Contractivity and Asymptotics in Wasserstein Metrics for Viscous Nonlinear Scalar Conservation Laws

José A. Carrillo; Marco Di Francesco; Corrado Lattanzio

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 2, page 277-292
  • ISSN: 0392-4033

Abstract

top
In this work, recent results concerning the long time asymptotics of one- dimensional scalar conservation laws with probability densities as initial data are reviewed and further applied to the case of viscous conservation laws with nonlinear degenerate diffusions. The non-strict contraction of the maximal transport distance together with a uniform expansion of the solutions lead to the existence of time-de- pendent asymptotic profiles for a large class of convection-diffusion problems with fully general nonlinearities and with degenerate diffusion.

How to cite

top

Carrillo, José A., Di Francesco, Marco, and Lattanzio, Corrado. "Contractivity and Asymptotics in Wasserstein Metrics for Viscous Nonlinear Scalar Conservation Laws." Bollettino dell'Unione Matematica Italiana 10-B.2 (2007): 277-292. <http://eudml.org/doc/290403>.

@article{Carrillo2007,
abstract = {In this work, recent results concerning the long time asymptotics of one- dimensional scalar conservation laws with probability densities as initial data are reviewed and further applied to the case of viscous conservation laws with nonlinear degenerate diffusions. The non-strict contraction of the maximal transport distance together with a uniform expansion of the solutions lead to the existence of time-de- pendent asymptotic profiles for a large class of convection-diffusion problems with fully general nonlinearities and with degenerate diffusion.},
author = {Carrillo, José A., Di Francesco, Marco, Lattanzio, Corrado},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {277-292},
publisher = {Unione Matematica Italiana},
title = {Contractivity and Asymptotics in Wasserstein Metrics for Viscous Nonlinear Scalar Conservation Laws},
url = {http://eudml.org/doc/290403},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Carrillo, José A.
AU - Di Francesco, Marco
AU - Lattanzio, Corrado
TI - Contractivity and Asymptotics in Wasserstein Metrics for Viscous Nonlinear Scalar Conservation Laws
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/6//
PB - Unione Matematica Italiana
VL - 10-B
IS - 2
SP - 277
EP - 292
AB - In this work, recent results concerning the long time asymptotics of one- dimensional scalar conservation laws with probability densities as initial data are reviewed and further applied to the case of viscous conservation laws with nonlinear degenerate diffusions. The non-strict contraction of the maximal transport distance together with a uniform expansion of the solutions lead to the existence of time-de- pendent asymptotic profiles for a large class of convection-diffusion problems with fully general nonlinearities and with degenerate diffusion.
LA - eng
UR - http://eudml.org/doc/290403
ER -

