Seminars and Colloquia by Series

Stability of periodic waves for 1D NLS

Series
PDE Seminar
Time
Tuesday, March 31, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Stephen GustafsonUBC
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.

Global well-posedness for some cubic dispersive equations

Series
PDE Seminar
Time
Tuesday, March 24, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Benjamin DodsonJohns Hopkins University
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.

Onsager's Conjecture

Series
PDE Seminar
Time
Tuesday, March 10, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tristan BuckmasterCourant Institute, NYU
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.

Hölder Continuous Euler Flows

Series
PDE Seminar
Time
Tuesday, March 3, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
skiles 006
Speaker
Phillip IsettMIT
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.

On Splash and splat singularities for incompressible fluid interfaces

Series
PDE Seminar
Time
Tuesday, February 24, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Diego Cordoba GazolazICMAT
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.

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, YaguangShanghai 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 MarzuolaUniversity 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 TrivisaUniversity 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 ChenGeorgia 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.

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