Seminars and Colloquia by Series

Series: PDE Seminar
Wednesday, February 18, 2015 - 11:05 , Location: Skiles 170 (Special) , Wang, Yaguang , Shanghai Jiaotong University , Organizer:
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.
Series: PDE Seminar
Tuesday, February 17, 2015 - 15:05 , Location: Skiles 006 , Robin Young , University of Massachusetts, Amherst , Organizer:
Series: PDE Seminar
Tuesday, January 27, 2015 - 15:05 , Location: Skiles 006 , Jeremy Marzuola , University of North Carolina at Chapel Hill , Organizer:
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.
Series: PDE Seminar
Tuesday, January 13, 2015 - 15:05 , Location: Skiles 006 , Konstantina Trivisa , University of Maryland , Organizer:
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.
Series: PDE Seminar
Tuesday, December 9, 2014 - 15:05 , Location: Skiles 006 , Geng Chen , Georgia Tech , Organizer:
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. 
Series: PDE Seminar
Tuesday, December 2, 2014 - 15:00 , Location: Skiles 006 , Pierre Germain , Courant Institute , Organizer:
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.
Series: PDE Seminar
Tuesday, November 25, 2014 - 15:05 , Location: Skiles 006 , Changyou Wang , Purdue University , Organizer:
 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). 
Series: PDE Seminar
Tuesday, November 4, 2014 - 15:00 , Location: Skiles 006 , Michael Shearer , North Carolina State University , Organizer:
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.
Series: PDE Seminar
Tuesday, October 28, 2014 - 15:05 , Location: Skiles 006 , Albert Fathi , École Normale Supérieure de Lyon, France , Organizer:
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. 
Series: PDE Seminar
Tuesday, October 7, 2014 - 15:00 , Location: Skiles 006 , Xuwen Chen , Brown University , Organizer:
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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