References

top
  1. AGUEH, M., Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory, Adv. Differential Equations, 10 (2005), 309-360. Zbl1103.35051MR2123134
  2. AMBROSIO, L. - GIGLI, N. - SAVARÉ, G., Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zurich. Birkhauser Verlag, Basel, 2005. MR2129498
  3. ANDREUCCI, D., Degenerate parabolic equations with initial data measures, Trans. Amer. Math. Soc., 349 (10) (1997), 3911-3923. Zbl0885.35056MR1333384DOI10.1090/S0002-9947-97-01530-4
  4. BÉNILAN, P. - BOUILLET, J. E., On a parabolic equation with slow and fast diffusions, Nonlinear Anal., 26 (4) (1996), 813-822. Zbl0840.35052MR1362754DOI10.1016/0362-546X(94)00321-8
  5. BOLLEY, F. - BRENIER, Y. - LOEPER, G., Contractive metrics for scalar conservation laws, J. Hyperbolic Differ. Equ., 2 (2005), 91-107. Zbl1071.35081MR2134955DOI10.1142/S0219891605000397
  6. BOLLEY, F. - CARRILLO, J. A., Tanaka Theorem for Inelastic Maxwell Models, To appear in Comm. Math. Phys. MR2346391DOI10.1007/s00220-007-0336-x
  7. BRENIER, Y., Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations, Comm. Pure Appl. Math., 52 (1999), 411-452, 1999. Zbl0910.35098MR1658919DOI10.1002/(SICI)1097-0312(199904)52:4<411::AID-CPA1>3.0.CO;2-3
  8. BRENIER, Y., L 2 formulation of multidimensional scalar conservation laws, to appear in Arch. Ration. Mech. Anal. Zbl1180.35346MR2506069DOI10.1007/s00205-009-0214-0
  9. CARLEN, E. - GANGBO, W., Constrained steepest descent in the 2-Wasserstein metric, Ann. of Math., 157 (2003), 807-846. Zbl1038.49040MR1983782DOI10.4007/annals.2003.157.807
  10. CARRILLO, J., Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147 (1999), 269-361. Zbl0935.35056MR1709116DOI10.1007/s002050050152
  11. CARRILLO, J. A. - DI FRANCESCO, M. - GUALDANI, M. P., Semidiscretization and long- time asymptotics of nonlinear diffusion equations, Comm. Math. Sci., 1 (2007), 21-53. Zbl1145.35028MR2301287DOI10.4310/CMS.2007.v5.n5.a4
  12. CARRILLO, J. A. - DI FRANCESCO, M. - LATTANZIO, C., Contractivity of Wasserstein metrics and asymptotic profiles for scalar conservation laws, J. Differential Equations, 231 (2006), 425-458. Zbl1168.35390MR2287892DOI10.1016/j.jde.2006.07.017
  13. CARRILLO, J. A. - DI FRANCESCO, M. - TOSCANI, G., Intermediate asymptotics beyond homogeneity and self-similarity: long time behavior for u t = Δ ϕ ( u ) , Arch. Rational Mech. Anal., 180 (2006), 127-149. Zbl1096.35015MR2211709DOI10.1007/s00205-005-0403-4
  14. CARRILLO, J. A. - GUALDANI, M. P. - TOSCAN, G., Finite speed of propagation in porous media by mass transportation methods, C. R. Acad. Sci. Paris, 338 (2004), 815-818. MR2059493DOI10.1016/j.crma.2004.03.025
  15. CARRILLO, J. A., MCCANN, R. J. - VILLANI, C., Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Rational Mech. Anal., 179 (2006), 217-264. Zbl1082.76105MR2209130DOI10.1007/s00205-005-0386-1
  16. CARRILLO, J. A. - TOSCANI, G., Wasserstein metric and large-time asymptotics of nonlinear diffusion equations, In New trends in mathematical physics, World Sci. Publ., Hackensack, NJ, 2004, 234-244. Zbl1089.76055MR2163983
  17. CARRILLO, J. A. - VÁZQUEZ, J. L., Asymptotic Complexity in Filtration Equations, To appear in J. Evol. Equ. MR2328935DOI10.1007/s00028-006-0298-z
  18. CRANDALL, M. - PIERRE, M., Regularizing effects for u t + A ϕ ( u ) = 0 in L 1 , J. Funct. Anal., 45 (1982), 194-212. Zbl0483.35076MR647071DOI10.1016/0022-1236(82)90018-0
  19. CULLEN, M. - GANGBO, W., A variational approach for the 2-dimensional semi- geostrophic shallow water equations, Arch. Ration. Mech. Anal., 156 (2001) 241-273. Zbl0985.76008MR1816477DOI10.1007/s002050000124
  20. DI FRANCESCO, M. - MARKOWICH, P. A., Entropy dissipation and Wasserstein metric methods for the viscous Burgers' equation: convergence to diffusive waves, In Partial Differential Equations and Inverse Problems, 362 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2004, 145-165. Zbl1062.35111MR2091496DOI10.1090/conm/362/06610
  21. DI FRANCESCO, M. - WUNSCH, M., Large time behavior in Wasserstein spaces and relative entropy for bipolar drift-diffusion-Poisson models, to appear in Monat. Math. Zbl1159.35008MR2395521DOI10.1007/s00605-008-0532-6
  22. EVJE, S. - KARLSEN, K. H., Viscous splitting approximation of mixed hyperbolic- parabolic convection-diffusion equations, Numer. Math., 83 (1) (1999), 107-137. Zbl0961.65084MR1702599DOI10.1007/s002110050441
  23. ESCOBEDO, M. - VÁZQUEZ, J. L. - ZUAZUA, E., Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal., 124 (1) (1993), 43-65. MR1233647DOI10.1007/BF00392203
  24. HOPF, E., The partial differential equation u t + u u x = μ u x x , Comm. Pure Appl. Math., 3 (1950), 201-230. Zbl0039.10403MR47234DOI10.1002/cpa.3160030302
  25. KALASHNIKOV, A. S., Some problems of the qualitative theory of second-order non-linear degenerate parabolic equations, Uspekhi Mat. Nauk, 42 (2(254)) (1987), 135- 176, 287. MR898624
  26. LAURENÇOT, P. - SIMONDON, F., Long-time behaviour for porous medium equations with convection, Proc. Roy. Soc. Edinburgh Sect. A, 128 (2) (1998), 315-336. Zbl0906.35050MR1621331DOI10.1017/S0308210500012816
  27. LIU, T. P. - PIERRE, M., Source-solutions and asymptotic behavior in conservation laws, J. Differential Equations, 51 (1984), 419-441. Zbl0545.35057MR735207DOI10.1016/0022-0396(84)90096-2
  28. LOTT, J. - VILLANI, C., Ricci curvature for metric-measure spaces via optimal transport, To appear in Ann. of Math. Zbl1178.53038MR2480619DOI10.4007/annals.2009.169.903
  29. MCCANN, R. J., Stable rotating binary stars and fluid in a tube, Houston J. Math., 32 (2006), 603-632. Zbl1096.85006MR2219334
  30. OTTO, F., The geometry of dissipative evolution equation: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. Zbl0984.35089MR1842429DOI10.1081/PDE-100002243
  31. OTTO, F. - WESTDICKENBERG, M., Eulerian calculus for the contraction in the wasserstein distance, SIAM J. Math. Anal., 37 (2005), 1227-1255. Zbl1094.58016MR2192294DOI10.1137/050622420
  32. STURM, K. T., Convex functionals of probability measures and nonlinear diffusions on manifolds, J. Math. Pures Appl., 84 (2005) 149-168. Zbl1259.49074MR2118836DOI10.1016/j.matpur.2004.11.002
  33. TOSCANI, G., A central limit theorem for solutions of the porous medium equation, J. Evol. Equ., 5 (2005), 185-203. Zbl1082.35091MR2133441DOI10.1007/s00028-005-0183-1
  34. VÁZQUEZ, J. L., The Porous Medium Equation. New contractivity results, Progress in Nonlinear Differential Equations and Their Applications, 63 (205) (Volume in honor of H. Brezis, Proceedings of Gaeta Congress, June 2004), 433-451. MR2176734DOI10.1007/3-7643-7384-9_42
  35. VILLANI, C., Topics in optimal transportation, 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. MR1964483DOI10.1007/b12016
  36. VILLANI, C., Optimal transport, old and new, Lecture Notes for the 2005 Saint-Flour summer school, to appear in Springer2007. MR2815763DOI10.1007/978-3-642-21216-1

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.