Noticias en español
Asymptotic stability manifolds for solitons in the generalized Good Boussinesq equation.
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.
Read MoreAbout infinite energy solutions to the incompressible Navier-Stokes equations.
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.
Read MoreVirtual levels and virtual states of operators in Banach spaces.
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...
Read MoreAnisotropic harmonic maps and Ginzburg-Landau type relaxation.
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.
Read MoreEigenvalue splitting of polynomial order for a system of Schrödinger operators with energy-level crossing.
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...
Read MoreSolitons and multilinear Harmonic Analysis with potentials.
Abstract: We study some classes of nonlinear dispersive PDEs with potentials in both 1 and 3 dimensions, motivated by questions on the stability of solitons and topological solitons. Our approach is based on the so-called “distorted Fourier transform” adapted to Schrodinger operators, and the development of multilinear harmonic analysis in this setting. This approach allows us to treat low power nonlinearities and describe the global space-time behavior of solutions, also by capturing singularities in frequency space which may arise due to various coherent phenomena. Several...
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