How parabolic free boundaries approximate hyperbolic fronts

Gilding, Brian; Natalini, Roberto; Tesei, Alberto

doi:10.1090/S0002-9947-99-02236-9

How parabolic free boundaries approximate hyperbolic fronts
HTML articles powered by AMS MathViewer

by Brian H. Gilding, Roberto Natalini and Alberto Tesei PDF

Trans. Amer. Math. Soc. 352 (2000), 1797-1824 Request permission

Abstract:

A rather complete study of the existence and qualitative behaviour of the boundaries of the support of solutions of the Cauchy problem for nonlinear first-order and second-order scalar conservation laws is presented. Among other properties, it is shown that, under appropriate assumptions, parabolic interfaces converge to hyperbolic ones in the vanishing viscosity limit.

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
L. Alvarez and J. I. Diaz, Sufficient and necessary initial mass conditions for the existence of a waiting time in nonlinear-convection processes, J. Math. Anal. Appl. 155 (1991), no. 2, 378–392. MR 1097289, DOI 10.1016/0022-247X(91)90008-N
L. Alvarez, J. I. Diaz, and R. Kersner, On the initial growth of the interfaces in nonlinear diffusion-convection processes, Nonlinear diffusion equations and their equilibrium states, I (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 12, Springer, New York, 1988, pp. 1–20. MR 956055, DOI 10.1007/978-1-4613-9605-5_{1}
Sigurd Angenent, Analyticity of the interface of the porous media equation after the waiting time, Proc. Amer. Math. Soc. 102 (1988), no. 2, 329–336. MR 920995, DOI 10.1090/S0002-9939-1988-0920995-1
D. G. Aronson, The porous medium equation, Nonlinear diffusion problems (Montecatini Terme, 1985) Lecture Notes in Math., vol. 1224, Springer, Berlin, 1986, pp. 1–46. MR 877986, DOI 10.1007/BFb0072687
D. G. Aronson and J. L. Vázquez, Eventual $C^\infty$-regularity and concavity for flows in one-dimensional porous media, Arch. Rational Mech. Anal. 99 (1987), no. 4, 329–348. MR 898714, DOI 10.1007/BF00282050
P. Bénilan, Evolution Equations and Accretive Operators, Lecture Notes taken by S. Lenhart, University of Kentucky, 1981.
Tung Chang and Ling Hsiao, The Riemann problem and interaction of waves in gas dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 41, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR 994414
Carmen Cortázar and Manuel Elgueta, Localization and boundedness of the solutions of the Neumann problem for a filtration equation, Nonlinear Anal. 13 (1989), no. 1, 33–41. MR 973366, DOI 10.1016/0362-546X(89)90032-1
C. M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26 (1977), no. 6, 1097–1119. MR 457947, DOI 10.1512/iumj.1977.26.26088
C. M. Dafermos, Regularity and large time behaviour of solutions of a conservation law without convexity, Proc. Roy. Soc. Edinburgh Sect. A 99 (1985), no. 3-4, 201–239. MR 785530, DOI 10.1017/S0308210500014256
Jesus Ildefonso Diaz and Robert Kersner, Non existence d’une des frontières libres dans une équation dégénérée en théorie de la filtration, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 12, 505–508 (French, with English summary). MR 702558
Jesus Ildefonso Diaz and Robert Kersner, On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium, J. Differential Equations 69 (1987), no. 3, 368–403. MR 903393, DOI 10.1016/0022-0396(87)90125-2
N. O. Maksimova, A theorem of Phragmén-Lindelöf type for generalized solutions of quasilinear parabolic equations of nonstationary filtration type, Problems in applied mathematics and information science (Russian), “Nauka”, Moscow, 1987, pp. 262–274, 296 (Russian). MR 939825
