# American Institute of Mathematical Sciences

April  2013, 33(4): 1657-1697. doi: 10.3934/dcds.2013.33.1657

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

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

Received  September 2011 Revised  July 2012 Published  October 2012

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.
Citation: Flavia Smarrazzo, Andrea Terracina. Sobolev approximation for two-phase solutions of forward-backward parabolic problems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1657-1697. doi: 10.3934/dcds.2013.33.1657
##### References:
 [1] G. Anzellotti, Pairings between measures and functions and compensated compactness,, Ann. Mat. Pura ed Appl., 135 (1983), 293.  doi: 10.1007/BF01781073.  Google Scholar [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.  doi: 10.1137/0524082.  Google Scholar [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.  doi: 10.3934/dcds.2006.16.783.  Google Scholar [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.  doi: 10.1063/1.451190.  Google Scholar [5] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Applied Mathematical Sciences, 121 (1996).   Google Scholar [6] G. Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws,, Arch. Ration. Mech. Anal., 147 (1999), 89.  doi: 10.1007/s002050050146.  Google Scholar [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.  doi: 10.1007/BF01759299.  Google Scholar [8] L. C. Evans and M. Portilheiro, Irreversibility and hysteresis for a forward-backward diffusion equation,, Math. Mod. Meth. Appl. Sci., 14 (2004), 1599.  doi: 10.1142/S0218202504003763.  Google Scholar [9] P. C. Fife, Models for phase separation and their mathematics,, Electron. J. Differential Equations, 48 (2000).   Google Scholar [10] H. L. Frisch and J. Jäckle, Properties of a generalized diffusion equation with memory,, J. Chem. Phys., 85 (1986), 1621.  doi: 10.1063/1.451204.  Google Scholar [11] M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Phys. D, 92 (1996), 178.  doi: 10.1016/0167-2789(95)00173-5.  Google Scholar [12] M. Ghisi and M. Gobbino, Gradient estimates for the Perona-Malik equation,, Math. Ann., 337 (2007), 557.  doi: 10.1007/s00208-006-0047-1.  Google Scholar [13] B. H. Gilding and A. Tesei, The Riemann problem for a forward-backward parabolic equation,, Phys. D, 239 (2010), 291.  doi: 10.1016/j.physd.2009.10.006.  Google Scholar [14] K. Höllig, Existence of infinitely many solutions for a forward backward heat equation,, Trans. Amer. Math. Soc., 278 (1983), 299.  doi: 10.2307/1999317.  Google Scholar [15] K. Höllig and J. A. Nohel, A diffusion equation with a nonmonotone constitutive function,, in, (1983), 409.   Google Scholar [16] P. Lafitte and C. Mascia, Numerical exploration of a forward-backward diffusion equation,, Math. Models Methods. Appl. Sci., 22 (2012).   Google Scholar [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 (1967).   Google Scholar [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.   Google Scholar [19] C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equation,, Arch. Rational Mech. Anal., 163 (2002), 87.   Google Scholar [20] C. Mascia, A. Terracina and A. Tesei, Evolution of stable phases in forward-backward parabolic equations,, in, (2007), 47.   Google Scholar [21] C. Mascia, A. Terracina and A. Tesei, Two-phase entropy solutions of a forward-backward parabolic equation,, Arch. Ration. Mech., 194 (2009), 887.  doi: 10.1007/s00205-008-0185-6.  Google Scholar [22] A. Novick-Cohen and R. L. Pego, Stable patterns in a viscous diffusion equation,, Trans. Amer. Math. Soc., 324 (1991), 331.  doi: 10.1090/S0002-9947-1991-1015926-7.  Google Scholar [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.  doi: 10.1080/03605309808821353.  Google Scholar [24] P. I. Plotnikov, Passing to the limit with respect to viscosity in an equation with variable parabolicity direction,, Diff. Equ., 30 (1994), 614.   Google Scholar [25] P. I. Plotnikov, Equations with alternating direction of parabolicity and the hysteresis effect,, Russian Acad. Sci., 47 (1993), 604.   Google Scholar [26] P. I. Plotnikov, Forward-backward parabolic equations and hysteresis,, J. Math. Sci., 93 (1999), 747.  doi: 10.1007/BF02366851.  Google Scholar [27] F. Smarrazzo, On a class of equations with variable parabolicity direction,, Discrete Contin. Dyn. Syst., 22 (2007), 729.  doi: 10.3934/dcds.2008.22.729.  Google Scholar [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.  doi: 10.4171/IFB/239.  Google Scholar [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.  doi: 10.1137/090763561.  Google Scholar [30] A. Terracina, Qualitative behavior of the two-phase entropy solution of a forward-backward parabolic problem,, SIAM J. Math. Anal., 43 (2011), 228.  doi: 10.1137/090778833.  Google Scholar [31] J. L. Vázquez, "Porous Medium Equation. Mathematical Theory,'', Oxford University Press, (2006).   Google Scholar [32] A. Visintin, Forward-backward parabolic equations and hysteresis,, Calc. Var. Partial Differential Equations, 15 (2002), 115.  doi: 10.1007/s005260100120.  Google Scholar [33] K. Zhang, Existence of infinitely many solutions for the one-dimensional Perona-Malik model,, Calc. Var., 26 (2006), 171.  doi: 10.1007/s00526-005-0363-4.  Google Scholar

