Seminars and Colloquia by Series

Global Existence and Long Time Behavior in the 1+1 dimensional Principal Chiral Model with Applications to Solitons

Series
PDE Seminar
Time
Tuesday, February 7, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jessica Trespalacios JulioUniversidad de Chile

We consider the 1+1 dimensional vector valued Principal Chiral Field model (PCF) obtained as a simplification of the Vacuum Einstein Field equations under the Belinski-Zakharov symmetry. PCF is an integrable model, but a rigorous description of its evolution is far from complete. Here we provide the existence of local solutions in a suitable chosen energy space, as well as small global smooth solutions under a certain non degeneracy condition. We also construct virial functionals which provide a clear description of decay of smooth global solutions inside the light cone. Finally, some applications are presented in the case of PCF solitons, a first step towards the study of its nonlinear stability. 

Optimal control of stochastic delay differential equations

Series
PDE Seminar
Time
Tuesday, January 31, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Filippo de FeoPolitecnico di Milano

In this talk we will discuss an optimal control problem for stochastic differential delay equations. We will only consider the case with delays in the state. We will show how to rewrite the problem in a suitable infinite-dimensional Hilbert space. Then using the dynamic programming approach we will characterize the value function of the problem as the unique viscosity solution of an infinite dimensional Hamilton-Jacobi-Bellman equation.  We will discuss partial C^{1}-regularity of the value function. This regularity result is particularly interesting since it permits to construct a candidate optimal feedback map which may allow to find an optimal feedback control. Finally we will discuss some ideas about the case in which delays also appear in the controls.

This is a joint work with S. Federico and A. Święch.

Smooth ergodic theory for evolutionary PDE

Series
PDE Seminar
Time
Tuesday, January 24, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex BluementhGeorgia Tech

Smooth ergodic theory provides a framework for studying systems exhibiting dynamical chaos, features of which include sensitive dependence with respect to initial conditions, correlation decay (even for deterministic systems!) and complicated fractal-like attractor geometry. This talk will be an overview of some of these ideas as they apply to evolutionary PDE, with an emphasis on dissipative semilinear parabolic problems, and a discussion of some of my work in this direction, joint with: Lai-Sang Young and Sam Punshon-Smith. 

Non-uniqueness and convex integration for the forced Euler equations

Series
PDE Seminar
Time
Tuesday, January 17, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Stan PalasekUCLA

This talk is concerned with α-Hölder-continuous weak solutions of the incompressible Euler equations. A great deal of recent effort has led to the conclusion that the space of Euler flows is flexible when α is below 1/3, the famous Onsager regularity. We show how convex integration techniques can be extended above the Onsager regularity to all α<1/2 in the case of the forced Euler equations. This leads to a new non-uniqueness theorem for any initial data. This work is joint with Aynur Bulut and Manh Khang Huynh.

On the Optimal Control of McKean Vlasov SDE and Mean Field Games in Infinite Dimension

Series
PDE Seminar
Time
Tuesday, January 10, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Fausto GozziLuiss University

In this talk we report on recent works (with A. Cosso, I. Kharroubi, H. Pham, M. Rosestolato) on the optimal control of (possibly path dependent) McKean-Vlasov equations valued in Hilbert spaces. On the other side we present the first ideas of a work with S. Federico, D. Ghilli and M. Rosestolato, on Mean Field Games in infinite dimension.

We will begin by presenting some examples for both topics and their relations. Then most of the time will be devoted to the first topic and the main results (the dynamic programming principle, the law invariance property of the value function, the Ito formula and the fact that the value function is a viscosity solution of the HJB equation, a first comparison result).

We conclude, if time allows, with the first ideas on the solution of the HJB-FKP system arising in mean Field Games in infinite dimension.

Global-in-space stability of self-similar blowup for supercritical wave maps

Series
PDE Seminar
Time
Tuesday, November 15, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Online: https://gatech.zoom.us/j/95574359880?pwd=cGpCa3J1MFRkY0RUeU1xVFJRV0x3dz09
Speaker
Irfan GlogićUniversity of Vienna

A distinctive feature of nonlinear evolution equations is the possibility of breakdown of solutions in finite time. This phenomenon, which is also called singularity formation or blowup, has both physical and mathematical significance, and, as a consequence, predicting blowup and understanding its nature is a central problem of the modern analysis of nonlinear PDEs.

In this talk we concentrate on wave maps – a geometric nonlinear wave equation – and we discuss the existence and stability of self-similar solutions, as in all higher dimensions they appear to drive the generic blowup behavior. We outline a novel framework for studying global-in-space stability of such solutions; we then men-tion some long-awaited results that we thereby obtained, and, finally, we discuss the new mathematical challenges that our approach generates.

Uniform linear inviscid damping near monotonic shear flows in the whole space

Series
PDE Seminar
Time
Tuesday, November 8, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hao JiaUniversity of Minnesota

In recent years tremendous progress was made in understanding the ``inviscid damping" phenomenon near shear flows and vortices, which are steady states for the 2d incompressible Euler equation, especially at the linearized level. However, in real fluids viscosity plays an important role. It is natural to ask if incorporating the small but crucial viscosity term (and thus considering the Navier Stokes equation in a high Reynolds number regime instead of Euler equations) could change the dynamics in any dramatic way. It turns out that for the perturbative regime near a spectrally stable monotonic shear flows in an infinite periodic channel (to avoid boundary layers and long wave instabilities), we can prove uniform-in-viscosity inviscid damping. The proof introduces techniques that provide a unified treatment of the classical Orr-Sommerfeld equation in a way analogous to Rayleigh equations. 

Generic Mean Curvature Flow with Cylindrical Singularities

Series
PDE Seminar
Time
Tuesday, November 1, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ao SunUniversity of Chicago

We study the local and global dynamics of mean curvature flow with cylindrical singularities. We find the most generic dynamic behavior of such singularities, and show that the singularities with the most generic dynamic behavior are robust. We also show that the most generic singularities are isolated and type-I. Among applications, we prove that the singular set structure of the generic mean convex mean curvature flow has certain patterns, and the level set flow starting from a generic mean convex hypersurface has low regularity. This is joint work with Jinxin Xue (Tsinghua University)

Explicit formula of multi-solitary waves of the Benjamin–Ono equation

Series
PDE Seminar
Time
Tuesday, October 25, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ruoci SunGeorgia Tech

Every multi-soliton manifold of the Benjamin–Ono equation on the line is invariant under the Benjamin–Ono flow. Its generalized action–angle coordinates allow to solve this equation by quadrature and we have the explicit expression of every multi-solitary wave.

The existence of Prandtl-Batchelor flows on disk and annulus

Series
PDE Seminar
Time
Tuesday, October 4, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Speaker
Zhiwu LinGeorgia Tech

For steady two-dimensional incompressible flows with a single eddy (i.e. nested closed streamlines), Prandtl (1905) and Batchelor (1956) proposed that in the limit of vanishing viscosity the vorticity is constant in an inner region separated from the boundary layer. By constructing higher order approximate solutions of the Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on a disk with the wall velocity slightly different from the rigid-rotation. The leading order term of the flow is the constant vorticity solution (i.e. rigid rotation) satisfying the Batchelor-Wood formula. For an annulus with wall velocities slightly different from the rigid-rotation, we also constructed Prandtl-Batchelor flows, whose leading order terms are rotating shear flows. This is a joint work with Chen Gao, Mingwen Fei and Tao Tao. 

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