Stable patterns in a viscous diffusion equation

Authors:
A. Novick-Cohen and R. L. Pego

Journal:
Trans. Amer. Math. Soc. **324** (1991), 331-351

MSC:
Primary 35K70; Secondary 35K55, 80A15

MathSciNet review:
1015926

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a pseudoparabolic regularization of a forward-backward nonlinear diffusion equation , motivated by the problem of phase separation in a viscous binary mixture. The function is non-monotone, so there are discontinuous steady state solutions corresponding to arbitrary arrangements of phases. We find that any bounded measurable steady state solution satisfying , a.e. is dynamically stable to perturbations in the sense of convergence in measure. In particular, smooth solutions may achieve discontinuous asymptotic states. Furthermore, stable states need not correspond to absolute minimizers of free energy, thus violating Gibbs' principle of stability for phase mixtures.

**[A]**E. C. Aifantis,*On the problem of diffusion in solids*, Acta Mech.**37**(1980), no. 3-4, 265–296 (English, with German summary). MR**586062**, 10.1007/BF01202949**[AB]**G. Andrews and J. M. Ball,*Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity*, J. Differential Equations**44**(1982), no. 2, 306–341. Special issue dedicated to J. P. LaSalle. MR**657784**, 10.1016/0022-0396(82)90019-5**[BFJ]**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.**[C]**J. W. Cahn,*On spinodal decomposition*, Acta Metall.**9**(1961), 795.**[CG]**P. J. Chen and M. E. Gurtin,*On a theory of heat conduction involving two temperatures*, Z. Angew. Math. Phys.**19**(1968), 614-627.**[D]**J. Dieudonné,*Foundations of modern analysis*, Academic Press, New York-London, 1969. Enlarged and corrected printing; Pure and Applied Mathematics, Vol. 10-I. MR**0349288****[Du]**C. J. Dürning,*Differential sorption in viscoelastic fluids*, J. Polym. Sci. Polym. Phys. Ed.**23**(1985), 1831.**[H]**Daniel Henry,*Geometric theory of semilinear parabolic equations*, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR**610244****[JF1]**J. Jäckle and H. L. Frisch,*Relaxation of chemical potential and a generalized diffusion equation*, J. Polym. Sci. Polym. Phys. Ed.**23**(1985), 675-682.**[JF2]**-,*Properties of a generalized diffusion equation with a memory*, J. Chem. Phys.**85**(1986), 1621-27.**[KA]**K. Kuttler and Elias C. Aifantis,*Existence and uniqueness in nonclassical diffusion*, Quart. Appl. Math.**45**(1987), no. 3, 549–560. MR**910461****[P]**Robert L. Pego,*Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability*, Arch. Rational Mech. Anal.**97**(1987), no. 4, 353–394. MR**865845**, 10.1007/BF00280411**[PW]**Murray H. Protter and Hans F. Weinberger,*Maximum principles in differential equations*, Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original. MR**762825****[S]**G. B. Stephenson,*Spinodal decomposition in amorphous systems*, J. Non-Cryst. Sol.**66**(1984), 393-427.**[T1]**Tsuan Wu Ting,*Parabolic and pseudo-parabolic partial differential equations*, J. Math. Soc. Japan**21**(1969), 440–453. MR**0264231****[T2]**Tsuan Wu Ting,*A cooling process according to two-temperature theory of heat conduction*, J. Math. Anal. Appl.**45**(1974), 23–31. MR**0330771****[T3]**Tsuan Wu Ting,*Certain non-steady flows of second-order fluids*, Arch. Rational Mech. Anal.**14**(1963), 1–26. MR**0153255****[T]**Hans Triebel,*Interpolation theory, function spaces, differential operators*, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978. MR**503903**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1991-1015926-7

Keywords:
Pattern formation,
viscosity,
phase separation,
dissipative system,
pseudoparabolic,
nonlinear stability

Article copyright:
© Copyright 1991
American Mathematical Society