Abstract : Vorticity filaments (one-dimensional structures where vorticity concentrates) play a central role in understanding turbulence generation and energy transfer in fluids. In this talk, I will discuss about how these structures evolve in a viscous fluid. I will consider initial data given by a vorticity measure supported on an infinite smooth curve in R^3. I will show that, for short enough time, the solution consists of a leading-order Lamb–Oseen vortex centered around a curve that evolves according to the binormal flow, a second-order term reflecting the local curvature of the filament, and a small nonlocal correction. For this, I will need to assume that $\Gamma$ , the circulation around the vortex, is sufficiently small.

The talk will be held in person

Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/93324747064?pwd=bzbZ2ADIpsi2ye6t00fJnwWOLJ4JLy.1

Abstract : We introduce a mechanism (Conformal Renormalization) to cancel divergences in Einstein gravity for asymptotically hyperbolic Einstein (AHE) spaces. In the bulk, the procedure amounts to embedding Einstein gravity in Conformal Gravity, whose action is given by a conformal invariant in four dimensions. This scheme is proved to be equivalent to both holographic techniques (for physicists) and the notion of Renormalized Volume (for mathematicians).

In turn, for surfaces anchored to the conformal boundary of AHE spaces, its area and other co-dimension 2 functionals also exhibit a divergent behavior. We show how Conformal Gravity in the bulk induces a finite surface/energy functional which, for suitable conditions, reduces to the formulas of Renormalized Area, Reduced Hawking Mass and Willmore Energy.

The talk will be held in person
Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/93324747064?pwd=bzbZ2ADIpsi2ye6t00fJnwWOLJ4JLy.1

Abstract : In the last decades, there has been fascinating progress in the variational theory for the area functional – that is, the codimension 1 volume – using tools from PDEs and Geometric Measure Theory, and in connection with the problem of finding prescribed mean curvature (PMC) hypersurfaces.

In this talk, we describe some recent contributions from joint work with Jared Marx-Kuo (Rice University) in which we construct infinitely many PMCs for a large class of prescribing functions in a compact Riemannian manifold containing a strictly stable minimal hypersurface.

Zoom: https://uchile.zoom.us/j/93324747064?pwd=bzbZ2ADIpsi2ye6t00fJnwWOLJ4JLy.1

Abstract :-abstractMetodos-2

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Abstract :

Zoom: https://uchile.zoom.us/j/93324747064?pwd=bzbZ2ADIpsi2ye6t00fJnwWOLJ4JLy.1

Abstract : We consider a non-isotropically perturbed nonlinear Schrödinger equation posed on two-dimensional cylindrical domains of the form T×R T and R×T. This equation arises in models describing wave propagation in fiber arrays.

In this talk, we present several well-posedness results for initial data belonging to Sobolev spaces. For the cylindrical domain T×R, we establish global well-posedness in L^2xL^2 for small initial data by proving an L^4 – L^2 Strichartz-type inequality. In the case of the domain R×T, we were unable to adapt the same estimate, so we employed a different approach to obtain well-posedness for data with regularity above L^2 regularity.

These results are part of a joint work with M. Panthee (UNICAMP, Brazil) and M. Nogueira (Federal University of Itajubá, Brazil).

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Abstract :We study the limiting case in dimension one for the fractional Caffarelli-Kohn-Nirenberg inequality, obtaining Onofri’s inequality in the unit disk as a limit. One difficulty here is the lack of an explicit expression for the extremal. An important aspect is the study of solutions of the weighted Liouville equation for the half-Laplacian in dimension one. This is joint work with A. Hyder (TIFR Bangalore) and M. Saez (U. Pontificia Chile).

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Abstract : We consider scalar field theories on the line with Ginzburg-Landau (double-well) self-interaction potentials. Prime examples include the ϕ4 model and the sine-Gordon model. These models feature simple examples of topological solitons called kinks. The study of their asymptotic stability leads to a rich class of problems owing to the combination of weak dispersion in one space dimension, low power nonlinearities, and intriguing spectral features of the linearized operators such as threshold resonances or internal modes. We present a perturbative proof of the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major difficulty is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the modulated kink. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations. The entire framework of our proof, including the systematic development of the distorted Fourier theory, is general and not specific to the sine-Gordon model.

This is a joint work with Jonas Lührmann (Texas A&M)

Zoom: https://uchile.zoom.us/j/93324747064?pwd=bzbZ2ADIpsi2ye6t00fJnwWOLJ4JLy.1

Abstract : The steady motion of a viscous incompressible fluid in a junction of unbounded channels with sources and sinks is modeled through the Navier-Stokes equations under inhomogeneous Dirichlet boundary conditions. In contrast to many previous works, the domain is not assumed to be simply–connected and the fluxes are not assumed to be small. In this very general setting, we prove the existence of a solution with a uniformly bounded Dirichlet integral in every compact subset. This is a generalization of the classical Ladyzhenskaya–Solonnikov result obtained under the additional assumption of zero boundary conditions. For small data of the problem we also prove the unique solvability and attainability of Couette-Poiseuille flows at infinity. The main novelty of our approach
is the proof of the corresponding Leray-Hopf inequality by Leray’s_reductio ad absurdum_ argument. For this contradiction approach, we use some fine properties of weak solutions to the Euler system based on Morse–Sard-type theorems in Sobolev spaces obtained by Bourgain, Korobkov & Kristensen.

Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/93324747064?pwd=bzbZ2ADIpsi2ye6t00fJnwWOLJ4JLy.1

Abstract : In this talk, we investigate the existence of weak solutions for elliptic problems involving Choquard nonlinearity. These equations have attracted significant attention due to their ability to model long-range interactions in various real-world applications. A key concept in solving PDEs is that of weak solutions. These solutions satisfy the integral form of the PDE and are useful when classical solutions may not exist or are challenging to compute. This makes them exceedingly valuable in practical applications. We will use Variational methods to solve the PDEs. This technique essentially transforms the problem of solving a PDE into the problem of finding critical points of the associated functional.

Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/93324747064?pwd=bzbZ2ADIpsi2ye6t00fJnwWOLJ4JLy.1