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Inertial manifolds for models of compressible gas dynamics. (English) Zbl 0685.76029

The connection between infinite dimensional and finite dimensional dynamical systems, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Boulder/CO 1987, Contemp. Math. 99, 165-179 (1989).
Summary: [For the entire collection see Zbl 0681.00010.]
We present some results relevant to the chaotic dynamics which prevail in the propagation phase boundaries for a van der Waals compressible fluid. The physical problem is characterized by a non-convex equation of state relating pressure to density. This generates possible chaotic spinoidal decomposition regions (that is, chaotically mixed liquid-gas phases) within the non-convex part of the equation of state. Mathematically, the major problem is to construct an inertial manifold for such a slightly dissipative hyperbolic system of conservation laws. The general methods outlined in this workshop [see the articles by P. Constantin and R. Temam; this Proc. Vol.] do not readily insure the existence of an inertial manifold within such a context. In the process, we shall outline new ideas in constructing inertial manifolds for slightly dissipative Hamiltonian systems (of which our model is a special case); primary applications are problems of nonlinear elasticity.

MSC:

76N15 Gas dynamics (general theory)
35L65 Hyperbolic conservation laws

Citations:

Zbl 0681.00010