Seminars and Colloquia by Series

Stability of Three-dimensional Prandtl Boundary Layers

Series
PDE Seminar
Time
Wednesday, February 18, 2015 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 170 (Special)
Speaker
Wang, Yaguang – Shanghai Jiaotong University
In this talk, we shall study the stability of the Prandtl boundary layer equations in three space variables. First, we obtain a well-posedness result of the three-dimensional Prandtl equations under some constraint on its flow structure. It reveals that the classical Burgers equation plays an important role in determining this type of flow with special structure, that avoids the appearance of the complicated secondary flow in the three-dimensional Prandtl boundary layers. Second, we give an instability criterion for the Prandtl equations in three space variables. Both of linear and nonlinear stability are considered. This criterion shows that the monotonic shear flow is linearly stable for the three dimensional Prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable, which is an exact complement to the above well-posedness result for a special flow. This is a joint work with Chengjie Liu and Tong Yang.

Quasilinear Schrödinger equations

Series
PDE Seminar
Time
Tuesday, January 27, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jeremy Marzuola – University of North Carolina at Chapel Hill
We survey some recent results by the speaker, Jason Metcalfe and Daniel Tataru for small data local well-posedness of quasilinear Schrödinger equations. In addition, we will discuss some applications recently explored with Jianfeng Lu and recent progress towards the large data short time problem. Along the way, we will attempt to motivate analysis of the problem with connections to problems from Density Functional Theory.

On kinetic models for the collective self-organization of agents

Series
PDE Seminar
Time
Tuesday, January 13, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Konstantina Trivisa – University of Maryland
A class of kinetic models for the collective self-organization of agents is presented. Results on the global existence of weak solutions as well as a hydrodynamic limit will be discussed. The main tools employed in the analysis are the velocity averaging lemma and the relative entropy method. This is joint work with T. Karper and A. Mellet.

Large solutions for compressible Euler equations in one space dimension

Series
PDE Seminar
Time
Tuesday, December 9, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Geng Chen – Georgia Tech
The existence of large BV (total variation) solution for compressible Euler equations in one space dimension is a major open problem in the hyperbolic conservation laws, where the small BV existence was first established by James Glimm in his celebrated paper in 1964. In this talk, I will discuss the recent progress toward this longstanding open problem joint with my collaborators. The singularity (shock) formation and behaviors of large data solutions will also be discussed.

Infinite volume limit for the Nonlinear Schrodinger Equation and Weak Turbulence

Series
PDE Seminar
Time
Tuesday, December 2, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Pierre Germain – Courant Institute
Abstract: the theory of weak turbulence has been put forward by appliedmathematicians to describe the asymptotic behavior of NLS set on a compactdomain - as well as many other infinite dimensional Hamiltonian systems.It is believed to be valid in a statistical sense, in the weaklynonlinear, infinite volume limit. I will present how these limits can betaken rigorously, and give rise to new equations.

Everywhere differentiability of viscosity solutions to a class of Aronsson's equations

Series
PDE Seminar
Time
Tuesday, November 25, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Changyou Wang – Purdue University
For a $C^{1,1}$-uniformly elliptic matrix $A$, let $H(x,p)=$ be the corresponding Hamiltonian function. Consider the Aronsson equation associated with $H$: $$(H(x,Du))x H_p(x,Du)=0.$$ In this talk, I will indicate everywhere differentiability of any viscosity solution of the above Aronsson's equation. This extends an important theorem by Evans and Smart on the infinity harmonic functions (i.e. $A$ is the identity matrix).

Shock wave solutions of conservation laws and their regularization by dissipation and dispersion.

Series
PDE Seminar
Time
Tuesday, November 4, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael Shearer – North Carolina State University
Shock waves are idealizations of steep spatial gradients of physical quantities such as pressure and density in a gas, or stress in an elastic solid. In this talk, I outline the mathematics of shock waves for nonlinear partial differential equations that are simple models of physical systems. I will focus on non-classical shocks and smooth waves that they approximate. Of particular interest are comparisons between nonlinear traveling waves influenced strongly by dissipative effects such as viscosity or surface tension, and spreading waves generated by the balance between dispersion and nonlinearity, when the nonlinearity is non-convex.

Regularity of Solutions of Hamilton-Jacobi Equation on a Domain

Series
PDE Seminar
Time
Tuesday, October 28, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Albert Fathi – École Normale Supérieure de Lyon, France
In this lecture, we will explain a new method to show that regularity on the boundary of a domain implies regularity in the inside for PDE's of the Hamilton-Jacobi type. The method can be applied in different settings. One of these settings concerns continuous viscosity solutions $U : T^N\times [0,+\infty[ \rightarrow R$ of the evolutionary equation $\partial_t U(x, t) + H(x, \partial_x U(x, t) ) = 0,$ where $T^N = R^N / Z^N$, and $H: T^N \times R^N$ is a Tonelli Hamiltonian, i.e. H(x, p) is $C^2$, strictly convex superlinear in p. Let D be a compact smooth domain with boundary $\partial D$ contained in $T^N \times ]0,+\infty[$ . We show that if U is differentiable at each point of $\partial D$, then this is also the case on the interior of D. There are several variants of this result in different settings. To make the result accessible to the layman, we will explain the method on the function distance to a closed subset of an Euclidean space. This example contains all the ideas of the general case.

The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-body Evolution

Series
PDE Seminar
Time
Tuesday, October 7, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Xuwen Chen – Brown University
We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is focusing and matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D N-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to infinity. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D to 1D coupling constant.

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