J. I. Diaz and S. I. Shmarëv, On the behaviour of the interface in nonlinear processes with convection dominating diffusion via Lagrangian coordinates, Adv. Math. Sci. Appl. 1 (1992), no. 1, 19–45. MR 1161482
C. J. van Duyn and L. A. Peletier, A class of similarity solutions of the nonlinear diffusion equation, Nonlinear Anal. 1 (1976/77), no. 3, 223–233. MR 511671, DOI 10.1016/0362-546X(77)90032-3
I. M. Gel′fand, Some problems in the theory of quasi-linear equations, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 87–158 (Russian). MR 0110868
B. H. Gilding, Properties of solutions of an equation in the theory of infiltration, Arch. Rational Mech. Anal. 65 (1977), no. 3, 203–225. MR 447847, DOI 10.1007/BF00280441
B. H. Gilding, A nonlinear degenerate parabolic equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 3, 393–432. MR 509720
B. H. Gilding, The occurrence of interfaces in nonlinear diffusion-advection processes, Arch. Rational Mech. Anal. 100 (1988), no. 3, 243–263. MR 918796, DOI 10.1007/BF00251516
J. Horn, Über eine hypergeometrische Funktion zweier Veränderlichen, Monatsh. Math. Phys. 47 (1939), 359–379 (German). MR 91, DOI 10.1007/BF01695508
B. H. Gilding, Localization of solutions of a nonlinear Fokker-Planck equation with Dirichlet boundary conditions, Nonlinear Anal. 13 (1989), no. 10, 1215–1240. MR 1020728, DOI 10.1016/0362-546X(89)90008-4
B. H. Gilding and R. Kersner, Instantaneous shrinking in nonlinear diffusion-convection, Proc. Amer. Math. Soc. 109 (1990), no. 2, 385–394. MR 1007496, DOI 10.1090/S0002-9939-1990-1007496-9
Brian H. Gilding and Robert Kersner, Diffusion-convection-réaction, frontières libres et une équation intégrale, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 11, 743–746 (French, with English summary). MR 1139830
B. H. Gilding and R. Kersner, The characterization of reaction-convection-diffusion processes by travelling waves, J. Differential Equations 124 (1996), no. 1, 27–79. MR 1368061, DOI 10.1006/jdeq.1996.0002
Edwige Godlewski and Pierre-Arnaud Raviart, Hyperbolic systems of conservation laws, Mathématiques & Applications (Paris) [Mathematics and Applications], vol. 3/4, Ellipses, Paris, 1991. MR 1304494
R. E. Grundy, Asymptotic solution of a model nonlinear convective diffusion equation, IMA J. Appl. Math. 31 (1983), no. 2, 121–137. MR 728117, DOI 10.1093/imamat/31.2.121
M. Guedda, D. Hilhorst, and C. Picard, The one-dimensional porous medium equation with convection: continuous differentiability of interfaces after the waiting time, Appl. Math. Lett. 5 (1992), no. 1, 59–62. MR 1144369, DOI 10.1016/0893-9659(92)90137-X
K. Höllig and H.-O. Kreiss, $C^\infty$-regularity for the porous medium equation, Math. Z. 192 (1986), no. 2, 217–224. MR 840825, DOI 10.1007/BF01179424
A. S. Kalašnikov, Construction of generalized solutions of quasi-linear equations of first order without convexity conditions as limits of solutions of parabolic equations with a small parameter, Dokl. Akad. Nauk SSSR 127 (1959), 27–30 (Russian). MR 0108651
A. S. Kalašnikov, The nature of the propagation of perturbations in processes that can be described by quasilinear degenerate parabolic equations, Trudy Sem. Petrovsk. Vyp. 1 (1975), 135–144 (Russian). MR 0425367
A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Uspekhi Mat. Nauk 42 (1987), no. 2(254), 135–176, 287 (Russian). MR 898624
Stanley Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math. 16 (1963), 305–330. MR 160044, DOI 10.1002/cpa.3160160307
R. Keršner, Conditions for the localization of heat perturbations in a semibounded moving medium in the presence of absorption, Vestnik Moskov. Univ. Ser. I Mat. Meh. 31 (1976), no. 4, 52–58 (Russian, with English summary). MR 0425360
