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

[1]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[2]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

[3]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[4]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[5]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[6]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[7]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHum approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055

[8]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[9]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[10]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[11]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[12]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[13]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[14]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[15]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[16]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[17]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[18]

Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020350

[19]

Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074

[20]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]