Local exponential attractors for models of phase change for compressible gas dynamics

The authors investigate a model for dynamic phase transitions in a van der Waals compressible fluid. As the pressure is given by a nonconvex equation of state, which also blows up for a finite volume, the corresponding initial value problem is of mixed hyperbolic-elliptic type. Therefore, it generates nontrivial dynamics. The system of conservation laws when regularized with capillarity terms excludes the appearance of shocks but keeps most of the interesting dynamics. By introducing the appropriate Hamiltonian function, local invariant domains are constructed that avoid the blow-up and at the same time allow solutions of mixed type. They show that, even in regions of mixed type, the initial value problem exhibits finite-dimensional dynamical behaviour by establishing the existence of local attractors and of exponential attractors of finite fractal dimension.