[35] Mccann Robert J., A Convexity Principle for Interacting Gases, Adv. Math. 128 (1997) 153-179. | MR 1451422 | Zbl 0901.49012

[36] Mccann Robert J., Exact Solutions to the Transportation Problem on the Line, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 455 (1984) (1999) 1341-1380. | MR 1701760 | Zbl 0947.90010

[37] Mihalas D., Mihalas B., Foundations of Radiation Hydrodynamics, Oxford University Press, 1984. | MR 781346 | Zbl 0651.76005

[38] Felix Otto, Doubly degenerate diffusion equations as steepest descent, preprint, 1996. | MR 1415045

[39] Rachev Svetlozar T., Rüschendorf Ludger, Mass Transportation Problems, Vols. I. and II, Probab. Appl. (N. Y.), Springer-Verlag, New York, 1998. | MR 1619170 | Zbl 0990.60500

[40] Rosenau Philip, Tempered Diffusion: a Transport Process With Propagating Fronts and Initial Delay, Phys. Rev. A 46 (1992) 7371-7374.

[41] Simon Jacques, Compact sets in the space ${L}^{p}(0,T;B)$, Ann. Mat. Pura Appl. 146 (1987) 65-96. | MR 916688 | Zbl 0629.46031

[42] Temam Roger, Navier-Stokes Equation. Theory and Numerical Analysis, Stud. Math. Appl., vol. 2, third ed., North-Holland Publishing Co., Amsterdam, 1984. | MR 769654 | Zbl 0568.35002

[43] Villani Cedric, Topics in Optimal Transportation, Grad. Stud. Math., vol. 58, Amer. Math. Soc., 2003. | MR 1964483 | Zbl 1106.90001