Nonlinear Partial Differential Equations
Spring 2023
Date: | January 17, 2023 |
Time: | 3:00pm |
Location: | Zoom |
Speaker: | Abdon Moutinho, Université Paris 13 |
Title: | Dynamics of two interacting kinks for the phi6 model |
Abstract: | We consider the $\phi^6$ model given by the one-dimensional nonlinear wave equation $\partial_t^2 \phi - \partial_x^2 \phi + 2 \phi - 8 \phi^3 + 6 \phi^5 = 0$. In this talk, we will describe the long time behavior of all the solu- tions of this partial differential equation satisfying the boundary condition $\lim_{x \to \infty} \phi(t,x) = 1$, $\lim_{x \to -\infty} \phi(t,x) = -1$ and having energy slightly larger than the minimum possible. Indeed, we will also show that this set of solutions can be written explicitly as a sum of two moving kinks (topological solitons) with a remainder with low energy norm. Our estimate for the energy norm of the remainder is close to the optimal during the large time interval we study. |
Date: | January 24, 2023 |
Time: | 3:00pm |
Location: | BLOC 302 |
Speaker: | Benjamin Dodson, Johns Hopkins University |
Title: | Rigidity for the one dimensional, quintic nonlinear Schrodinger equation |
Abstract: | In this talk, we prove rigidity for the one dimensional, quintic nonlinear Schrodinger equation with mass equal to the mass of the soliton. We prove that non-scattering solutions are either the soliton or a pseudoconformal transformation of the soliton. |
Date: | February 14, 2023 |
Time: | 3:00pm |
Location: | BLOC 302 |
Speaker: | Graham Cox, Memorial University of Newfoundland |
Title: | Turing instability via the generalized Maslov index |
Abstract: | The Maslov index is a topological invariant naturally associated to a Hamiltonian system. This is a useful tool for linear stability analysis, provided the eigenvalue equation can be rewritten in a Hamiltonian form. In this talk I will describe a recent generalization of the Maslov index to non-Hamiltonian systems. This generalization allows one to study reaction-diffusion system of activator-inhibitor type. As an application, I will show how this new index can be used to characterize the Turing instability. |
Date: | March 21, 2023 |
Time: | 3:00pm |
Location: | BLOC 302 |
Speaker: | Xin Liu, Texas A&M University |
Title: | A revisit to the rigorous justification of the quasi-geostrophic approximation |
Abstract: | The quasi-geostrophic approximation is used to model large-scale atmospheric/oceanic flows closed to the geostrophic balance, i.e., the Coriolis force, the pressure, and the gravity are in balance. Such an approximation for inviscid flows has been investigated in the case without boundary or without oscillating fast waves. In this talk, I will (1) review the classical mathematical results of the QG approximation, (2) point out the possible boundary layer when fast rotation is not present, and (3) show that with fast rotation, there is not boundary layer. In particular, we rigorously justify the QG approximation with both boundary and oscillating fast waves. This is done by introducing a new generalized potential vorticity, obtaining uniform estimates, and passing the weak limit. Our result demonstrates the stabilizing effect of rotation by suppressing the boundary layer. This is joint work with C. Bardos and E. Titi. |
Date: | March 28, 2023 |
Time: | 3:00pm |
Location: | BLOC 302 |
Speaker: | Boualem Khouider, University of Vicotria |
Title: | Optimal transport for particle image velocimetry |
Abstract: | Particle image velocimetry (PIV) is a non intrusive method used to measure the velocity field in laboratory experiments. Small particles, immersed in the fluid, are illuminated by a pulsating laser gun and a suspended camera records the particles' positions from the reflected laser light. Cross correlation algorithms are traditionally used to retrieve the flow velocity from the successive images by measuring the average displacement of look-a-like clusters belonging to two successive images. We propose a new method for PIV, based on the L2 optimal mass transportation (OT) problem. The suspended particles are modelled by a network of Gaussian-like distributions and the flow fluid is approximated by the optimal transport map of distribution networks associated with successive images. We derive rigorous bounds on the approximation error in terms of the model parameters, namely the size of the Gaussians and the noise level. To obtain the numerical solution of the OT problem, we solve the associated Monge-Ampere equation using a PDE based Newton-like method combined with an efficient spectral method for the underlying linearized PDE. Numerical experiments based on two synthetic flow fields, consisting of a plane shear and an array of vortices are used to validate the methodology. We also consider the case of particles with different masses that are randomly seeded and compare the OT-PIV method with a typical cross correlation algorithm for the case of real data. Using a combination of theory and numerical experiments, we demonstrate that, in the presence of particles with different weights/brightness, the OT method is more accurate for the largest/brightest particles and it is more faithful when the particles are far enough from each other, making it more suitable for the so-called particle tracking regime of PIV, i.e, when the seeding density is low. In deed, using numerical experiments, we demonstrate that for low seeding densities, the OT method performs better than the traditional cross correlation algorith |
Date: | April 4, 2023 |
Time: | 3:00pm |
Location: | BLOC 302 |
Speaker: | Christos Mantoulidis, Rice University |
Title: | Generic regularity of minimizing hypersurfaces in dimensions 9 and 10 |
Abstract: | In joint work with Otis Chodosh and Felix Schulze we showed that the problem of finding a least-area compact hypersurface with prescribed boundary or homology class has a smooth solution for generic data in dimensions 9 and 10. In this talk I will explain the main ideas in the proof. |
Date: | April 11, 2023 |
Time: | 3:00pm |
Location: | ZOOM |
Speaker: | Xu Yuan, Chinese University of Hong Kong |
Title: | Construction of multi-solitons for the energy-critical wave equation |
Abstract: | We will review some results on the construction and interaction of solitary waves for the energy-critical focusing wave equation. After discussing briefly the well-known conjecture of soliton resolution, we will present recent results of the existence of multi-solitary waves in the case of weak interactions. |
Date: | April 25, 2023 |
Time: | 3:00pm |
Location: | BLOC 302 |
Speaker: | Juhi Jang, University of Southern California |
Title: | Dynamics of Newtonian stars |
Abstract: | A classical model to describe the dynamics of Newtonian stars is the gravitational Euler-Poisson system. The Euler-Poisson system admits a wide range of star solutions that are in equilibrium or expand for all time or collapse in a finite time or rotate. In this talk, I will discuss some recent progress on those star solutions with focus on expansion and collapse. If time permits, I will also discuss the non-radial stability of self-similarly expanding Goldreich-Weber star solutions. |