Inserted: 1 sep 2006
Last Updated: 25 feb 2009
Journal: Arch. Ration. Mech. Anal.
We prove the global existence of nonnegative variational solutions to the ``drift diffusion'' evolution equation under variational boundary condition.
Despite the lack of a maximum principle for fourth order equations, nonnegative solutions can be obtained as a limit of a variational approximation scheme by exploiting the particular structure of this equation, which is the gradient flow of the Fisher Information functional with respect to the Kantorovich-Rubinstein-Wasserstein distance between probability measures. We also study long time behaviour of the solutions, proving their exponential decay to the equilibrium state in many important cases.
Keywords: Optimal transport, Fisher information, quantum drift diffusion, Wasserstein distance, nonnegative solutions, fourth order evolution equations, entropy, log-concave measures