# Spring 2017 Analysis Seminars

# Analysis Seminar Schedule Spring 2017

date | speaker | title | host(s) | |
---|---|---|---|---|

January 17, Math Department Colloquium | Fabio Pusateri (Princeton) | The Water Waves Problem | Sigurd Angenent | |

January 24, Joint Analysis/Geometry Seminar | Tamás Darvas (Maryland) | Existence of constant scalar curvature Kähler metrics and properness of the K-energy | Jeff Viaclovsky | |

Monday, January 30, 3:30, VV901 (PDE Seminar) | Serguei Denissov (UW Madison) | Instability in 2D Euler equation of incompressible inviscid fluid | ||

February 7 | Andreas Seeger (UW Madison) | The Haar system in Sobolev spaces | ||

February 21 | Jongchon Kim (UW Madison) | Some remarks on Fourier restriction estimates | Andreas Seeger | |

March 7, Mathematics Department Distinguished Lecture | Roger Temam (Indiana) | On the mathematical modeling of the humid atmosphere | Leslie Smith | |

Wednesday, March 8, Joint Applied Math/PDE/Analysis Seminar | Roger Temam (Indiana) | Weak solutions of the Shigesada-Kawasaki-Teramoto system | Leslie Smith | |

March 14 | Xianghong Chen (UW Milwaukee) | Restricting the Fourier transform to some oscillating curves | Andreas Seeger | |

March 21 | SPRING BREAK |
| ||

Monday, March 27 (joint PDE/Analysis Seminar), 3:30, VV901 | Sylvia Serfaty (NYU) | Mean Field Limits for Ginzburg Landau Vortices | Hung Tran | |

March 28 | Brian Cook (Fields Institute) | Twists on the twisted ergodic theorems | Andreas Seeger | |

Friday, March 31, 4:00 p.m., B139 | Laura Cladek (UBC) | Endpoint bounds for the lacunary spherical maximal operator | Andreas Seeger | |

April 4 | Francesco Di Plinio (Virginia) | Sparse domination of singular integral operators | Andreas Seeger | |

April 11 | Xianghong Gong (UW Madison) | Hoelder estimates for homotopy operators on strictly pseudoconvex domains with C^2 boundary | ||

April 25 (joint PDE/Analysis Seminar) | Chris Henderson (University of Chicago) | A local-in-time Harnack inequality and applications to reaction-diffusion equations | Jessica Lin |

# Abstracts

### Fabio Pusateri

*The Water Waves problem*

We will begin by introducing the free boundary Euler equations which are a system of nonlinear PDEs modeling the motion of fluids, such as waves on the surface of the ocean. We will discuss several works done on this system in recent years, and how they fit into the broader context of the study of nonlinear evolution problems. We will then focus on the question of global regularity for water waves, present some of our main results - obtained in collaboration with Ionescu and Deng-Ionescu-Pausader - and sketch some of the main ideas.

### Tamás Darvas

*Existence of constant scalar curvature Kähler metrics and properness of the K-energy*

Given a compact Kähler manifold $(X,\omega)$, we show that if there exists a constant scalar curvature Kähler metric cohomologous to $\omega$ then Mabuchi's K-energy is J-proper in an appropriate sense, confirming a conjecture of Tian from the nineties. The proof involves a careful study of weak minimizers of the K-energy, and involves a surprising amount of analysis. This is joint work with Robert Berman and Chinh H. Lu.

### Serguei Denissov

*Instability in 2D Euler equation of incompressible inviscid fluid*

We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.

### Andreas Seeger

*The Haar system in Sobolev spaces*

We consider the Haar system on Sobolev spaces and ask: When is it a Schauder basis? When is it an unconditional basis? Some answers are given in recent joint work Tino Ullrich and Gustavo Garrigós.

### Jongchon Kim

*Some remarks on Fourier restriction estimates*

The Fourier restriction problem, raised by Stein in the 1960’s, is a hard open problem in harmonic analysis. Recently, Guth made some impressive progress on this problem using polynomial partitioning, a divide and conquer technique developed by Guth and Katz for some problems in incidence geometry. In this talk, I will introduce the restriction problem and the polynomial partitioning method. In addition, I will present some sharp L^p to L^q estimates for the Fourier extension operator that use an estimate of Guth as a black box.

### Roger Temam (Colloquium)

*On the mathematical modeling of the humid atmosphere*

The humid atmosphere is a multi-phase system, made of air, water vapor, cloud-condensate, and rain water (and possibly ice / snow, aerosols and other components). The possible changes of phase due to evaporation and condensation make the equations nonlinear, non-continuous (and non-monotone) in the framework of nonlinear partial differential equations. We will discuss some modeling aspects, and some issues of existence, uniqueness and regularity for the solutions of the considered problems, making use of convex analysis, variational inequalities, and quasi-variational inequalities.