R. Kersner, R. Natalini, and A. Tesei, Shocks and free boundaries: the local behaviour, Asymptotic Anal. 10 (1995), no. 1, 77–93. MR 1319013
S. N. Kružkov, Results on the nature of the continuity of solutions of parabolic equations, and certain applications thereof, Mat. Zametki 6 (1969), 97–108 (Russian). MR 249832
S. N. Kružkov, Generalized solutions of the Cauchy problem in the large for first order nonlinear equations, Dokl. Akad. Nauk. SSSR 187 (1969), 29–32 (Russian). MR 0249805
S. N. Kružkov, The Cauchy problem for certain classes of quasilinear parabolic equations, Mat. Zametki 6 (1969), 295–300 (Russian). MR 259371
S. N. Kružkov, First order quasilinear equations with several independent variables. , Mat. Sb. (N.S.) 81 (123) (1970), 228–255 (Russian). MR 0267257
S. N. Kružkov and F. Hil′debrand, The Cauchy problem for first order quasilinear equations in the case when the domain of dependence on the initial data is infinite, Vestnik Moskov. Univ. Ser. I Mat. Meh. 29 (1974), no. 1, 93–100 (Russian, with German summary). Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ. (Russian). MR 0344653
S. N. Kruzhkov and E. Yu. Panov, First-order conservative quasilinear laws with an infinite domain of dependence on the initial data, Dokl. Akad. Nauk SSSR 314 (1990), no. 1, 79–84 (Russian); English transl., Soviet Math. Dokl. 42 (1991), no. 2, 316–321. MR 1118483
L. C. Young, On an inequality of Marcel Riesz, Ann. of Math. (2) 40 (1939), 567–574. MR 39, DOI 10.2307/1968941
Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0350216
Hiroshi Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 2, 401–441. MR 672070
L. A. Peletier, The porous media equation, Applications of nonlinear analysis in the physical sciences (Bielefeld, 1979), Surveys Reference Works Math., vol. 6, Pitman, Boston, Mass.-London, 1981, pp. 229–241. MR 659697
D. H. Sattinger, On the total variation of solutions of parabolic equations, Math. Ann. 183 (1969), 78–92. MR 251379, DOI 10.1007/BF01361263
S. I. Shmarëv, The instantaneous origin of singularities in a solution of a degenerate parabolic equation, Sibirsk. Mat. Zh. 31 (1990), no. 4, 166–179 (Russian); English transl., Siberian Math. J. 31 (1990), no. 4, 671–682 (1991). MR 1083744, DOI 10.1007/BF00970640
S. I. Shmarëv, Properties of free boundaries for a class of degenerate parabolic equations in filtration theory, Dokl. Akad. Nauk 326 (1992), no. 3, 431–435 (Russian); English transl., Russian Acad. Sci. Dokl. Math. 46 (1993), no. 2, 296–300. MR 1203223
S. I. Shmarëv, On a degenerate parabolic equation in filtration theory: monotonicity and $C^\infty$-regularity of interface, Adv. Math. Sci. Appl. 5 (1995), no. 1, 1–29. MR 1325956
Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
A. N. Stokes, Intersections of solutions of nonlinear parabolic equations, J. Math. Anal. Appl. 60 (1977), no. 3, 721–727. MR 463695, DOI 10.1016/0022-247X(77)90011-7
Juan Luis Vázquez, Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium, Trans. Amer. Math. Soc. 277 (1983), no. 2, 507–527. MR 694373, DOI 10.1090/S0002-9947-1983-0694373-7
Juan Luis Vázquez, Regularity of solutions and interfaces of the porous medium equation via local estimates, Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), no. 1-2, 1–13. MR 1007534, DOI 10.1017/S0308210500028146
Juan Luis Vázquez, An introduction to the mathematical theory of the porous medium equation, Shape optimization and free boundaries (Montreal, PQ, 1990) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 380, Kluwer Acad. Publ., Dordrecht, 1992, pp. 347–389. MR 1260981