zbMATH — the first resource for mathematics

On the Cahn-Hilliard equation with a logarithmic free energy. (English) Zbl 0831.35088
Let \(\Omega \subset \mathbb{R}^n\) be a bounded domain with smooth boundary, \(\nu > 0\) and \(0 < \theta < \theta_c < \infty\). The authors show that \[ \partial_t u - \Delta \left[ \nu \Delta u - \theta_c u + {\theta \over 2} \ln {1 + u \over 1 - u} \right] = 0 \quad \text{on } \Omega \times (0, \infty),\;\partial_nu \equiv 0 \quad \text{on } \partial \Omega \times (0, \infty) \] induces a solution semiflow on \(\{\gamma \in L^\infty (\Omega) : |\gamma |\leq 1\), \(|\int_\Omega \gamma |< \text{meas} (\Omega)\}\) equipped with the \(L^2\)- topology and on a corresponding \(H^1\)-set. They also establish that this semiflow has a global attractor and give estimates for the dimension of that attractor. The case of periodic boundary conditions is treated at the same time.
Reviewer: G.Hetzer (Auburn)

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
47H20 Semigroups of nonlinear operators
Full Text: DOI
[1] Cahn, J.W., On spinodal decomposition, Acta metall., 9, 795-801, (1961)
[2] Cahn, J.W.; Hilliard, J.E., Free energy of a non-uniform system I. interfacial free energy, J. chem. phys., 2, 258-267, (1958)
[3] ELLIOT C. M. & LUCKHAUSS S., A generalized equation for phase separation of a multi-component mixture with interfacial free energy, (to appear).
[4] Novick-Cohen, A.; Segel, L.A., Nonlinear aspects of the Cahn-Hilliard equation, Physica D, 10, 277-298, (1984)
[5] NICOLAENKO B. & SCHEURER B., Low dimensional behavior of the pattern formation Cahn-Hilliard equation, in Trends in the Theory and Practice of Nonlinear Analysis. · Zbl 0581.35041
[6] Nicolaenko, B.; Scheurer, B.; Temam, R., Some global dynamical properties of a class of pattern formation equations, Communs. partial diff. eqns, 14, 2, 245-297, (1985) · Zbl 0691.35019
[7] Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, (1988), Springer New York · Zbl 0662.35001
[8] Hale, J.K., Asymptotic behavior of dissipative systems, () · Zbl 0117.30403
[9] Hale, J.K.; Raugel, G., Lower semicontinuity of attractors of gradient systems and applications, Annali mat. pura appl., CLIV, 281-326, (1989) · Zbl 0712.47053
[10] Constantin, P.; Foias, C.; Temam, R., Attractors representing turbulent flows, Mem. am. math. soc., 53, 314, (1985) · Zbl 0567.35070
[11] Babin, A.V.; Vishik, M.I., Regular attractors of semigroups and evolution equations, J. math. pures appl., 62, 441-491, (1983) · Zbl 0565.47045
[12] Lions, J.-L., Quelques méthodes de résolution des problémes aux limites non linéaires, (1969), Dunod Paris · Zbl 0189.40603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.