Section: New Results

Miscellaneous results

In [24] A. Benoit proved that for linear hyperbolic systems of equations in the quarter space a violent instability can be caused by the accumulation of an arbitrary large number of weak instabilities. The proof of this result is based on the construction of the WKB expansions for hyperbolic corner problems with self-interacting phases and is a continuiation of [45] .

In [9] , C. Cancès, T. Gallouët, and L. Monsaingeon gave a gradient flow interpretation for incompressible immiscible two-phase flows in porous media. With C. Chainais-Hillairet, T. Gallouët characterized the pseudo-stationary state for a corrosion model in [31] .

In [8] , D. Bonheure, E. Moreira dos Santos, M. Ramos and H. Tavares construct least energy nodal solutions of Hamiltonian elliptic systems. The construct is tricky since the functional associated to Hamiltonian elliptic systems is strongly indefinite. The proof uses a dual variational argument and an approximation scheme with some ideas of Gama-convergence type.

In [27] , D. Bonheure, P. D’Avenia and A. Pomponio aim to derive rigorously the PDE formulation of the Born-Infeld model in the electrostatic case. This nonlinear model of electromagnetism was introduced by Born and Infeld who proposed a new Lagrangian which theoretically assumes the existence of a maximal field intensity, likewise Einstein’s Lagragian of special relativity opposed to Newton’s Lagrangian of classical mechanics. The paper contains new results and new insights on the model. It covers several relevant particular cases but we are still far from the full understanding of the problem.

In [29] and [30] , D. Bonheure and coauthors study patterns and phase transitions in a fourth order extension of the famous Allen-Cahn model. In [29] , some rigidity results à la Gibbons are proved while [30] concerns qualitative properties of positive patterns with Navier boundary conditions. A conjecture related to De Giorgi’s famous one concerning the one dimensionality of monotone phase transition in the classical Allen Cahn model is proposed in [29] .

In [28] , D. Bonheure and collaborators study multi-layer solutions of the Lin-Ni-Takagi model, which comes from the Keller-Segel model of chemotaxis in a specific case. 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 this model.

In [53] , M. Duerinckx proved a new mean-field limit result for the gradient flow evolution of particle systems with pairwise Riesz interactions, in dimensions 1 and 2, in cases for which this problem was still open. The proof is based on a method introduced by Serfaty [66] in the context of the Ginzburg-Landau vortices, using regularity and stability properties of the limiting equation.