### Roger Temam (Seminar)

*Weak solutions of the Shigesada-Kawasaki-Teramoto system*

We will present a result of existence of weak solutions to the Shigesada-Kawasaki-Teramoto system, in all dimensions. The method is based on new a priori estimates, the construction of approximate solutions and passage to the limit. The proof of existence is completely self-contained and does not rely on any earlier result. Based on an article with Du Pham, to appear in Nonlinear Analysis.

### Xianghong Chen

*Restricting the Fourier transform to some oscillating curves*

I will talk about Fourier restriction to some compact smooth curves. The problem is relatively well understood for curves with nonvanishing torsion due to work of Drury from the 80's, but is less so for curves that contain 'flat' points (i.e. vanishing torsion). Sharp results are known for some monomial-like or finite type curves by work of Bak-Oberlin-Seeger, Dendrinos-Mueller, and Stovall, where a geometric inequality (among others) plays an important role. Such an inequality fails to hold if the torsion demonstrates strong sign-changing behavior, in which case endpoint restriction bounds may fail. In this talk I will present how one could obtain sharp non-endpoint results for certain space curves of this kind. Our approach uses a covering lemma for smooth functions that strengthens a variation bound of Sjolin, who used it to obtain a similar result for plane curves. This is joint work with Dashan Fan and Lifeng Wang.

### Sylvia Serfaty

*Mean Field Limits for Ginzburg Landau Vortices*

Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.

### Brian Cook

*Twists on the twisted ergodic theorems*

The classical pointwise ergodic theorem has been adapted to include averages twisted by a phase polynomial, primary examples being the ergodic theorems of Wiener-Wintner and Lesigne. Certain uniform versions of these results are also known. Here uniformity refers to the collection of polynomials of degree less than some prescribed number. In this talk we wish to consider weakening the hypothesis in these latter results by considering uniformity over a smaller class of polynomials, which is naturally motivated when considering certain applications related to the circle method.

### Laura Cladek

*Endpoint bounds for the lacunary spherical maximal operator*

Define the lacunary spherical maximal operator as the maximal operator corresponding to averages over spheres of radius 2^k for k an integer. This operator may be viewed as a model case for studying more general classes of singular maximal operators and Radon transforms. It is a classical result in harmonic analysis that this operator is bounded on L^p for p>1, but the question of weak-type (1, 1) boundedness (which would correspond to pointwise convergence of lacunary spherical averages for functions in L^1 has remained open. Although this question still remains open, we discuss some new endpoint bounds for the operator near L^1 that allows us to conclude almost everywhere pointwise convergence of lacunary spherical means for functions in a slightly smaller space than L\log\log\log L. This is based on joint work with Ben Krause.

### Francesco di Plinio

*Sparse domination of singular integral operators*

Singular integral operators, which are a priori signed and non-local, can be dominated in norm, pointwise, or dually, by sparse averaging operators, which are in contrast positive and localized. The most striking consequence is that weighted norm inequalities for the singular integral follow from the corresponding, rather immediate estimates for the averaging operators. In this talk, we present several positive sparse domination results of singular integrals falling beyond the scope of classical Calderón-Zygmund theory; notably, modulation invariant multilinear singular integrals including the bilinear Hilbert transforms, variation norm Carleson operators, matrix-valued kernels, rough homogeneous singular integrals and critical Bochner-Riesz means, and singular integrals along submanifolds with curvature. Collaborators: Amalia Culiuc, Laura Cladek, Jose Manuel Conde-Alonso, Yen Do, Yumeng Ou and Gennady Uraltsev.

### Xianghong Gong

*Hoelder estimates for homotopy operators on strictly pseudoconvex domains with C^2 boundary*

Abstract: We derive a new homotopy formula for a bounded strictly pseudoconvex domain of C^2 boundary by using a method of Lieb and Range, and we obtain estimates for its homotopy operator. We show that the d-bar equation on the domain admits a solution gaining half-derivative in the Hoelder-Zygmund spaces. The estimates are also applied to obtain a boundary regularity for D-solutions on a suitable product domain in the Levi-flat Euclidean spaces.

### Chris Henderson

*A local-in-time Harnack inequality and applications to reaction-diffusion equations*

Abstract: The classical Harnack inequality requires one to look back in time to obtain a uniform lower bound on the solution to a parabolic equation. In this talk, I will introduce a Harnack-type inequality that allows us to remove this restriction at the expense of a slightly weaker bound. I will then discuss applications of this bound to (time permitting) three non-local reaction-diffusion equations arising in biology. In particular, in each case, this inequality allows us to show that solutions to these equations, which do not enjoy a maximum principle, may be compared with solutions to a related local equation, which does enjoy a maximum principle. Precise estimates of the propagation speed follow from this.