Sobolev approximation for two-phase solutions of forward-backward parabolic problems

Flavia Smarrazzo; Andrea Terracina

doi:10.3934/dcds.2013.33.1657

Article Contents

2013, Volume 33, Issue 4: 1657-1697. Doi: 10.3934/dcds.2013.33.1657

This issue Previous Article Non-integrability of generalized Yang-Mills Hamiltonian system Next Article Well-posedness for a modified two-component Camassa-Holm system in critical spaces

Sobolev approximation for two-phase solutions of forward-backward parabolic problems

Flavia Smarrazzo^1, and
Andrea Terracina^1,

1.
Dipartimento di Matematica "G. Castelnuovo,", Sapienza Università di Roma, Piazzale A. Moro 5, 00185 Roma

Received: September 2011

Revised: July 2012

Published: April 2013

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Abstract

We discuss some properties of a forward-backward parabolic problem that arises in models of phase transition in which two stable phases are separated by an interface. Here we consider a formulation of the problem that comes from a Sobolev approximation of it. In particular we prove uniqueness of the previous problem extending to nonlinear diffusion function a result obtained in [21] in the piecewise linear case. Moreover, we analyze the third order partial differential problem that approximates the forward-backward parabolic one. In particular, for some classes of initial data, we obtain a priori estimates that generalize that proved in [22]. Using these results we study the singular limit of the Sobolev approximation, as a consequence we obtain existence of the forward-backward problem for a class of initial data.

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Mathematics Subject Classification: Primary: 35K20, 35D99; Secondary: 35K70.

