UVA Harmonic Analysis & PDEs Seminar, Spring 2017
Organizers: Francesco Di Plinio, Yen Do, Aseel Farhat, Zoran Grujic
Regular time and location: Tuesdays, 3:304:30pm, 317 Kerchof Hall
This semester we meet Mondays 1111:50 in 317 Kerchof Hall.
Jan 23, 2017
(Kerchof 326,
1111:50)

Organizational meeting

Jan 30, 2017
(Kerchof 326, 1111:50)

Juraj Foldes (UVA)
Title: Entropy, Euler equation, and Variational methods
Abstract: Two dimensional turbulent flows for large Reynold's numbers can be approximated by solutions of incompressible Euler's equation. As time increases, the solutions of Euler's equation are increasing their disorder; however, at the same time, they are limited by the existence of infinitely many invariants. Hence, it is natural to assume that the limit profiles are functions which maximize an entropy given the values of conserved quantities. Such solutions are described by methods of Statistical Mechanics and are called maximal entropy solutions. Nevertheless, there is no general agreement in the literature on what is the right notion of the entropy. We will show that on symmetric domains, independently of the choice of entropy, the maximal entropy solutions with small energy respect the geometry of the domain.
This is a joint work with Vladimir Sverak (University of Minnesota).

Feb 06, 2017
(Kerchof 317, 1111:50)

Aseel Farhat (UVA)
Title: Mixing By Incompressible Flows
Abstract: We will go over two recent papers on mixing by (1) G. Alberti, G. Crippa and A. Mazzucato and by (2) Y. Yao and A. Zlatos. We will present the two constructions of incompressible flows in L^\infty(W^{1,1}) that rmixes meanzero initial data to scale r in O(log(r)) time. That is, that the bound in the rearrangement cost conjecture by Bressan is optimal when p=1.

Feb 08, 2017
(Kerchof 326, 12pm)

Aseel Farhat (UVA)
Title: Mixing By Incompressible Flows (part II)
(cancelled, moved to next week, see below)

Feb 13, 2017
(Kerchof 317, 1112)

Aseel Farhat (UVA)
Title: Mixing By Incompressible Flows (part II)

Feb 20, 2017


Feb 27, 2017
(Kerchof 317, 1112)

Peter Perry (U Kentucky)
Title: Soliton Resolution for the Derivative Nonlinear Schr\"{o}dinger Equation
Abstract: This talk reports on joint work with Robert Jenkins, Jiaqi Liu, and Catherine Sulem. The derivative nonlinear Schr\"{o}dinger equation (DNLS) is a dispersive nonlinear PDE in one space and one time dimension which describes the propagation of Alfv\'{e}n waves in plasmas, is known to admit solitary wave solutions, and was shown to be completely integrable by Kaup and Newell. We'll outline a proof using the inverse scattering method that this equation has solutions global in time for generic initial data in a weighted Sobolev space and show that for such data, the solution is asymptotic to the sum of a finite number of solitons plus radiation.

March 06, 2017

Spring Break

March 13, 2017
(Kerchof 317, 1111:50)

Liaosha Xu (UVA)
Titte: Pointwise Multipliers In Weighted BMOs And L log^k L Estimates For Vorticity
Abstract:
In this talk we first review some results in harmonic analysis and a theorem on L log L bound for vorticity by Zachary and Zoran. Then we attempt to generalize the theorem by studying L log^k L bound for the vorticity. The main challenge is to extend the pointwise multiplier theorem and the Logcharacterization of BMO for more general weight functions. If time allows, there will also be some intuition behind this generalization, i.e. how this could improve the results in a paper on subcritical blowup scenarios for 3D NSE.

March 20, 2017

Jared Whitehead (BYU)
Titte: Convective systems with additive noise
Abstract:
We investigate some of the mathematical and physical properties of a convective system with additive stochastic noise in the temperature evolution equation. In particular, we develop a methodology to evaluate the stability of a base conductive state even in the presence of a timedependent white noise. This is joint work with Juraj Foldes, Nathan GlattHoltz, and Geordie Richards.

March 27, 2017
(Kerchof 317, 1111:50)

Liaosha Xu (UVA)
Titte: Pointwise Multipliers In Weighted BMOs And L log^k L Estimates For Vorticity, part II

April 03, 2017

Armen Shirikyan
Title: TBA
Abstract: TBA

April 10, 2017

Mark Lewers (UVA)
Title: Lp Outer Measure Theory and (Outer) Carleson Embeddings
Abstract: This talk is review of what we went over in a series of talks this past fall (based off a paper by Y. Do and C. Thiele). We will go over the theory of Lp spaces using outer measures rather than classical measure theory. Unlike previously developed theories based on super level sets, we'll go over the theory developed by Do and Thiele which uses predefined averages over generating sets of the outer measure, called super level measures. Besides going over some basic intuitive results in outer Lp spaces, we'll investigate how Carleson's embedding map can be extended to the outer Lp case on the upper half plane.

April 17, 2017


April 24, 2017
(Kerchof 317, 1111:50)

Kelsey Goodin (UVA)
Title: An Introduction to the Calculus of Variations
Abstract: We can consider a special case of partial differential equations, Euler Lagrange Equations, whose solutions can be characterized as minimum values of functional equations. To solve these equations, we appeal to calculus of variations, which is a functional analytic analogue of the optimization concepts from undergrad calculus. In this talk we’ll introduce variational methods for solving PDE’s and discuss techniques for tackling nonconstant boundary conditions and additional constraints. Time permitting; we’ll discuss the Weierstrass construction and fields.

*indicates different day, time, and/or location
Past Talks (Click here for current semester)
