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We investigate the Cauchy problem and the diffusion asymptotics for a spatially inhomogeneous kinetic model associated to a nonlinear Fokker-Planck operator. Its solution describes the density evolution of interacting particles whose mobility is hampered by their aggregation. When the initial data lies below a Maxwellian, we derive the global well-posedness with instantaneous smoothness. The proof relies on hypoelliptic analogue of the classical parabolic theory, as well as a positivity-spreading result based on the Harnack inequality and barrier function methods. Moreover, the scaled equation leads to the fast diffusion flow under the low field limit. The relative phi-entropy method enables us to see the connection between the overdamped dynamics of the nonlinearly coupled kinetic model and the correlated fast diffusion. The global in time quantitative diffusion asymptotics is then derived by combining entropic hypocoercivity, relative phi-entropy and barrier function methods.
We derive a diffusion approximation for the kinetic Vlasov-Fokker-Planck equation in bounded spatial domains with specular reflection type boundary conditions. The method of proof involves the construction of a particular class of test functions to b
This paper mainly investigates the Cauchy problem of the spatially weighted dissipative equation with initial data in the weighted Lebesgue space. A generalized Hankel Transform is introduced to derive the analytical solution and a special Youngs Ine
We consider a Fokker-Planck equation which is coupled to an externally given time-dependent constraint on its first moment. This constraint introduces a Lagrange-multiplier which renders the equation nonlocal and nonlinear. In this paper we exploit
We study the long time behaviour of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fkker-Planck equation. We prove a uniform (in the number of particles) exponential convergence to equilibrium for t
Let $H$ be a norm of ${bf R}^N$ and $H_0$ the dual norm of $H$. Denote by $Delta_H$ the Finsler-Laplace operator defined by $Delta_Hu:=mbox{div},(H( abla u) abla_xi H( abla u))$. In this paper we prove that the Finsler-Laplace operator $Delta_H$ acts