The relativistic Vlasov-Maxwell system describes the evolution of a collisionless plasma. The problem of linear instability of this system is considered in two physical settings: the so-called "one and onehalf" dimensional case, and the three dimensional case with cylindrical symmetry. Sufficient conditions for instability are obtained in terms of the spectral properties of certain Schrödinger operators that act on the spatial variable alone (and not in full phase space). An important aspect of these conditions is that they do not require any boundedness assumptions on the domains, nor do they require monotonicity of the equilibrium.
In this paper, we establish uniqueness of the solution of the Vlasov-Poisson system with spatial density belonging to a certain class of Orlicz spaces. This extends the uniqueness result of Loeper [11] (which holds for density in L ∞ ) and of the paper [15]. Uniqueness is a direct consequence of our main result, which provides a quantitative stability estimate for the Wasserstein distance between two weak solutions with spatial density in such Orlicz spaces, in the spirit of Dobrushin's proof of stability for mean-field PDEs. Our proofs are built on the second-order structure of the underlying characteristic system associated to the equation.
We consider the problem of resource allocation in a decentralized market where users and suppliers trade for a single commodity. Due to the lack of strict concavity, convergence to the optimal solution by means of classical gradient type dynamics for the prices and demands, is not guaranteed. In the paper we explicitly characterize in this case the asymptotic behaviour of trajectories and provide an exact characterization of the limiting oscillatory solutions. Methods of modifying the dynamics are also given, such that convergence to an optimal solution is guaranteed, without requiring additional information exchange among the users.
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