### Eulerian dynamics with alignment interactions

- Series
- PDE Seminar
- Time
- Tuesday, March 12, 2019 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Changhui Tan – University of South Carolina – tan@math.sc.edu

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- Series
- PDE Seminar
- Time
- Tuesday, March 12, 2019 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Changhui Tan – University of South Carolina – tan@math.sc.edu

The Euler-Alignment system arises as a macroscopic representation of the Cucker-Smale model, which describes the flocking phenomenon in animal swarms. The nonlinear and nonlocal nature of the system bring challenges in studying global regularity and long time behaviors. In this talk, I will discuss the global wellposedness of the Euler-Alignment system with three types of nonlocal alignment interactions: bounded, strongly singular, and weakly singular interactions. Different choices of interactions will lead to different global behaviors. I will also discuss interesting connections to some fluid dynamics systems, including the fractional Burgers equation, and the aggregation equation.

- Series
- PDE Seminar
- Time
- Tuesday, March 5, 2019 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Professor Shouhong Wang – Indiana University – showang@indiana.edu

First, we introduce a new field theoretical interpretation of quantum mechanical wave functions, by postulating that the wave function is the common wave function for all particles in the same class determined by the external potential V, of the modulus of the wave function represents the distribution density of the particles, and the gradient of phase of the wave function provides the velocity field of the particles. Second, we show that the key for condensation of bosonic particles is that their interaction is sufficiently weak to ensure that a large collection of boson particles are in a state governed by the same condensation wave function field under the same bounding potential V. For superconductivity, the formation of superconductivity comes down to conditions for the formation of electron-pairs, and for the electron-pairs to share a common wave function. Thanks to the recently developed PID interaction potential of electrons and the average-energy level formula of temperature, these conditions for superconductivity are explicitly derived. Furthermore, we obtain both microscopic and macroscopic formulas for the critical temperature. Third, we derive the field and topological phase transition equations for condensates, and make connections to the quantum phase transition, as a topological phase transition. This is joint work with Tian Ma.

- Series
- PDE Seminar
- Time
- Tuesday, February 26, 2019 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Professor Hongjie Dong – Brown University – Hongjie_Dong@Brow.edu

I will first give a short introduction of the Navier-Stokes equations, then review some previous results on theconditional regularity of solutions to the incompressible Navier-Stokes equations in the critical Lebesguespaces, and finally discuss some recent work which mainly addressed the boundary regularity issue.

- Series
- PDE Seminar
- Time
- Tuesday, February 12, 2019 - 15:00 for 1 hour (actually 50 minutes)
- Location
- skiles 006
- Speaker
- Ting Zhang – Zhejiang University – zhangting79@zju.edu.cn

Abstract: In this talk, we consider the Cauchy problem of the N-dimensional incompressible viscoelastic fluids with N ≥ 2. It is shown that, in the low frequency part, this system possesses some dispersive properties derived from the one parameter group e∓itΛ. Based on this dispersive effect, we construct global solutions with large initial velocity concentrating on the low frequency part. Such kind of solution has never been seen before in the literature even for the classical incompressible Navier-Stokes equations. The proof relies heavily on the dispersive estimates for the system of acoustics, and a careful study of the nonlinear terms. And we also obtain the similar result for the isentropic compressible Navier-Stokes equations. Here, the initial velocity with arbitrary B⋅N 2 −1 2,1 norm of potential part P⊥u0 and large highly oscillating are allowed in our results. (Joint works with Daoyuan Fang and Ruizhao Zi)

- Series
- PDE Seminar
- Time
- Tuesday, January 22, 2019 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Matias Delgadino – Imperial College

In this talk we will introduce two models for the movement of a small droplet over a substrate: the thin film equation and the quasi static approximation. By tracking the motion of the apparent support of solutions to the thin film equation, we connect these two models. This connection was expected from Tanner's law: the edge velocity of a spreading thin film on a pre-wetted solid is approximately proportional to the cube of the slope at the inflection. This is joint work with Prof. Antoine Mellet.

- Series
- PDE Seminar
- Time
- Tuesday, November 27, 2018 - 15:00 for 1 hour (actually 50 minutes)
- Location
- skiles 006
- Speaker
- Yilun(Allen) Wu – The University of Oklahoma – allenwu@ou.edu

A rotating star may be modeled as gas under self gravity with a given total mass and prescribed angular velocity. Mathematically this leads to the Euler-Poisson system. In this talk, we present an existence theorem for such stars that are rapidly rotating, depending continuously on the speed of rotation. No previous results using continuation methods allowed rapid rotation. The key tool for the result is global continuation theory via topological degree, combined with a delicate limiting process. The solutions form a connected set $\mathcal K$ in an appropriate function space. Take an equation of state of the form $p = \rho^\gamma$; $6/5 < \gamma < 2$, $\gamma\ne 4/3$. As the speed of rotation increases, we prove that either the density somewhere within the stars becomes unbounded, or the supports of the stars in $\mathcal K$ become unbounded. Moreover, the latter alternative must occur if $\frac43<\gamma<2$. This result is joint work with Walter Strauss.

- Series
- PDE Seminar
- Time
- Tuesday, November 13, 2018 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Prof. Shigeaki Koike – Tohoku University, Japan

We discuss bilateral obstacle problems for fully nonlinear second order
uniformly elliptic partial differential equations (PDE for short) with
merely continuous obstacles. Obstacle problems arise not only in
minimization of energy functionals under restriction by obstacles but
also stopping time problems in stochastic optimal control theory. When
the main PDE part is of divergence type, huge amount of works have been
done. However, less is known when it is of non-divergence type.
Recently, Duque showed that the Holder continuity of viscosity solutions
of bilateral obstacle problems, whose PDE part is of non-divergence
type, and obstacles are supposed to be Holder continuous. Our purpose is
to extend his result to enable us to apply a much wider class of PDE.
This is a joint work with Shota Tateyama (Tohoku University).

- Series
- PDE Seminar
- Time
- Tuesday, November 6, 2018 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Hao Jia – University of Minnesota – jia@umn.edu

The two dimensional Euler equation is globally wellposed, but the long time behavior of solutions is not well understood. Generically, it is conjectured that the vorticity, due to mixing, should weakly but not strongly converge as $t\to\infty$. In an important work, Bedrossian and Masmoudi studied the perturbative regime near Couette flow $(y,0)$ on an infinite cylinder, and proved small perturbation in the Gevrey space relaxes to a nearby shear flow. In this talk, we will explain a recent extension to the case of a finite cylinder (i.e. a periodic channel) with perturbations in a critical Gevrey space for this problem. The main interest of this extension is to consider the natural boundary effects, and to ensure that the Couette flow in our domain has finite energy. Joint work with Alex Ionescu.

- Series
- PDE Seminar
- Time
- Tuesday, October 16, 2018 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Prof. Yanyan Li – Rutgers University – yyli@math.rutgers.edu

We give derivative estimates for solutions to divergence form elliptic equations with piecewise
smooth coefficients. The novelty of these estimates is that, even though they depend on the shape
and on the size of the surfaces of discontinuity of the coefficients, they are independent of the
distance between these surfaces.

- Series
- PDE Seminar
- Time
- Thursday, October 11, 2018 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Marc Sedjro – African Institute for Mathematical Sciences, Tanzania

In this talk, we introduce several models of the so-called forward-forward Mean-Field Games (MFGs). The forward-forward models arise in the study of numerical schemes to approximate stationary MFGs. We establish a link between these models and a class of hyperbolic conservation laws. Furthermore, we investigate the existence of solutions and examine long-time limit properties. Joint work with Diogo Gomes and Levon Nurbekyan.

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