Category Archives: Past seminar

Integrable Geometries in AdS_{3}

Speaker: Marcela Cárdenas

Universidad Andrés Bello (Chile)

Date: Monday, June 5, 2023 at 12 Santiago time

Abstract:In this talk we discuss the geometrization of 1+1 integrable systems included in the AKNS integrable system, which contains the Korteweg de-Vries (KDV), modified KDV, sine-Gordon and non-linear Schrödinger equations. This is possible through the construction of a broad class of asymptotic conditions for the gravitational field in three dimensions, reproducing the properties of the AKNS dynamics. We study the consistency, asymptotic symmetry algebra and integrability properties of these novel boundary conditions.

Venue: Online via Zoom / Sala de Seminarios, 5th floor, Beauchef 851
Chair: Rayssa Caju

Steady-state Navier-Stokes flow in an obstructed pipe under mixed boundary conditions and with a prescribed transversal flux rate

Speaker: Gianmarco Sperone

Politecnico di Milano

Date: Monday, May 22, 2023 at 12 Santiago time

Abstract: The steady motion of a viscous incompressible fluid in an obstructed finite pipe is modeled through the Navier-Stokes equations with mixed boundary conditions involving the Bernoulli pressure and the tangential velocity on the inlet and outlet of the tube, while a transversal flux rate F is prescribed along the pipe. Existence of a weak solution to such Navier-Stokes system is proved without any restriction on the data by means of the Leray-Schauder Principle, in which the required a priori estimate is obtained by a contradiction argument based on Bernoulli’s law. Through variational techniques and with the use of an exact flux carrier, an explicit upper bound on F (in terms of the viscosity, diameter and length of the tube) ensuring the uniqueness of such weak solution is given. This upper bound is shown to converge to zero at a given rate as the length of the pipe goes to infinity. In an axially symmetric framework, we also prove the existence of a weak solution displaying rotational symmetry.

Venue: Online via Zoom / Sala de Seminarios, 5th floor, Beauchef 851
Chair: Rayssa Caju

 

 

YouTube video (in Spanish)

 

Lipschitz regularity of almost minimizers for a degenerate one-phase Bernoulli-type functional

Speaker: Nicolò Forcillo

University of Rome Tor Vergata, Italy

Date: Monday, May 15, 2023 at 12 Santiago time

Abstract: In this talk, we deal with almost minimizers for the energy functional $$J_p\left(u, \Omega\right):=\int_{\Omega}\left( |\nabla u(x)|^p + \chi_{\{u>0\}}(x)\right) dx,\quad  p > 1,\quad (1)$$ where \(\Omega\) is a bounded domain in \(\mathbb R^n\) and \(u \geq 0\). The functional \(J_p\) is a generalization to each \(p > 1\) of the classical one-phase (Bernoulli) energy functional, corresponding to \(p = 2\) in (1). Almost minimizers of \(J_2\) were investigated recently in [2, 1]. However, in [4] D. De Silva and O. Savin provided a different approach with respect to [2, 1], based on nonvariational techniques, to deal with almost minimizers of \(J_2\) and their free boundaries. Precisely, inspired by [5], they showed that almost minimizers of  \(J_2\) are “viscosity solutions” in a more general sense. This property roughly means that almost minimizers satisfy comparison in a neighborhood of a touching point whose size depends on the properties of the test functions. Once this property was established, the regularity of the free boundary for almost minimizers followed via the techniques developed by De Silva in [3].

In this talk, we present an optimal Lipschitz continuity result for almost minimizers of \(J_p\), \(p > max \left\{\frac{2n}{n+2} , 1\right\}\). Our approach is inspired by the method introduced in [4]. The talk is based on a joint work with S. Dipierro, F. Ferrari, and E. Valdinoci, see [6].

References

[1]  G. David, M. Engelstein, and T. Toro. Free Boundary Regularity for Almost-Minimizers. Adv. Math., 350: 1109–1192, 2019.

