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A decay estimate for a class of hyperbolic pseudo-differential equations

Sandra Lucente, Guido Ziliotti (1999)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We consider the equation u t ⁢ − ⁢ i ⁢ Λ ⁢ u = 0 , where Λ = λ ⁢ D x is a first order pseudo-differential operator with real symbol λ ⁢ ξ . Under a suitable convexity assumption on λ we find the decay properties for u ⁢ t , x . These can be applied to the linear Maxwell system in anisotropic media and to the nonlinear Cauchy Problem u t ⁢ − ⁢ i ⁢ Λ ⁢ u = f ⁢ u , u ⁢ 0 , x = g ⁢ x . If f ⁢ u is a smooth function which satisfies f ⁢ u ≃ u p near u = 0 , and g is small in suitably Sobolev norm, we prove global existence theorems provided p is greater than a critical exponent.

A Fractional Analog of the Duhamel Principle

Umarov, Sabir, Saydamatov, Erkin (2006)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 35CXX, 26A33, 35S10The well known Duhamel principle allows to reduce the Cauchy problem for linear inhomogeneous partial differential equations to the Cauchy problem for corresponding homogeneous equations. In the paper one of the possible generalizations of the classical Duhamel principle to the time-fractional pseudo-differential equations is established.* This work partially supported by NIH grant P20 GMO67594.

An analysis of quantum Fokker-Planck models: a Wigner function approach.

Anton Arnold, José L. López, Peter A. Markowich, Juan Soler (2004)

Revista Matemática Iberoamericana

The analysis of dissipative transport equations within the framework of open quantum systems with Fokker-Planck-type scattering is carried out from the perspective of a Wigner function approach. In particular, the well-posedness of the self-consistent whole-space problem in 3D is analyzed: existence of solutions, uniqueness and asymptotic behavior in time, where we adopt the viewpoint of mild solutions in this paper. Also, the admissibility of a density matrix formulation in Lindblad form with Fokker-Planck...

Gradient flows of the entropy for jump processes

Matthias Erbar (2014)

Annales de l'I.H.P. Probabilités et statistiques

We introduce a new transport distance between probability measures on ℝ d that is built from a Lévy jump kernel. It is defined via a non-local variant of the Benamou–Brenier formula. We study geometric and topological properties of this distance, in particular we prove existence of geodesics. For translation invariant jump kernels we identify the semigroup generated by the associated non-local operator as the gradient flow of the relative entropy w.r.t. the new distance and show that the entropy is...

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