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Series: PDE Seminar

TBA

Series: PDE Seminar

TBA

Series: PDE Seminar

TBA

Series: PDE Seminar

In this talk we will review compactness results and singularity theorems related to harmonic maps. We first talk about maps from Riemann surfaces with tension fields bounded in a local Hardy space, then talk about stationary harmonic maps from higher dimensional manifolds, finally talk about heat flow of harmonic maps.

Series: PDE Seminar

In this talk we study master equations arising from mean field game

problems, under the crucial monotonicity conditions.

Classical solutions of such equations require very strong technical

conditions. Moreover, unlike the master equations arising from mean

field control problems, the mean field game master equations are

non-local and even classical solutions typically do not satisfy the

comparison principle, so the standard viscosity solution approach seems

infeasible. We shall propose a notion of weak solution for such

equations and establish its wellposedness. Our approach relies on a new

smooth mollifier for functions of measures, which unfortunately does not

keep the monotonicity property, and the stability result of master

equations. The talk is based on a joint work with Jianfeng Zhang.

Series: PDE Seminar

Series: PDE Seminar

I will consider the motion of a rigid body with an interior cavity that is completely filled with a viscous fluid. The equilibria of the system will be characterized and their stability properties are analyzed. It will be shown that the fluid exerts a stabilizing effect, driving the system towards a state where it is moving as a rigid body with constant angular velocity. In addition, I will characterize the critical spaces for the governing evolution equation, and I will show how parabolic regularization in time-weighted spaces affords great flexibility in establishing regularity and stability properties for the system. The approach is based on the theory of Lp-Lq maximal regularity. (Joint work with G. Mazzone and J. Prüss).

Series: PDE Seminar

In a joint work with Sameer Iyer, the validity of steady Prandtl layer expansion is established in a channel. Our result covers the celebrated Blasius boundary layer profile, which is based on uniform quotient estimates for the derivative Navier-Stokes equations, as well as a positivity estimate at the flow entrance.

Series: PDE Seminar

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

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.