Mathematical Theory: The Cauchy Problem For Perturbative Solutions
The rigorous mathematical theory is based on solving the Cauchy problem for the evolution equation (here the partial differential Vlasov-Poisson equation) and proving estimates on the solution.
First a rather complete linearized mathematical theory has been developed since Landau.
Going beyond the linearized equation and dealing with the nonlinearity has been a longstanding problem in the mathematical theory of Landau damping. Previously one mathematical result at the non-linear level was the existence of a class of exponentially damped solutions of the Vlasov-Poisson equation in a circle which had been proved in by means of a scattering technique (this result has been recently extended in). However these existence results do not say anything about which initial data could lead to such damped solutions.
In the recent paper the initial data issue is solved and Landau damping is mathematically established for the first time for the non-linear Vlasov equation. It is proved that solutions starting in some neighborhood (for the analytic or Gevrey topology) of a linearly stable homogeneous stationary solution are (orbitally) stable for all times and are damped globally in time. The damping phenomenon is reinterpreted in terms of transfer of regularity of as a function of and, respectively, rather than exchanges of energy. Large scale variations pass into variations of smaller and smaller scale in velocity space, corresponding to a shift of the Fourier spectrum of as a function of . This shift, well known in linear theory, proves to hold in the non-linear case.
Read more about this topic: Landau Damping
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