Studies in the Dynamics of a Plasma

Weitzner, Harold

doi:10.1002/cpa.21399

Cited by 1 publication

(1 citation statement)

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“…One should expect substantial differences when examining higher order regularity of the weak solutions. We hope that the theory presented here will be supplemented by an adequate regularity theory and that it may prove useful for obtaining reliable estimates for the energy absorption at points of possible plasma heating (an important question which remains open as noted recently by Weitzner in Section 4 of [29]). Equations of mixed Tricomi and Keldysh forms have also been studied recently by Chen [5], although the equations considered there are not in divergence form and hence do not have a natural variational structure in general.…”

Section: Introductionmentioning

confidence: 94%

On the Dirichlet Problem of Mixed Type for Lower Hybrid Waves in Axisymmetric Cold Plasmas

Lupo

Monticelli

Payne

2014

Arch Rational Mech Anal

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For a class of linear second order partial differential equations of mixed elliptic-hyperbolic type, which includes a well known model for analyzing possible heating in axisymmetric cold plasmas, we give results on weak well-posedness of the Dirichlet problem and show that such solutions are characterized by a variational principle. The weak solutions are shown to be saddle points of natural functionals suggested by the divergence form of the PDEs. Moreover, the natural domains of the functionals are the weighted Sobolev spaces to which the solutions belong. In addition, all critical levels will be characterized in terms of global extrema of the functionals restricted to suitable infinite dimensional linear subspaces. These subspaces are defined in terms of a robust spectral theory with weights which is associated to the linear operator and is developed herein. Similar characterizations for the weighted eigenvalue problem and nonlinear variants will also be given. Finally, topological methods are employed to obtain existence results for nonlinear problems including perturbations in the gradient which are then applied to the well-posedness of the linear problem with lower order terms.

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Section: Introductionmentioning

confidence: 94%