It has been observed in several recent works that, for some classes of linear time-delay systems, spectral values of maximal multiplicity are dominant, a property known as multiplicity-induced-dominancy (MID). This paper starts the investigation of whether MID holds for delay differential-algebraic systems by considering a single-delay system compo...
The purpose of this article is to study quasi linear parabolic partial differential equations of second order, on a bounded junction, satisfying a nonlinear and non dynamical Neumann boundary condition at the junction point. We prove the existence and the uniqueness of a classical solution.
We consider the Schrödinger equation with a nondispersive logarithmic nonlinearity and a repulsive harmonic potential. For a suitable range of the coefficients, there exist two positive stationary solutions, each one generating a continuous family of solitary waves. These solutions are Gaussian, and turn out to be orbitally unstable. We also discus...
We consider the 1D linear Schrödinger equation, on a bounded interval, with Dirichlet boundary conditions and bilinear scalar control. The small-time local exact controllability around the ground state was proved in [BeaLau10], under an appropriate nondegeneracy assumption. Here, we work under a weaker nondegeneracy assumption and we prove the smal...
The multigroup neutron $SP_N$ equations, which are an approximation of the neutron transport equation, are used to model nuclear reactor cores. In their steady state, these equations can be written as a source problem or an eigenvalue problem. We study the resolution of those two problems with an $H^1$-conforming finite element method and a Discont...
The Bingham model for viscoplastic materials involves the minimization of a non-differentiable functional. The regularity of the associated solution is investigated here. The simplified scalar case is considered first: The total variation minimization problem seeks the unique minimizer $u ∈$ BV$(Ω)$ of bounded variation of the energy $\frac{1}{2} \...
The purpose of this article is to highlight mechanisms whereby small scales (or high frequencies) and phases can arise suddenly in nonlinear flows, with special emphasis on Hamilton-Jacobi equations. Such phenomena are derived from a multiscale and multiphase analysis of nonlinear differential equations (with oscillating source terms) for which we ...
This memoir proposes an introduction to the asymptotic mathematical modeling of surface and interfacial gravity water waves. It aims at providing a unified approach to the derivation and rigorous justification of many standard and less-standard models.
We define the Anderson Hamiltonian H on a two-dimensional manifold using high order paracontrolled calculus. It is a self-adjoint operator with pure point spectrum. We get lower and upper bounds on its eigenvalues which imply an almost sure Weyl-type law for H.
Boundary conditions for Bismut's hypoelliptic Laplacian which naturally correspond to Dirichlet and Neumann boundary conditions for Hodge Laplacians are considered. Those are related with specific boundary conditions for the differential and its various adjoints. Once the closed realizations of those operators are well understood, the commutation o...
