Low-dimensional behavior of the pattern formation Cahn-Hilliard equation. (English) Zbl 0581.35041

Nicolaenko, Basil; Scheurer, Bruno

Trends in the theory and practice of non-linear analysis, Proc. 6th Int. Conf., Arlington/Tex. 1984, North-Holland Math. Stud. 110, 323-336 (1985).

[For the entire collection see Zbl 0564.00006.]
We investigate the fourth-Hilliard parabolic partial differential equation which describes pattern formation in phase transition. Neumann and periodic boundary conditions are considered for a domain in \(R^ n\), \(1\leq n\leq 3\). This equation is characterized by a negative (backward) second order diffusion and multiple steady states for the appropriate range of parameters. We establish compactness of the orbits in \(H^ 1(\Omega)\) and convergence to some steady state. We demonstrate that the Cohn-Hilliard equation admits an intrinsic low dimensional behavior: in \(R^ 1\), the number of determining modes (in a Galerkin expansion) is proportional to \(L^{3/2}\); where \(L\), the diameter of the domain, is also proportional to the number of unstable modes for the linearized equation. Similar results hold for \(n=2,3\).

MSC:

35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40	Asymptotic behavior of solutions to PDEs
35K20	Initial-boundary value problems for second-order parabolic equations

Keywords:

pattern formation; phase transition; Neumann; periodic boundary conditions; second order diffusion; multiple steady states; compactness of the orbits; Cohn-Hilliard equation

Citations:

Zbl 0564.00006