Section: New Results

Miscellaneous results

In [21] Mitia Duerinckx establishes the global well-posedness of a family of equations, which are obtained in certain regimes — in a joint work in preparation with Sylvia Serfaty — as the mean-field evolution of the supercurrent density in a (2D section of a) type-II superconductor with pinning and with imposed electric current. General vortex-sheet initial data are also considered, and the uniqueness and regularity properties of the solution are investigated.

In [33], [8], [11], [12], D. Bonheure, J.-B. Casteras and collaborators made bifurcation analysis and constructed multi-layer solutions of the Lin-Ni-Takagi and Keller-Segel equations, which come from the Keller-Segel system of chemotaxis in specific cases. A remarkable feature of the results is that the layers do not accumulate to the boundary of the domain but satisfy an optimal partition problem contrary to the previous type of solutions constructed for these models.

In [16], [17], [35], J.-B. Casteras and collaborators study different problems related to the existence of constant mean curvature hypersurfaces with prescribed asymptotic boundary on Cartan-Hadamard manifold. In particular, they obtained the first existence results for minimal graphs with prescribed asymptotic Dirichlet data under a pointwise pinching condition for sectionals curvatures.

S. De Bièvre and co-workers present in [67] a general approach to calculating the entanglement of formation for superpositions of two-mode coherent states, placed equidistantly on a circle in phase space. In the particular case of rotationally-invariant circular states the value of their entanglement is shown to be given by analytical expressions. They analyse the dependence of the entanglement on the radius of the circle and number of components in the superposition.

A. Benoit continues his analysis of hyperbolic equations in corner spaces. He addresses in [30] the rigorous construction of geometric optics expansions for weakly well-posed hyperbolic corner problems. He studies in [31] the semi-group stability for finite difference discretizations of hyperbolic systems of equations in corner domains, extending previous results of Coulombel & Gloria and Coulombel in the case of the halfspace.