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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.

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
Full Text: DOI
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