All-Time Existence of Smooth Solutions to PDEs of Mixed Type and the Invariant Subspace of Uniform States

AbstractWe prove a general result concerning the all-time existence of smooth solutions of the space-periodic Cauchy problem for a class of PDEs which involve the coupling of a linear with a nonlinear operator. The initial data are assumed to have small deviations from a constant state. Cases of particular interest are hyperbolic-parabolic systems. For their linear part, we develop simple algebraic conditions which guarantee the applicability of our general all-time existence result. Applications include a complex model of magnetogasdynamics, including dispersion due to Hall currents. Results for standard MHD and gasdynamic systems follow as special cases. We also treat multidimensional viscous Boussinesq equations, which are of third order in space. In these applications, the modes corresponding to spatially uniform states will not decay with increasing time, but are associated with a finite dimensional invariant subspace consisting of all constant states. This subspace may consist solely of stationary states or may contain nontrivial dynamics. An example of the latter is provided by the Boussinesq system if Coriolis forces are included. In either case, the technical complications arising from the invariant subspace of constant states makes our result different from corresponding all-time existence results on the whole space or with other boundary conditions