[2]  G. David and T. Toro. Regularity of almost minimizers with free boundary. Calc. Var. Partial Differential Equations, 54: 455–524, 2015.

[3]  D. De Silva. Free boundary regularity for a problem with right hand side. Interfaces and free boundaries, 13: 223–238, 2011.

[4]  D. De Silva and O. Savin. Almost minimizers of the one-phase free boundary problem. Comm. Partial Differential Equations, 45 (8): 913–930, 2020.

[5]  D. De Silva and O. Savin. Quasi-Harnack inequality. Amer. J. Math., 143 (1): 307–331, 2021.

[6]  S. Dipierro, F. Ferrari, N. Forcillo, and E. Valdinoci. Lipschitz regularity of almost minimizers in one-phase problems driven by the p-Laplace operator. To appear in Indiana University Mathematics Journal, arXiv:2206.03238.

Venue: Online via Zoom / Sala de Seminarios, 5th floor, Beauchef 851
Chair: Rayssa Caju

Hamilton–Jacobi problems on graphs and networks.

Speaker: Antonio Siconolfi

Sapienza University of Rome, Italy

Date: Monday, April 24, 2023 at 12 Santiago time

Abstract: The talk presents some Hamilton–Jacobi problems on networks. The pecu- liarity is that the Hamiltonians on different arcs are unrelated, without any compatibility condition at the vertices. Nevertheless, uniqueness result and comparison principles can be obtained, suitably exploiting the geometry of the network,

Our approach consists in associating to the problem on the network a discrete or semidis- crete equation posed on an underlying abstract graph. This allows testing separately the equations on any arc and proving comparison results without using the doubling variable technique. This is beneficial in the numerical approximation as well.

Venue: Online via Zoom / Sala de Seminarios, 5th floor, Beauchef 851
Chair: Rayssa Caju

 

 

Minimally implicit Runge-Kutta methods: relativistic resistive magnetohydrodynamic equations and neutrino M1 transport equations

Speaker: Isabel Cordero Carrión

Universidad de Valencia, Spain

Date: Tuesday, April 18, 2023 at 12 Santiago time

Abstract:  In this talk I will present the minimally implicit Runge-Kutta methods. I will show their application in two different hyperbolic systems of equations with stiff source terms. On one hand, these methods have been sucessfully applied in the evolution of the resistive relativistic magnetohydrodynamic equations following Komissarov (2007) approach. On the other hand, these schemes have been also sucessfully applied in the evolution of the neutrino transport equations within the M1 closure approximation, and used in supernovae simulations. I will conclude the talk with some general remarks.

Venue: Online via Zoom / Sala John von Neumann, 7th floor, Beauchef 851
Chair: Claudio Muñoz

Linear and non-linear stability of collisionless many-particle systems on black hole exteriors

Speaker: Renato Velozo

Sorbonne Université, France

Date: Monday, April 10, 2023 at 12 Santiago time

Abstract: I will present upcoming linear and non-linear stability results concerning the asymptotic behavior of collisionless many-particle systems on black hole exteriors. On the one hand, I will discuss decay properties for solutions to the massive Vlasov equation on Schwarzschild spacetime. On the other hand, I will discuss an asymptotic stability result for the exterior of Schwarzschild as a solution to the Einstein–massless Vlasov system, assuming spherical symmetry. I will explain the use of hyperbolic dynamics to obtain decay in time of the energy momentum tensor by considering a Vlasov equation with a trapping potential.

