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Inertial manifolds and inertial sets for the phase-field equations. (English) Zbl 0758.35040
The authors investigate the evolution system \((*)\) \(\tau\varphi_ t=\xi^ 2\Delta\varphi+\varphi-\varphi^ 3+2u\), \(u_ t+{1\over 2}a\varphi_ t=K\Delta u\) which is known as the “phase field equations”. \(u\) is the temperature, \(\varphi\) an order parameter familiar in the Landau-Ginzburg theory of phase transitions; \(a,\tau,\xi,K\) are physical constants. System \((*)\) is considered in a bounded domain \(\Omega\subseteq R^ n\), \(n\leq 3\) with smooth boundary \(\Gamma\).
One considers an initial boundary value problem in a suitable function space setting and seeks information about the asymptotic behaviour of solution trajectories of \((*)\). The initial data \(\varphi_ 0,u_ 0\) are in \(H^ 1(\Omega)\) and \(L^ 2(\Omega)\), respectively.
Then it is shown (Thm. 1.1) that there exists a unique global solution \(\varphi\in C(R_ +,H^ 1)\), \(u\in C(R_ +,L^ 2)\) which has a number of regularity properties. Thus a global, nonlinear semigroup \(S(t)\), \(t\geq 0\) is defined by this result. It is then shown (Thm. 1.3) that \(S(t)\), \(t\geq 0\) has a global, maximal attractor in \(H^ 2(\Omega)\times H^ 2(\Omega)\), compact and connected in \(H^ 1(\Omega)\times L^ 2(\Omega)\), attracting bounded sets in \(H^ 1(\Omega)\times L^ 2(\Omega)\). The system \((*)\) is then transformed into an equivalent one \((**)\) which is accessible to the methods of R. Temam [Infinite- dimensional dynamical systems in mechanics and physics (1988; Zbl 0662.35001)]. The existence of an inertial manifold for \((**)\) is then proved for dimensions \(n\leq 2\) and for the case of a nonsmooth rectangle. For \(n=3\) the proof breaks down because the spectral gap condition is not satisfied. As a substitute, the existence of a so called inertial set is proved (Thm. 3.2). Finally, the dynamics of the flow is discussed when (under other boundary conditions) it is restricted to surfaces of constant energy.

MSC:
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35K45 Initial value problems for second-order parabolic systems
58D25 Equations in function spaces; evolution equations
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
35R25 Ill-posed problems for PDEs
80A22 Stefan problems, phase changes, etc.
35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
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