Partial Differential Equations Seminar

When dealing with PDEs arising in fluid mechanics, bounded-energy weaksolutions are, in many cases, the only type of solutions for which one can guarantee global existence without imposing any restrictions on the size of the initial data or forcing terms. Understanding how to construct such solutions is also crucial for designing stable numerical schemes.

In this talk, we will explain the strategy for contructing weak solutions for the Navier-Stokes system for viscous compressible flows, emphasizing the difficulties encountered in the case of non-isotropic viscous stress tensors. In particular, I will present some results obtained in collaboration with Didier Bresch and Maja Szlenk.

We are going to study the Ginzburg-Landau energy for two specific geometries, related to the very experiments on fermionic condensates: annuli and strips 

The specific geometry of a strip provides connections between solitons and vortices, called solitonic vortices, which are vortices with a solitonic behaviour in the infinite direction of the strip. Therefore, they are very different from classical vortices which have an algebraic decay at infinity. We show that there exist stationary solutions to the Gross-Pitaevskii equation with k vortices on a transverse line, which bifurcate from the soliton solution as the width of the strip is increased. This is motivated by recent experiments on the instability of solitons by imposing a phase shift in an elongated condensate for bosonic or fermionic atoms.

For annuli, we prescribe a very large degree on the outer boundary and find that either there is a transition from a giant vortex to vortices also in the bulk but tending to the outer boundary.

This is joint work with Ph. Gravejat and E.Sandier for solitonice vortices and Remy Rodiac for annuli.