Dispersie evolution of the particles and scattering solutions of the Boltzman equation

In a seminal work, Perthame and Lions applied the velocity averaging method to solutions of the Kinetic-transport equation to prove that the total energy within any bounded set of the spatial variable is integrable over time thereby establishing an analogy to the Morawetz estimate for the nonlinear Schrodinger equation.  In this talk, I will sketch a new approach using only conservation laws, to establish a stronger version of the aforementioned result.  The new idea is based on an uncertainty principle, and the introduction of blind cones with respect to an observer.  Moreover, this framework leads to a new estimate analogous to the interaction-Morawetz estimate for the nonlinear Schrodinger equation. Classically, it is known that the integrability over time of the energy within a bounded set implies dispersion of the particles under certain assumptions on the solution. In the new framework, we will discuss some physically meaningful applications for these estimates by showing that interactions as well as the total mass of particles concentrate within a specific collection of arbitrarily acute blind cones with respect to any observer. In fact, as the uncertainty inevitably increases with time, particles will move away in an asymptotically radial manner from any fixed observer thereby establishing a more general notion of dispersion.  These results are also applicable to the Boltzmann equation. The scattering theory of mild solutions to the Boltzmann equation near an equilibrium over the whole space has been studied by Bardos, Gamba, Golse and Levermore for both soft and hard potentials.  For the case of the hard spheres, they proved that any solution of the Boltzmann equation that is uniformly bounded by a Maxwellian, will scatter to a unique linear state in $L^{1}$ norm and that the mapping from the initial values to their corresponding scattered linear states is injective. Therefore, the only solution that scatters to Maxwellian is itself. The new framework improves on their work by proving the asymptotic completeness of the solutions of the Boltzmann equation near an equilibrium in the  $L^\infty$ setting. In particular, for any solution to the transport equation, there are arbitrarily close in $L^\infty$  norm, scattered solutions of the Boltzmann equation. From the physical point of view, the fact that solutions of the Boltzmann equation defined over the whole space will not necessarily converge to a state of thermodynamic equilibrium challenges the popular thermodynamic hypothesis of the disappearance of free energy.