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References

[1]	G. Anzellotti, Pairings between measures and functions and compensated compactness, Ann. Mat. Pura ed Appl., 135 (1983), 293-318.doi: 10.1007/BF01781073.
[2]	G. I. Barenblatt, M. Bertsch, R. Dal Passo and M. Ughi, A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow, SIAM J. Math. Anal., 24 (1993), 1414-1439.doi: 10.1137/0524082.
[3]	G. Bellettini, G. Fusco and N. Guglielmi, A concept of solution and numerical experiments for forward-backward diffusion equations, Discrete Contin. Dyn. Syst., 16 (2006), 783-842.doi: 10.3934/dcds.2006.16.783.
[4]	K. Binder, H. L. Frisch and J. Jäckle, Kinetics of phase separation in the presence of slowly relaxing structural variables, J. Chem. Phys., 85 (1986), 1505-1512.doi: 10.1063/1.451190.
[5]	M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Applied Mathematical Sciences, 121, Springer-Verlag, New-York, 1996.
[6]	G. Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal., 147 (1999), 89-118.doi: 10.1007/s002050050146.
[7]	A. De Pablo and J. L. Vazquez, Regularity of solutions and interfaces of a generalized porous medium equation in $\mathbbR^N$, Ann. Mat. Pure Appl., 58 (1991), 51-74.doi: 10.1007/BF01759299.
[8]	L. C. Evans and M. Portilheiro, Irreversibility and hysteresis for a forward-backward diffusion equation, Math. Mod. Meth. Appl. Sci., 14 (2004), 1599-1620.doi: 10.1142/S0218202504003763.
[9]	P. C. Fife, Models for phase separation and their mathematics, Electron. J. Differential Equations, 48 (2000), pp. 26.
[10]	H. L. Frisch and J. Jäckle, Properties of a generalized diffusion equation with memory, J. Chem. Phys., 85 (1986), 1621-1627.doi: 10.1063/1.451204.
[11]	M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192.doi: 10.1016/0167-2789(95)00173-5.
[12]	M. Ghisi and M. Gobbino, Gradient estimates for the Perona-Malik equation, Math. Ann., 337 (2007), 557-590.doi: 10.1007/s00208-006-0047-1.
[13]	B. H. Gilding and A. Tesei, The Riemann problem for a forward-backward parabolic equation, Phys. D, 239 (2010), 291-311.doi: 10.1016/j.physd.2009.10.006.
[14]	K. Höllig, Existence of infinitely many solutions for a forward backward heat equation, Trans. Amer. Math. Soc., 278 (1983), 299-316.doi: 10.2307/1999317.
[15]	K. Höllig and J. A. Nohel, A diffusion equation with a nonmonotone constitutive function, in "Systems of Nonlinear Partial Differential Equations'' , Reidel, Dordrecht-Boston, Mass., (1983), 409-422.
[16]	P. Lafitte and C. Mascia, Numerical exploration of a forward-backward diffusion equation, Math. Models Methods. Appl. Sci., 22 (2012), 1250004 pp. 33.
[17]	O. A. Ladyzenskaja ,V. A. Solonnikov and N. N. Ural&ceva, "Linear and Quasi-linear Equations of Parabolic Type,'' Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R. I., 1967
[18]	H. 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), 401-441.
[19]	C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equation, Arch. Rational Mech. Anal., 163 (2002), 87-124
[20]	C. Mascia, A. Terracina and A. Tesei, Evolution of stable phases in forward-backward parabolic equations, in "Asymptotic Analysis and Singularities'' (edited by H. Kozono, T. Ogawa, K. Tanaka, Y. Tsutsumi and E. Yanagida), Advanced Studies in Pure Mathematics 47-2, Math. Soc. Japan, (2007), 451-478
[21]	C. Mascia, A. Terracina and A. Tesei, Two-phase entropy solutions of a forward-backward parabolic equation, Arch. Ration. Mech., 194 (2009), 887-925.doi: 10.1007/s00205-008-0185-6.
[22]	A. Novick-Cohen and R. L. Pego, Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc., 324 (1991), 331-351.doi: 10.1090/S0002-9947-1991-1015926-7.
[23]	V. Padrón, Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Comm. Partial Differential Equations, 23 (1998), 457-486.doi: 10.1080/03605309808821353.
[24]	P. I. Plotnikov, Passing to the limit with respect to viscosity in an equation with variable parabolicity direction, Diff. Equ., 30 (1994), 614-622.
[25]	P. I. Plotnikov, Equations with alternating direction of parabolicity and the hysteresis effect, Russian Acad. Sci., Dokl., Math., 47 (1993), 604-608.
[26]	P. I. Plotnikov, Forward-backward parabolic equations and hysteresis, J. Math. Sci., 93 (1999), 747-766.doi: 10.1007/BF02366851.
[27]	F. Smarrazzo, On a class of equations with variable parabolicity direction, Discrete Contin. Dyn. Syst., 22 (2007), 729-758.doi: 10.3934/dcds.2008.22.729.
[28]	F. Smarrazzo, Long-time behaviour of two-phase solutions to a class of forward-backward parabolic equations, Interface and Free Boundaries, 12 (2010), 369-408.doi: 10.4171/IFB/239.
[29]	F. Smarrazzo and A. Tesei, Long-time behaviour of solutions to a class of forward-backward parabolic equations, SIAM J. Math. Anal., 42 (2010), 1046-1093.doi: 10.1137/090763561.
[30]	A. Terracina, Qualitative behavior of the two-phase entropy solution of a forward-backward parabolic problem, SIAM J. Math. Anal., 43 (2011), 228-252.doi: 10.1137/090778833.
[31]	J. L. Vázquez, "Porous Medium Equation. Mathematical Theory,'' Oxford University Press, Oxford, 2006
[32]	A. Visintin, Forward-backward parabolic equations and hysteresis, Calc. Var. Partial Differential Equations, 15 (2002), 115-132.doi: 10.1007/s005260100120.
[33]	K. Zhang, Existence of infinitely many solutions for the one-dimensional Perona-Malik model, Calc. Var., 26 (2006), 171-199.doi: 10.1007/s00526-005-0363-4.