# Differential Equations

## Asymptotic stability manifolds for solitons in the generalized Good Boussinesq equation.

Event Date: Apr 15, 2021 in Differential Equations, Seminars

Abstract: In this talk, I shall consider the generalized Good-Boussinesq model in one dimension, with power nonlinearity and data in the energy space $H^1\times L^2$. I will present in more detail the long-time behavior of zero-speed solitary waves, or standing waves. By using virial identities, in the spirit of Kowalczyk, Martel, and Muñoz, we construct and characterize a manifold of even-odd initial data around the standing wave for which there is asymptotic stability in the energy space.

## About infinite energy solutions to the incompressible Navier-Stokes equations.

Event Date: Apr 01, 2021 in Differential Equations, Seminars

Abstract: We study estimates for the Navier–Stokes equations, in a sufficiently robust context to be applied to the construction of : 1) Discretely self-similar solutions, for initial data satisfying the weak condition to be locally square integrable. 2) Regular axially symmetrical solutions without swirl, for initial data which together with his gradient belong to a weighted Lebesgue space.

## Virtual levels and virtual states of operators in Banach spaces.

Event Date: Mar 25, 2021 in Differential Equations, Seminars

Abstract: Virtual levels admit several equivalent characterizations: (1) there are corresponding eigenstates from L^2 or a space “slightly weaker” than L^2; (2) there is no limiting absorption principle in the vicinity of a virtual level (e.g. no weights such that the “sandwiched” resolvent remains uniformly bounded); (3) an arbitrarily small perturbation can produce an eigenvalue. We develop a general approach to virtual levels in Banach spaces and provide applications to Schroedinger operators with non selfadjoint potentials and in any...

## Anisotropic harmonic maps and Ginzburg-Landau type relaxation.

Event Date: Mar 18, 2021 in Differential Equations, Seminars

Abstract: Consider maps $u:R^n\to R^k$ with values constrained in a fixed submanifold, and minimizing (locally) the energy $E(u)=\int W(\nablau)$. Here $W$ is a positive definite quadratic form on matrices. Compared to the isotropic case $W(\nabla u)=|\nabla u|^2$ (harmonic maps) this may look like a harmless generalization, but the regularity theory for general $W$’s is widely open. I will explain why, and describe results with Andres Contreras on a relaxed problem, where the manifold-valued constraint is replaced by an integral penalization.

## Eigenvalue splitting of polynomial order for a system of Schrödinger operators with energy-level crossing.

Event Date: Jan 12, 2021 in Differential Equations, Seminars

In this talk I will present recent results on the spectral analysis of a 1D semiclassical system of coupled Schrödinger operators each of which has a simple well potential. Such systems arise as important models in the Born-Oppenheimer approximation of molecular dynamics. Focusing on the cases where the two underlying classical periodic trajectories cross to each other which imply an energy-level crossing, we give Bohr-Sommerfeld type quantization rules for the eigenvalues of the system in both tangential and transversal crossing cases. Our main results consists in the eigenvalue splitting...