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We derive the hydrodynamic limit of a kinetic equation where the interactions in velocity are modeled by a linear operator (Fokker–Planck or linear Boltzmann) and the force in the Vlasov term is a stochastic process with high amplitude and short-range correlation. In the scales and the regime we consider, the hydrodynamic equation is a scalar second-order stochastic partial differential equation. Compared to the deterministic case, we also observe a phenomenon of enhanced diffusion.
Let be a normalised standard complex Gaussian vector, and project an Hermitian matrix onto the hyperplane orthogonal to . In a recent paper Faraut (Tunisian J. Math. 1 (2019), 585–606) has observed that the corresponding eigenvalue PDF has an almost identical structure to the eigenvalue PDF for the rank-1 perturbation , and asks for an explanation. We provide one by way of a common derivation involving the secular equations and associated Jacobians. This applies also in a related setting, for example when is a real Gaussian and Hermitian, and also in a multiplicative setting where are fixed unitary matrices with a multiplicative rank-1 deviation from unity, and is a Haar distributed unitary matrix. Specifically, in each case there is a dual eigenvalue problem giving rise to a PDF of almost identical structure.
Let be the K3 manifold. In this note, we discuss two methods to prove that certain generalized Miller–Morita–Mumford classes for smooth bundles with fiber are nonzero. As a consequence, we fill a gap in a paper of the first author, and prove that the homomorphism does not split. One of the two methods of proof uses a result of Franke on the stable cohomology of arithmetic groups that strengthens work of Borel, and may be of independent interest.
We present criteria for a certain coulombian interaction energy of infinitely many points in , , with a uniformly charged background, to be finite, as well as examples. We also show that in this unbounded setting, it is not always possible to project an vector field onto the set of gradients in a way that reduces its average norm on large balls.
We prove that the theorem of Mouhot and Villani on Landau damping near equilibrium for the Vlasov–Poisson equations on cannot, in general, be extended to high Sobolev spaces in the case of gravitational interactions. This is done by showing in every Sobolev space, there exists background distributions such that one can construct arbitrarily small perturbations that exhibit arbitrarily many isolated nonlinear oscillations in the density. These oscillations are known as plasma echoes in the physics community. For the case of electrostatic interactions, we demonstrate a sequence of small background distributions and asymptotically smaller perturbations in which display similar nonlinear echoes. This shows that in the electrostatic case, any extension of Mouhot and Villani’s theorem to Sobolev spaces would have to depend crucially on some additional nonresonance effect coming from the background — unlike the case of Gevrey- with regularity, for which results are uniform in the size of small backgrounds. In particular, the uniform dependence on small background distributions obtained in Mouhot and Villani’s theorem in the Gevrey class is false in Sobolev spaces. Our results also prove that the time-scale of linearized approximation obtained by previous work is sharp up to logarithmic corrections.
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