Kinetic transport equations with a given confining potential and non-linear relaxation type collision operators are considered. General (monotone) energy dependent equilibrium distributions are allowed with a chemical potential ensuring mass conservation. Existence and uniqueness of solutions is proven for initial data bounded by equilibrium distributions. The diffusive macroscopic limit is car...

We consider singular-degenerate, multivalued stochastic fast diffusion equations with multiplicative Lipschitz continuous noise. In particular, this includes the stochastic sign fast diffusion equation arising from the BakTang-Wiesenfeld model for self-organized criticality. A well-posedness framework based on stochastic variational inequalities (SVI) is developed, characterizing solutions to t...

The main purpose of present article is to find the fundamental solution of partial differential equations in the generalized theory of thermoelastic diffusion materials with double porosity in case of steady oscillations in terms of elementary functions.

In the presence of quantum measurements with direct photon detection the evolution of open quantum systems is usually described by stochastic master equations with jumps. Heuristically, diffusion models can be obtained from these equations as approximation. A condition for a general diffusion approximation for jump master equations is presented. This approximation is rigorously proved by using ...

Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regul...

We propose a rate equations model for the description of a multilongitudinal mode semiconductor laser. We show how usually disregarded effects such as parametric interaction, carrier-diffusion and interference between the fields of the different longitudinal modes play an important role in the final operating characteristics of the laser. Parametric interaction induces a non-trivial distributio...

In this article, the two-dimensional magneto-hydrodynamic (MHD) equations are considered with only magnetic diffusion. Here the magnetic diffusion is given by D a Fourier multiplier whose symbol m is given by m(ξ) = |ξ| log(e + |ξ|) . We prove that there exists an unique global solution in H(R) with s > 2 for these equations when β > 1. This result improves the previous works which require that...

In this paper we construct numerical schemes to approximate linear transport equations with slab geometry by diffusion equations. We treat both the case of pure diffusive scaling and the case where kinetic and diffusive scalings coexist. The diffusion equations and their data are derived from asymptotic and layer analysis which allows general scattering kernels and general data. We apply the ha...

The entropy production paradox for anomalous diffusion processes describes a phenomenon where one-parameter families of dynamical equations, falling between the diffusion and wave equations, have entropy production rates (Shannon, Tsallis or Renyi) that increase toward the wave equation limit unexpectedly. Moreover, also surprisingly, the entropy does not order the bridging regime between diffu...