show all references

##### References:
 [1] G. Anzellotti, Pairings between measures and functions and compensated compactness,, Ann. Mat. Pura ed Appl., 135 (1983), 293.  doi: 10.1007/BF01781073.  Google Scholar [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.  doi: 10.1137/0524082.  Google Scholar [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.  doi: 10.3934/dcds.2006.16.783.  Google Scholar [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.  doi: 10.1063/1.451190.  Google Scholar [5] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Applied Mathematical Sciences, 121 (1996).   Google Scholar [6] G. Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws,, Arch. Ration. Mech. Anal., 147 (1999), 89.  doi: 10.1007/s002050050146.  Google Scholar [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.  doi: 10.1007/BF01759299.  Google Scholar [8] L. C. Evans and M. Portilheiro, Irreversibility and hysteresis for a forward-backward diffusion equation,, Math. Mod. Meth. Appl. Sci., 14 (2004), 1599.  doi: 10.1142/S0218202504003763.  Google Scholar [9] P. C. Fife, Models for phase separation and their mathematics,, Electron. J. Differential Equations, 48 (2000).   Google Scholar [10] H. L. Frisch and J. Jäckle, Properties of a generalized diffusion equation with memory,, J. Chem. Phys., 85 (1986), 1621.  doi: 10.1063/1.451204.  Google Scholar [11] M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Phys. D, 92 (1996), 178.  doi: 10.1016/0167-2789(95)00173-5.  Google Scholar [12] M. Ghisi and M. Gobbino, Gradient estimates for the Perona-Malik equation,, Math. Ann., 337 (2007), 557.  doi: 10.1007/s00208-006-0047-1.  Google Scholar [13] B. H. Gilding and A. Tesei, The Riemann problem for a forward-backward parabolic equation,, Phys. D, 239 (2010), 291.  doi: 10.1016/j.physd.2009.10.006.  Google Scholar [14] K. Höllig, Existence of infinitely many solutions for a forward backward heat equation,, Trans. Amer. Math. Soc., 278 (1983), 299.  doi: 10.2307/1999317.  Google Scholar [15] K. Höllig and J. A. Nohel, A diffusion equation with a nonmonotone constitutive function,, in, (1983), 409.   Google Scholar [16] P. Lafitte and C. Mascia, Numerical exploration of a forward-backward diffusion equation,, Math. Models Methods. Appl. Sci., 22 (2012).   Google Scholar [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 (1967).   Google Scholar [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.   Google Scholar [19] C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equation,, Arch. Rational Mech. Anal., 163 (2002), 87.   Google Scholar [20] C. Mascia, A. Terracina and A. Tesei, Evolution of stable phases in forward-backward parabolic equations,, in, (2007), 47.   Google Scholar [21] C. Mascia, A. Terracina and A. Tesei, Two-phase entropy solutions of a forward-backward parabolic equation,, Arch. Ration. Mech., 194 (2009), 887.  doi: 10.1007/s00205-008-0185-6.  Google Scholar [22] A. Novick-Cohen and R. L. Pego, Stable patterns in a viscous diffusion equation,, Trans. Amer. Math. Soc., 324 (1991), 331.  doi: 10.1090/S0002-9947-1991-1015926-7.  Google Scholar [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.  doi: 10.1080/03605309808821353.  Google Scholar [24] P. I. Plotnikov, Passing to the limit with respect to viscosity in an equation with variable parabolicity direction,, Diff. Equ., 30 (1994), 614.   Google Scholar [25] P. I. Plotnikov, Equations with alternating direction of parabolicity and the hysteresis effect,, Russian Acad. Sci., 47 (1993), 604.   Google Scholar [26] P. I. Plotnikov, Forward-backward parabolic equations and hysteresis,, J. Math. Sci., 93 (1999), 747.  doi: 10.1007/BF02366851.  Google Scholar [27] F. Smarrazzo, On a class of equations with variable parabolicity direction,, Discrete Contin. Dyn. Syst., 22 (2007), 729.  doi: 10.3934/dcds.2008.22.729.  Google Scholar [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.  doi: 10.4171/IFB/239.  Google Scholar [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.  doi: 10.1137/090763561.  Google Scholar [30] A. Terracina, Qualitative behavior of the two-phase entropy solution of a forward-backward parabolic problem,, SIAM J. Math. Anal., 43 (2011), 228.  doi: 10.1137/090778833.  Google Scholar [31] J. L. Vázquez, "Porous Medium Equation. Mathematical Theory,'', Oxford University Press, (2006).   Google Scholar [32] A. Visintin, Forward-backward parabolic equations and hysteresis,, Calc. Var. Partial Differential Equations, 15 (2002), 115.  doi: 10.1007/s005260100120.  Google Scholar [33] K. Zhang, Existence of infinitely many solutions for the one-dimensional Perona-Malik model,, Calc. Var., 26 (2006), 171.  doi: 10.1007/s00526-005-0363-4.  Google Scholar
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