THE ZERO DIFFUSION LIMIT FOR NONLINEAR HYPERBOLIC SYSTEM WITH DAMPING AND DIFFUSION

We consider the Cauchy problem for a nonlinear hyperbolic system with damping and diffusion. Thanks to a suitably constructed corrector function, we can eliminate the layer at infinity and by using the energy method we establish the global existence of solutions if the initial data is a small perturbation around the corresponding linear diffusion waves. Furthermore, we study the zero diffusion limit and, precisely, we show that the sequence of solutions converges to the corresponding hyperbolic system as the diffusion parameter tends to zero.

THE VLASOV–POISSON EQUATIONS AS THE SEMICLASSICAL LIMIT OF THE SCHRÖDINGER–POISSON EQUATIONS: A NUMERICAL STUDY

In this paper, we numerically study the semiclassical limit of the Schrödinger–Poisson equations as a selection principle for the weak solution of the Vlasov–Poisson in one space dimension. Our numerical results show that this limit gives the weak solution that agrees with the zero diffusion limit of the Fokker–Planck equation. We also numerically justify the multivalued solution given by a moment system of the Vlasov–Poisson equations as the semiclassical limit of the Schrödinger–Poisson equations.

Gravito-inertial waves in a rotating stratified sphere or spherical shell

The properties of gravito-inertial waves propagating in a stably stratified rotating spherical shell or sphere are investigated using the Boussinesq approximation. In the perfect fluid limit, these modes obey a second-order partial differential equation of mixed type. Characteristics propagating in the hyperbolic domain are shown to follow three kinds of orbits: quasi-periodic orbits which cover the whole hyperbolic domain; periodic orbits which are strongly attractive; and finally, orbits ending in a wedge formed by one of the boundaries and a turning surface. To these three types of orbits, our calculations show that there correspond three kinds of modes and give support to the following conclusions. First, with quasi-periodic orbits are associated regular modes which exist at the zero-diffusion limit as smooth square-integrable velocity fields associated with a discrete set of eigenvalues, probably dense in some subintervals of [0, N], N being the Brunt–Väisälä frequency. Second, with periodic orbits are associated singular modes which feature a shear layer following the periodic orbit; as the zero-diffusion limit is taken, the eigenfunction becomes singular on a line tracing the periodic orbit and is no longer square-integrable; as a consequence the point spectrum is empty in some subintervals of [0, N]. It is also shown that these internal shear layers contain the two scales E1/3 and E1/4 as pure inertial modes (E is the Ekman number). Finally, modes associated with characteristics trapped by a wedge also disappear at the zero-diffusion limit; eigenfunctions are not square-integrable and the corresponding point spectrum is also empty.