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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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
DOI: https://doi.org/10.1090/S0002-9947-1991-1015926-7
MathSciNet review: 1015926
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Abstract: We consider a pseudoparabolic regularization of a forward-backward nonlinear diffusion equation $ {u_t} = \Delta (f(u) + \nu {u_t})$, motivated by the problem of phase separation in a viscous binary mixture. The function $ f$ 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 $ u(x)$ satisfying $ f(u) = {\text{constant}}$, $ f'(u(x)) > 0$ 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.


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DOI: https://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