Venue: Online via Zoom / Sala seminarios DIM, 5th floor, Beauchef 851
Chair: Rayssa Caju

Boundary controllability of critically singular parabolic equations on convex domains

Speaker: Bruno Vergara Biggio

Universitat de Barcelona, Spain

Date: Monday, March 27, 2023 at 12 Santiago time

Abstract:  In this talk I will discuss the null boundary control of heat-like equations on convex domains, featuring a singular potential that diverges as the inverse square of the distance to the boundary. For this purpose, I will establish global Carleman estimates for the associated operators by combining intermediate inequalities with distinct weights that involve non-smooth powers of the boundary distance. These estimates are sharp in the sense that they capture both the natural boundary conditions and the H^1-energy for the problem. Additionally, I will describe the role of the potential strength and the geometry of the domain in our results. This is based on joint work with A. Enciso (ICMAT) and A. Shao (QMUL).

Venue: Online via Zoom / Sala seminarios DIM, 5th floor, Beauchef 851
Chair: Rayssa Caju

Dynamics of Concentrated Vorticities In 2d and 3d Euler Flows

Speaker: Manuel del Pino 

University of Bath, UK

Date: Monday, March 20, 2023 at 12 Santiago time

Abstract: A classical problem that traces back to Helmholtz and Kirchhoff is the understanding of the dynamics of solutions to the Euler equations of an inviscid incompressible fluid when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent results on these solutions’ existence and asymptotic behavior. We describe, with precise asymptotics, interacting vortices, and traveling helices, and extension of these results for the 2d generalized SQG.  In particular we establish Helmholtz’ conjecture on leapfrogging vortex ring interaction. This is research in collaboration with J. Dávila, A. Fernández, M. Musso, and J. Wei.

Venue: Online via Zoom / Sala seminarios DIM, 5th floor, Beauchef 851
Chair: Rayssa Caju

Entropic and Fisher-Information type chaoticity for a family of rescaled states

Speaker: Hagop Tossounian

CMM – Universidad de Chile, Chile

Date: Monday, March 13, 2023 at 12 Santiago time

Abstract:  For a fixed probability measure f , and each N ≥ 2 we introduce an exchangeable random variable obtained from rescaling Y (Law(Y)=f ⊗N ) to the sphere ∑ xj 2 = N. It is known [2] that all the k-marginals of these processes converge weakly to f⊗k ,(a property known as chaoticity and used by Mark Kac [1]). The aim of the talk is to show that the chaos property of this sequence of rescaled r.v. can be strengthened to entropic chaos and to Fisher-information chaos, under mild assumptions on f . This work is j.w. Roberto Cortez and can be found in https://arxiv.org/abs/2204.05406.

[1] M. Kac. Foundations of kinetic theory. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, pages 171–197, Berkeley and Los Angeles, 1956. California U P

[2] Cortez, R., Tossounian, H. On a Thermostated Kac Model with Rescaling. Ann. Henri Poincar ́e 22, 1629–1668 (2021).

Venue: Online via Zoom / Sala seminarios DIM, 5th floor, Beauchef 851
Chair: Gabrielle Nornberg

Uniform a priori estimates for positive solutions of the Lane-Emden equation and system in the plane

Speaker: Boyan Sirakov

PUC-Rio, Brazil

Date: Monday, March 6, 2023 at 12 Santiago time

Abstract:  A few years ago we proved that positive solutions of the superlinear Lane-Emden equation in a two-dimensional smooth bounded domain are bounded independently of the exponent in the equation. Apart from being interesting in itself, this information plays a pivotal role in the asymptotic study of solutions for large exponents, as well as contributes to the old and hard conjecture of uniqueness of positive solutions in a convex domain. We recently took up a similar study for the Lane-Emden system and discovered that, contrary to initial intuition, the boundedness fails in general. This is compelling evidence of the richer nature of the system case. We prove the partial result that uniform boundedness holds provided the exponents in the system are comparable (while many open questions subsist). As a consequence, the energy of the solutions is uniformly bounded, and this has similar consequences as for the scalar equation.

This is a joint work with N. Kamburov from PUC-Chile.

Venue: Online via Zoom / Sala seminarios DIM, 5th floor, Beauchef 851
Chair: Gabrielle Nornberg