## Seminars and Colloquia by Series

Series: PDE Seminar
Tuesday, April 14, 2015 - 15:05 , Location: Skiles 006 , Pierre-Emmanuel Jabin , University of Maryland, College Park , Organizer:
We consider some recent models from stochastic or optimal control involving a very large number of agents. The goal is to derive mean field limits when the number of agents increases to infinity. This presents some new unique difficulties; the corresponding master equation is a non linear Hamilton-Jacobi equation for instance instead of the linear transport equations that are more typical in the usual mean field limits. We can nevertheless pass to the limit by looking at the problem from an optimization point of view and by using an appropriate kinetic formulation. This is a joint work with S. Mischler, E. Sere, D. Talay.
Series: PDE Seminar
Thursday, April 9, 2015 - 15:05 , Location: Skile 005 , Tian-Yi Wang , The Chinese University of Hong Kong , Organizer:
This is a special PDE seminar in Skiles 005. In this talk, we will introduce the compactness framework for approximate solutions to sonic-subsonic flows governed by the irrotational steady compressible Euler equations in arbitrary dimension. After that, similar results will be presented for the isentropic case. As a direct application, we establish several existence theorems for multidimensional sonic-subsonic Euler flows. Also, we will show the recent progress on the incompressible limits.
Series: PDE Seminar
Tuesday, April 7, 2015 - 15:05 , Location: Skiles 006 , Benoit Pausader , Princeton University , Organizer:
The Euler-Maxwell system describes the interaction between a compressible fluid of electrons over a background of fixed ions and the self-consistent electromagnetic field created by the motion.We show that small irrotational perturbations of a constant equilibrium lead to solutions which remain globally smooth and return to equilibrium. This is in sharp contrast with the case of neutral fluids where shock creation happens even for very nice initial data.Mathematically, this is a quasilinear dispersive system and we show a small data-global solution result. The main challenge comes from the low dimension which leads to slow decay and from the fact that the nonlinearity has some badly resonant interactions which force a correction to the linear decay. This is joint work with Yu Deng and Alex Ionescu.
Series: PDE Seminar
Tuesday, March 31, 2015 - 15:05 , Location: Skiles 006 , Stephen Gustafson , UBC , Organizer:
Cubic focusing and defocusing Nonlinear Schroedinger Equations admit spatially (and temporally) periodic standing wave solutions given explicitly by elliptic functions. A natural question to ask is: are they stable in some sense (spectrally/linearly, orbitally, asymptotically,...), against some class of perturbations (same-period, multiple-period, general...)? Recent efforts have slightly enlarged our understanding of such issues. I'll give a short survey, and describe an elementary proof of the linear stability of some of these waves. Partly joint work in progress with S. Le Coz and T.-P. Tsai.
Series: PDE Seminar
Tuesday, March 24, 2015 - 15:05 , Location: Skiles 006 , Benjamin Dodson , Johns Hopkins University , Organizer:
In this talk we examine the cubic nonlinear wave and Schrodinger equations. In three dimensions, each of these equations is H^{1/2} critical. It has been showed that such equations are well-posed and scattering when the H^{1/2} norm is bounded, however, there is no known quantity that controls the H^{1/2} norm. In this talk we use the I-method to prove global well posedness for data in H^{s}, s > 1/2.
Series: PDE Seminar
Tuesday, March 10, 2015 - 15:05 , Location: Skiles 006 , Tristan Buckmaster , Courant Institute, NYU , Organizer:
In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which dissipate energy. The first part of this conjecture has since been confirmed (cf. Eyink 1994, Constantin, E and Titi 1994). During this talk we will discuss recent work by Camillo De Lellis, László Székelyhidi Jr., Philip Isett and myself related to resolving the second component of Onsager's conjecture. In particular, we will discuss the construction of weak non-conservative solutions to the Euler equations whose Hölder $1/3-\epsilon$ norm is Lebesgue integrable in time.
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 , , ICMAT , , 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: