Stable patterns in a viscous diffusion equation
Authors:
A. NovickCohen and R. L. Pego
Journal:
Trans. Amer. Math. Soc. 324 (1991), 331351
MSC:
Primary 35K70; Secondary 35K55, 80A15
MathSciNet review:
1015926
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We consider a pseudoparabolic regularization of a forwardbackward nonlinear diffusion equation , motivated by the problem of phase separation in a viscous binary mixture. The function is nonmonotone, 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. 34, 265–296 (English, with
German summary). MR 586062
(81g:80003), http://dx.doi.org/10.1007/BF01202949
 [AB]
G.
Andrews and J.
M. Ball, Asymptotic behaviour and changes of phase in
onedimensional nonlinear viscoelasticity, J. Differential Equations
44 (1982), no. 2, 306–341. Special issue
dedicated to J. P. LaSalle. MR 657784
(83m:73046), http://dx.doi.org/10.1016/00220396(82)900195
 [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), 15051512.
 [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), 614627.
 [D]
J.
Dieudonné, Foundations of modern analysis, Academic
Press, New YorkLondon, 1969. Enlarged and corrected printing; Pure and
Applied Mathematics, Vol. 10I. MR 0349288
(50 #1782)
 [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, SpringerVerlag, BerlinNew
York, 1981. MR
610244 (83j:35084)
 [JF1]
J. Jäckle and H. L. Frisch, Relaxation of chemical potential and a generalized diffusion equation, J. Polym. Sci. Polym. Phys. Ed. 23 (1985), 675682.
 [JF2]
, Properties of a generalized diffusion equation with a memory, J. Chem. Phys. 85 (1986), 162127.
 [KA]
K.
Kuttler and Elias
C. Aifantis, Existence and uniqueness in nonclassical
diffusion, Quart. Appl. Math. 45 (1987), no. 3,
549–560. MR
910461 (89m:73006)
 [P]
Robert
L. Pego, Phase transitions in onedimensional nonlinear
viscoelasticity: admissibility and stability, Arch. Rational Mech.
Anal. 97 (1987), no. 4, 353–394. MR 865845
(87m:73037), http://dx.doi.org/10.1007/BF00280411
 [PW]
Murray
H. Protter and Hans
F. Weinberger, Maximum principles in differential equations,
SpringerVerlag, New York, 1984. Corrected reprint of the 1967 original. MR 762825
(86f:35034)
 [S]
G. B. Stephenson, Spinodal decomposition in amorphous systems, J. NonCryst. Sol. 66 (1984), 393427.
 [T1]
Tsuan
Wu Ting, Parabolic and pseudoparabolic partial differential
equations, J. Math. Soc. Japan 21 (1969),
440–453. MR 0264231
(41 #8827)
 [T2]
Tsuan
Wu Ting, A cooling process according to twotemperature theory of
heat conduction, J. Math. Anal. Appl. 45 (1974),
23–31. MR
0330771 (48 #9108)
 [T3]
Tsuan
Wu Ting, Certain nonsteady flows of secondorder fluids,
Arch. Rational Mech. Anal. 14 (1963), 1–26. MR 0153255
(27 #3224)
 [T]
Hans
Triebel, Interpolation theory, function spaces, differential
operators, NorthHolland Mathematical Library, vol. 18,
NorthHolland Publishing Co., AmsterdamNew York, 1978. MR 503903
(80i:46032b)
 [A]
 E. C. Aifantis, On the problem of diffusion in solids, Acta Mech. 37 (1980), 265296. MR 586062 (81g:80003)
 [AB]
 G. Andrews and J. M. Ball, Asymptotic behaviour and changes of phase in onedimensional nonlinear viscoelasticity, J. Differential Equations 44 (1982), 306341. MR 657784 (83m:73046)
 [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), 15051512.
 [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), 614627.
 [D]
 J. Dieudonné, Foundations of modern analysis, Academic Press, New York, 1969. MR 0349288 (50:1782)
 [Du]
 C. J. Dürning, Differential sorption in viscoelastic fluids, J. Polym. Sci. Polym. Phys. Ed. 23 (1985), 1831.
 [H]
 D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math. vol. 840, Springer, New York, 1981. MR 610244 (83j:35084)
 [JF1]
 J. Jäckle and H. L. Frisch, Relaxation of chemical potential and a generalized diffusion equation, J. Polym. Sci. Polym. Phys. Ed. 23 (1985), 675682.
 [JF2]
 , Properties of a generalized diffusion equation with a memory, J. Chem. Phys. 85 (1986), 162127.
 [KA]
 K. Kuttler and E. C. Aifantis, Existence and uniqueness in nonclassical diffusion, Quart. Appl. Math. 45 (1987), 549560. MR 910461 (89m:73006)
 [P]
 R. L. Pego, Phase transitions in onedimensional nonlinear viscoelasticity: admissibility and stability, Arch. Rational Mech. Anal. 97 (1987), 353394. MR 865845 (87m:73037)
 [PW]
 M. Protter and H. Weinberger, Maximum principles in differential equations, SpringerVerlag, New York, 1984. MR 762825 (86f:35034)
 [S]
 G. B. Stephenson, Spinodal decomposition in amorphous systems, J. NonCryst. Sol. 66 (1984), 393427.
 [T1]
 T. W. Ting, Parabolic and pseudoparabolic partial differential equations, J. Math. Soc. Japan 21 (1969), 440453. MR 0264231 (41:8827)
 [T2]
 , A cooling process according to twotemperature theory of heat conductions, J. Math. Anal. Appl. 45 (1974), 2331. MR 0330771 (48:9108)
 [T3]
 , Certain nonsteady flows of second order fluids, Arch. Rational Mech. Anal. 14 (1963), 126. MR 0153255 (27:3224)
 [T]
 H. Triebel, Interpolation theory, function spaces and differential operators, NorthHolland, Amsterdam, 1978. MR 503903 (80i:46032b)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
35K70,
35K55,
80A15
Retrieve articles in all journals
with MSC:
35K70,
35K55,
80A15
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199110159267
PII:
S 00029947(1991)10159267
Keywords:
Pattern formation,
viscosity,
phase separation,
dissipative system,
pseudoparabolic,
nonlinear stability
Article copyright:
© Copyright 1991
American Mathematical Society
