Category Archives: Past seminar

Solutions of the Navier-Stokes equations with forced rapid spatial decay

Past seminar

Speaker:  Matthieu Pageard

Université Claude Bernard Lyon 1

Date:  Monday, May 27th at 4:20 p.m.

Abstract:

I will present a recently obtained result on spatial decay for the incompressible Navier-Stokes equations in R^n. It is well-known that generic mild solutions cannot decay faster than |x|^(-n-1). This is a consequence of the inner structure of the equations, through the decay properties of the so-called Oseen kernel. Here, we propose an approach to go beyond this rate of decay. The idea is to introduce an external forcing in the system. This will allow to eliminate the slowly decaying term in an asymptotic profile previously obtained by Brandolese and Vigneron. We will begin by a theorem of existence of mild solutions in some well-suited function spaces, and then obtain a rapidly decaying solution to a Navier-Stokes system with a specific force. The proof is based on an algorithmic construction of external forces, inspired by a recent work of Brandolese and Okabe.
Venue: DIM seminar room, Beauchef 851, 5th floor.

 

Chair: Ricardo Freire

Study and simulation of traffic waves: Jamiton collisions

Past seminar

Speaker: Sebastián Tapia

Universidad de Chile

Date: Monday, May 13th at 4:20 p.m.

Abstract: A jamiton corresponds to a traveling wave type solution that appears from the phantom congestion phenomenon and theoretically in macroscopic traffic equations. The objective was to study numerically the exit velocities and other emergent properties in jamiton collisions on a circular route, motivated by kink-antikink collisions in the phi^4 equation. For this, a numerical method based on finite volumes is established to simulate the jamiton collision with good accuracy. From the simulations, it is conjectured that the collision of two jamitons originates a new jamiton. Then, 274 jamitons of different sizes were collided and it was observed that the exit velocity presents a different behavior than phi^4. Also, the following were studied: The amplitude, length and driver reaction time dependencies.

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

The Riemann Zeta Flow and Generalizations

Past seminar

Speaker: Vicente Salinas

Universidad de Chile

Date: Monday, May 6th at 4:20 p.m.

Abstract: The study of the Riemann Zeta function is of utmost importance, particularly for the connection with the distribution of prime numbers and the Riemann Hypothesis. This work presents a different approach by addressing this problem in Number Theory using techniques from partial differential equations (PDEs). To establish this connection, we examine a heat equation with the Riemann Zeta function as the source term.

On this occasion, our focus will be on the two main theorems we have proven. Firstly, we will discuss the theorem on local existence of solutions for small time intervals, and its extension to a family of functions including Zeta, the Dirichlet L-functions. Additionally, we will provide conditions to ensure the global existence and the asymptotic behavior of the flow.

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

 

On fractional quasilinear equations with elliptic degeneracy

Past seminar

Speaker: Disson dos Prazeres

Universidade Federal de Sergipe

Date:  Monday, April 29th at 4:20 p.m.

Abstract: In this talk, we present a systematic approach to investigate the existence, multiplicity, and local gradient regularity of solutions for nonlocal quasilinear equations with local gradient degeneracy. Our method involves an interactive geometric argument that interplays with uniqueness property for the corresponding homogeneous problem, leading with gradient Hölder regularity estimates. This approach is intrinsically developed for nonlocal scenarios, where uniqueness holds for the local homogeneous problem.

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

 

Local Energy control in the presence of a zero-energy resonance

Past seminar

Speaker: José Manuel Palacios

University of Toronto

Date: Wednesday, April 17 at 4:20 p.m

Abstract: We consider the problem of stability and local energy decay for co-dimension one perturbations of the soliton of the cubic Klein-Gordon equation in 1+1 dimensions. Our main result gives a weighted time-averaged control of the local energy over a time interval which is exponentially long in the size of the initial (total) energy. A major difficulty is the presence of a zero-energy resonance in the linearized operator, which is a well-known obstruction to improved local decay properties. We address this issue by using virial estimates that are frequency-localized in a time-dependent way and introducing a “singular virial functional” with time-dependent weights to control the mass of the perturbation projected away from small frequencies.

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

 

On a variational problem related to the Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities.

Past seminar

Speaker: Tobias Ried

(TU Munich)

Date: Monday, April  15 at 4:15 p.m. 

Abstract: In this talk I will introduce a variational problem arising in the derivation of upper bounds on the optimal constants in the Cwikel-Lieb-Rozenblum (CLR) inequality based upon a substantial refinement of Cwikel’s original proof. The approach we developed with D. Hundertmark, P. Kunstmann and S. Vugalter in [Invent. Math. 231 (2023), no.1, 111-167] highlights a natural but overlooked connection of optimal bounds on the CLR constant with bounds for maximal Fourier multipliers from harmonic analysis.
I will show how, through a variational characterization of the L1 norm of the Fourier transform of a function and convex duality, this variational problem can be reformulated in terms of a variant of the classical Hadamard three lines lemma. By studying Hardy-like spaces of holomorphic functions in a strip in the complex plane, together with T. Carvalho Corso, we were able to derive an analytic formula for the minimizers, and use it to get the best possible upper bounds for the optimal constants in the CLR and LT inequalities achievable by the method of Hundertmark-Kunstmann-Vugalter and myself.

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

A variational and numerical approach to model inverse problems applied in subduction earthquakes

Past seminar

Speaker: Jorge Aguayo

CMM- Universidad de Chile

Date: Monday, March 25 at 4:15 p.m. 

Abstract: Inspired by some experimental (numerical) works on fractional diffusion PDEs, we develop a rigorous framework to prove that solutions to the fractional Navier-Stokes equations, which involve the fractional Laplacian operator, converge to a solution of the classical case. Precisely, in the setting of mild solutions, we prove uniform convergence in both the time and spatial variables and derive a precise convergence rate, revealing some phenomenological effects.

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

 

From non-local to local Navier-Stokes equations.

Past seminar

Speaker: Oscar Jarrín

Universidad de la Ámericas, Ecuador

Date: Monday, March 18 at 4:15 p.m. at Santiago time

Abstract: Inspired by some experimental (numerical) works on fractional diffusion PDEs, we develop a rigorous framework to prove that solutions to the fractional Navier-Stokes equations, which involve the fractional Laplacian operator, converge to a solution of the classical case. Precisely, in the setting of mild solutions, we prove uniform convergence in both the time and spatial variables and derive a precise convergence rate, revealing some phenomenological effects.

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

 

The logarithmic Sobolev inequality and the Sobolev inequality in large dimensions

Past seminar

Speaker: Jean Dolbeault

(Université Paris-Dauphine, France)

Date: March 11th at 16:15 hs  at Santiago time

Abstract: The logarithmic Sobolev inequality can be considered as the infinite dimensional limit of ]the Sobolev inequality. This lecture is devoted to a review of some recent results in this direction concerning gradient flow methods (carré du champ) and stability results.

References:
1.⁠ ⁠J. Dolbeault, M. J. Esteban, A. Figalli, R. L. Frank, M. Loss, Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence, Preprint arXiv: 2209.08651 and hal-03780031, (2023).
2.⁠ ⁠G. Brigati, J. Dolbeault, And N. Simonov, On Gaussian interpolation inequalities, C. R. Math. Acad. Sci. Paris, 362 (2024), pp. 21–44.
3.⁠ ⁠G. Brigati, J. Dolbeault, And N. Simonov, Stability for the logarithmic Sobolev inequality, Preprint arXiv: 2303.12926, (2023).