Seminars and Colloquia by Series

Series: PDE Seminar
Tuesday, March 3, 2015 - 15:05 , Location: skiles 006 , Phillip Isett , MIT , Organizer:
Motivated by the theory of hydrodynamic turbulence, L. Onsager conjectured in 1949 that solutions to the incompressible Euler equations with Holder regularity less than 1/3 may fail to conserve energy.  C. De Lellis and L. Székelyhidi, Jr. have pioneered an approach to constructing such irregular flows based on an iteration scheme known as convex integration.  This approach involves correcting “approximate solutions" by adding rapid oscillations, which are designed to reduce the error term in solving the equation.  In this talk, I will discuss an improved convex integration framework, which yields solutions with Holder regularity 1/5- as well as other recent results.
Series: PDE Seminar
Tuesday, February 24, 2015 - 15:00 , Location: Skiles 006 , Prof. Diego Cordoba Gazolaz , ICMAT , dcg@icmat.es , Organizer: Ronghua Pan
For the water waves system we have shown the formation in finite time of splash and splat singularities. A splash singularity is when the interface remain smooth but self-intersects at a point and a splat singularity is when it self-intersects along an arc. In this talk I will discuss new results on stationary splash singularities for water waves and in the case of a parabolic system a splash can also develop but not a splat singularity. 
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.

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