Fokker-Planck models for Bose-Einstein particles
Seminar Room 1, Newton Institute
AbstractWe study nonnegative, measure-valued solutions of the initial value problem for one-dimensional drift-diffusion equations where the linear drift has a driving potential with a quadratic growth at infinity, and the nonlinear diffusion is governed by an increasing continuous and bounded function. The initial value problem is studied in correspondence to initial densities that belong to the space of nonnegative Borel measures with finite mass and finite quadratic momentum and it is the gradient flow of a suitable entropy functional with respect to the Wasserstein distance. Due to the degeneracy of diffusion for large densities, concentration of masses can occur, whose support is transported by the drift. We shall show that the large-time behavior of solutions depends on a critical mass which can be explicitly characterized in terms of the diffusion function and of the drift term. If the initial mass is less than the critical mass, the entropy has a unique minimizer which is absolutely continuous with respect to the Lebesgue measure. Conversely, when the total mass of the solutions is greater than the critical one, the steady state has a singular part in which the exceeding mass is accumulated.
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