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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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