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Large-time behavior of periodic entropy solutions to anisotropic degenerate parabolic-hyperbolic equations

Authors: Gui-Qiang Chen and Benoît Perthame
Journal: Proc. Amer. Math. Soc. 137 (2009), 3003-3011
MSC (2000): Primary 35K65, 35K15, 35B10, 35B40, 35D99; Secondary 35K10, 35B30, 35B41, 35M10, 35L65
Published electronically: April 10, 2009
MathSciNet review: 2506459
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Abstract: We are interested in the large-time behavior of periodic entropy solutions in $ L^\infty$ to anisotropic degenerate parabolic-hyperbolic equations of second order. Unlike the pure hyperbolic case, the nonlinear equation is no longer self-similar invariant, and the diffusion term in the equation significantly affects the large-time behavior of solutions; thus the approach developed earlier, based on the self-similar scaling, does not directly apply. In this paper, we develop another approach for establishing the decay of periodic solutions for anisotropic degenerate parabolic-hyperbolic equations. The proof is based on the kinetic formulation of entropy solutions. It involves time translations and a monotonicity-in-time property of entropy solutions and employs the advantages of the precise kinetic equation for the solutions in order to recognize the role of nonlinearity-diffusivity of the equation.

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  • 1. M. Bendahmane, M. Langlais, and M. Saad, On some anisotropic reaction-diffusion systems with $ L\sp 1$-data modeling the propagation of an epidemic disease, Nonlinear Anal. 54 (2003), no. 4, 617-636. MR 1983439 (2004c:35204)
  • 2. M. C. Bustos, F. Concha, R. Bürger, and E. M. Tory, Sedimentation and Thickening: Phenomenological Foundation and Mathematical Theory, Kluwer Academic Publishers: Dordrecht, Netherlands, 1999. MR 1747460 (2002b:76101)
  • 3. J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Rational Mech. Anal. 147 (1999), 269-361. MR 1709116 (2000m:35132)
  • 4. G. Chavent and J. Jaffre, Mathematical Models and Finite Elements for Reservoir Simulation, North Holland: Amsterdam, 1986.
  • 5. G.-Q. Chen and E. DiBenedetto, Stability of entropy solutions to the Cauchy problem for a class of hyperbolic-parabolic equations, SIAM J. Math. Anal. 33 (2001), 751-762. MR 1884720 (2002m:35125)
  • 6. G.-Q. Chen and H. Frid, Decay of entropy solutions of nonlinear conservation laws, Arch. Rational Mech. Anal. 146(2) (1999), 95-127. MR 1718482 (2000h:35099)
  • 7. G.-Q. Chen and B. Perthame, Well-posedness for nonisotropic degenerate parabolic-hyperbolic equations, Annales de l'Institut Henri Poincaré Analyse Non Linéaire 20 (2003), 645-668. MR 1981403 (2004c:35235)
  • 8. C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, second edition, Springer-Verlag: Berlin, 2005. MR 2169977 (2006d:35159)
  • 9. A.-L. Dalibard and B. Perthame, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc. 361 (2009), 2319-2335.
  • 10. C. De Lellis, F. Otto, and M. Westdickenberg, Structure of entropy solutions for multi-dimensional scalar conservation laws, Arch. Rational Mech. Anal. 170 (2003), 137-184. MR 2017887 (2005c:35191)
  • 11. B. Engquist and W. E, Large time behavior and homogenization of solutions of two-dimensional conservation laws, Comm. Pure Appl. Math. 46 (1993), 1-26. MR 1193341 (94a:35077)
  • 12. M. S. Espedal, A. Fasano, and A. Mikelić, Filtration in Porous Media and Industrial Applications, Lecture Notes in Math. 1734, Springer-Verlag: Berlin, 2000. MR 1816142 (2001i:76096)
  • 13. F. Golse, P.-L. Lions, B. Perthame, and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 76 (1988), 110-125. MR 923047 (89a:35179)
  • 14. P.-E. Jabin, F. Otto, and B. Perthame, Line-energy Ginzburg-Landau models: zero-energy states, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), 187-202. MR 1994807 (2005k:35070)
  • 15. P.-E. Jabin and B. Perthame, Regularity in kinetic formulations via averaging lemmas. A tribute to J. L. Lions, ESAIM Control Optim. Calc. Var. 8 (2002), 761-774 (electronic). MR 1932972 (2003j:35207)
  • 16. K. H. Karlsen and N. H. Risebro, Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients, M2AN Math. Model. Numer. Anal. 35(2) (2001), 239-269. MR 1825698 (2002b:35138)
  • 17. P.-L. Lions, B. Perthame, and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1994), 169-191. MR 1201239 (94d:35100)
  • 18. A. Michel and J. Vovelle, Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods, SIAM J. Numer. Anal. 41 (2003), 2262-2293 (electronic). MR 2034615 (2004k:35228)
  • 19. F. Murat, Compacité par compensation, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), 489-507. MR 506997 (80h:46043a)
  • 20. J. Nolen, G. Papanicolaou, and O. Pironneau, A framework for adaptive multiscale methods for elliptic problems, Multiscale Model. Simul. 7 (2008), 171-196. MR 2399542
  • 21. B. Perthame, Kinetic Formulation of Conservation Laws, Oxford Univ. Press: Oxford, 2002. MR 2064166 (2005d:35005)
  • 22. B. Perthame and P. E. Souganidis, A limiting case for velocity averaging, Ann. Sci. École Norm. Sup. (4) 31 (1998), 591-598. MR 1634024 (99h:82064)
  • 23. B. Perthame and P. E. Souganidis, Dissipative and entropy solutions to non-isotropic degenerate parabolic balance laws, Arch. Rational Mech. Anal. 170 (2003), 359-370. MR 2022136 (2006c:35162a)
  • 24. D. Serre, Systems of Conservation Laws, Cambridge University Press: Cambridge, 2000. MR 1775057 (2001c:35146)
  • 25. E. Tadmor and T. Tao, Velocity averaging, kinetic formulations and regularizing effects in quasilinear PDEs, Comm. Pure Appl. Math. 60 (2007), 1488-1521. MR 2342955 (2008g:35011)
  • 26. L. Tartar, Compensated compactness and applications to partial differential equations, In: Research Notes in Mathematics, 39, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. 4, ed. R.J. Knops, Pitman Press: Boston-London, 1979. MR 584398 (81m:35014)

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Additional Information

Gui-Qiang Chen
Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China – and – Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2730

Benoît Perthame
Affiliation: Université Pierre et Marie Curie, Université Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu, 75005, Paris, France – and – Institut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, France

Keywords: Periodic solutions, entropy solutions, decay, large-time behavior, kinetic formulation, degenerate parabolic equations, anisotropic diffusion, nonlinearity-diffusivity
Received by editor(s): October 16, 2008
Published electronically: April 10, 2009
Additional Notes: The first author’s research was supported in part by the National Science Foundation under Grants DMS-0244473, DMS-0807551, DMS-0720925, and DMS-0505473; an Alexander von Humboldt Foundation Fellowship; and the Natural Science Foundation of China under Grant NSFC-10728101. This paper was written as part of the International Research Program on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during the academic year 2008–09.
Communicated by: Walter Craig
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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