Speaker
Description
Around 2005, Degond, Mehats and Ringhofer proposed a novel approach for the derivation of quantum fluid models (known as QHD, QET etc) from first principles: they apply a moment method to the quantum Boltzmann equation, using a BGK collision operator for moment closure. In the simplest case, the resulting fluid model is the non-local quantum drift diffusion equation nlQDD. It turns out that this equation is formally the Wasserstein gradient flow of the relative von Neumann entropy.
In our (so far unsuccessful) attempt to understand the appearance of nlQDD's gradient flow structure, we have re-done the derivation by moment closure consistently on the level of spatial discretization, and we are able to replicate (albeit still not understand) the variational form in the discrete setting, along with essentially all the relevant estimates. We can further pass to the continuous limit, thereby generalizing en passant the so far only result on existence of weak solutions from close-to-equilibrium to large data. Finally, I will discuss a discretized DLSS equation that arises in the semi-classical expansion of nlQDD, and even bears two gradient flow structures: the Wasserstein one and another, second order one. This second gradient flow structure is more deeply analyzed in the presentation of Andre Schlichting.
This is mainly joint work with Eva-Maria Rott; the DLSS part is a joint work with Andre Schlichting and Giuseppe Savare.
