NCSU Differential Equations/Nonlinear Analysis Seminar Schedule Spring 2023

Wednesday, March 22, 15:00-16:00, SAS 4201

Speaker: Ayman Rimah Said, Duke University
Title: Logarithmic spirals in 2d perfect fluids
Abstract: In this talk I will present recent results with In-Jeong from Seoul national university where we study logarithmic spiraling solutions to the 2d incompressible Euler equations which solve a nonlinear transport system on $\mathbb{S}$. We show that this system is locally well-posed in $L^p, p\geq 1$ as well as for atomic measures, that is logarithmic spiral vortex sheets. In particular, we realize the dynamics of logarithmic vortex sheets as the well-defined limit of logarithmic solutions which could be smooth in the angle. Furthermore, our formulation not only allows for a simple proof of existence and bifurcation for non-symmetric multi branched logarithmic spiral vortex sheets but also provides a framework for studying asymptotic stability of self-similar dynamics.

We also give a complete characterization of the long time behavior of logarithmic spirals. We prove global well-posedness for bounded logarithmic spirals as well as data that admit at most logarithmic singularities. This is due to the observation that the local circulation of the vorticity around the origin is a strictly monotone quantity of time. We are then able to show a dichotomy in the long time behavior, solutions either blow up (either in finite or infinite time) or completely homogenize. In particular, bounded logarithmic spirals should converge to constant steady states. For logarithmic spiral sheets, the dichotomy is shown to be even more drastic where only finite time blow up or complete homogenization of the fluid can and does occur.

Wednesday, April 05, 15:00-16:00

Zoom meeting: Link
Speaker: Mihaela Ifrim, University of Wisconsin Madison, USA
Title: Global solutions for 1D cubic defocusing dispersive equations: Part I
Abstract: This article is devoted to a general class of one dimensional NLS problems with a cubic nonlinearity. The question of obtaining scattering, global in time solutions for such problems has attracted a lot of attention in recent years, and many global well-posedness results have been proved for a number of models under the assumption that the initial data is both \emph{small} and \emph{localized}. However, except for the completelyintegrable case, no such results have been known for small but non-localized initial data. In this article we introduce a new, nonperturbative method, to prove global well-posedness and scattering for L2 initial data which is \emph{small} but \emph{non-localized}. Our main structural assumption is that our nonlinearity is \emph{defocusing}. However, we do not assume that our problem has any exact conservation laws. Our method is based on a robust reinterpretation of the idea of interaction Morawetz estimates, developed almost 20 years ago by the I-team.     In terms of scattering, we prove that our global solutions satisfy both global L6 Strichartz estimates and bilinear L2 bounds. This is a Galilean invariant result, which is new even for the classical defocusing cubic NLS. There, by scaling our result also admits a large data counterpart.

Wednesday, April 12, 15:00-16:00

Zoom meeting: Link
Speaker: Mikhail Perepelitsa, University of Houston, USA
Title: Kinetic modeling of Myxobacteria motion with nematic alignment
Abstract: Motivated by motion of myxobacteria, we review several kinetic approaches for modeling motion of self-propelled, interacting rods. We will focus on collisional models of Boltzmann type and discuss the derivation of the governing equations, the range of their validity, and present some analytical and numerical results. We will show that collisional models have natural connection to classical mean-field models of nematic alignment.

Wednesday, April 19, 15:00-16:00

Zoom meeting: Link
Speaker: Fabio Ancona, University of Padova, Italy
Title: Hard congestion limit of the p-system in the BV setting
Abstract: We are concerned with the rigorous justification of the  so-called hard congestion limit from a compressible system with singular pressure towards a mixed  compressible-incompressible system modeling partially congested dynamics, in the framework of BV solutions. We will consider small BV perturbations of reference solutions constituted by (possibly interacting) large interfaces, and we will  analyze the dynamics of the corresponding solutions constructed by a  front tracking algorithm. This is part of a research in collaboration  with R. Bianchini (IAC-CNR, Rome) and C. Perrin (CNRS, Aix Marseille Univ.).