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
Famous quotes containing the words mathematical, problem and/or solutions:
“What he loved so much in the plant morphological structure of the tree was that given a fixed mathematical basis, the final evolution was so incalculable.”
—D.H. (David Herbert)
“How much atonement is enough? The bombing must be allowed as at least part-payment: those of our young people who are concerned about the moral problem posed by the Allied air offensive should at least consider the moral problem that would have been posed if the German civilian population had not suffered at all.”
—Clive James (b. 1939)
“Every man is in a state of conflict, owing to his attempt to reconcile himself and his relationship with life to his conception of harmony. This conflict makes his soul a battlefield, where the forces that wish this reconciliation fight those that do not and reject the alternative solutions they offer. Works of art are attempts to fight out this conflict in the imaginative world.”
—Rebecca West (1892–1983)