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For Zoom URL please contactLaura Keller

Autumn Semester 2023

Date / Time Speaker Title Location
26 September 2023
15:15-16:15		 Dr. Stefano Decio
Institute for Advanced Study, Princeton
Event Details

Analysis Seminar

Title Zeros of Steklov eigenfunctions
Speaker, Affiliation Dr. Stefano Decio,Institute for Advanced Study, Princeton
Date, Time 26 September 2023, 15:15-16:15
Location HG G 43
Abstract A Steklov eigenfunction in a bounded domain is a harmonic function whose normal derivative at the boundary is proportional to the function itself. I will tell you most of what I know about the zero sets of such functions. A nice fact is that there are many zeros near the boundary: I will give a gentle proof of this in the first part of the talk. In the second, perhaps a little less gentle, part I will discuss some lower and upper bounds for the Hausdorff measure of the zero set; several questions remain unanswered. Comparisons with the (slightly) better understood case of eigenfunctions of the Laplace-Beltrami operator will also be provided.
Zeros of Steklov eigenfunctionsread_more
HG G 43
24 October 2023
15:15-16:15		 Dr. Mitchell Taylor
ETH Zurich, Switzerland
Event Details

Analysis Seminar

Title Low regularity well-posedness for the general quasilinear Schrödinger equation
Speaker, Affiliation Dr. Mitchell Taylor,ETH Zurich, Switzerland
Date, Time 24 October 2023, 15:15-16:15
Location HG G 43
Abstract We present a new and relatively simple method for proving large data local well-posedness in low regularity Sobolev spaces for general quasilinear Schrödinger equations of the form \begin{equation*} \begin{cases} &i\partial_tu+g^{jk}(u,\overline{u},\nabla u,\nabla\overline{u})\partial_j\partial_k u=F(u,\overline{u},\nabla u,\nabla\overline{u}),\hspace{5mm} u:\mathbb{R}\times\mathbb{R}^d\to\mathbb{C}^m, \\ &u(0,x)=u_0(x), \end{cases} \end{equation*} assuming only non-degeneracy of the metric, nontrapping and mild regularity/decay of the initial data. As a consequence, we remove the uniform ellipticity assumption from the main result of Marzuola, Metcalfe and Tataru (Arch. Ration. Mech. Anal. 2021) and substantially weaken the regularity/decay assumptions from the pioneering works of Kenig, Ponce, Rolvung and Vega. This is based on joint work with Ben Pineau (UC Berkeley).
Low regularity well-posedness for the general quasilinear Schrödinger equationread_more
HG G 43
14 November 2023
15:15-16:15		 Ben Pineau
UC Berkeley
Event Details

Analysis Seminar

Title Title: Sharp Hadamard well-posedness for the incompressible free boundary Euler equations
Speaker, Affiliation Ben Pineau,UC Berkeley
Date, Time 14 November 2023, 15:15-16:15
Location HG G 43
Abstract I will talk about a recent preprint in which we establish an optimal local well-posedness theory in $H^s$ based Sobolev spaces for the free boundary incompressible Euler equations on a connected fluid domain. Some components of this result include: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: A uniqueness result which holds at the level of the Lipschitz norm of the velocity and the $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove essentially scale invariant energy estimates for solutions, relying on a newly constructed family of refined elliptic estimates; (v) Continuation criterion: We give the first proof of a continuation criterion at the same scale as the classical Beale-Kato-Majda criterion for the Euler equations on the whole space. Roughly speaking, we show that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of the construction of regular solutions. \\ Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is based on joint work with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.
Title: Sharp Hadamard well-posedness for the incompressible free boundary Euler equationsread_more
HG G 43
19 December 2023
15:15-16:15		 Prof. Dr. Richard Sowers
University of Illinois Urbana-Champaign
Event Details

Analysis Seminar

Title Lateral boundary conditions for a Kolmogorov-type PDE
Speaker, Affiliation Prof. Dr. Richard Sowers,University of Illinois Urbana-Champaign
Date, Time 19 December 2023, 15:15-16:15
Location HG G 43
Abstract 我们考虑一个hypoelliptic Kolmogorov-type PDE软木responding to a particle under white noise force. We are interested in imposing Dirichlet conditions at a side boundary. We construct a simple Gaussian heat kernel inside the domain, and investigate a boundary-layer kernel. We show that this boundary layer heat kernel has a novel jump condition. We outline a polynomial expansion of the heat kernels, and then construct a Volterra equation.This Volterra equation has a periodic structure resulting from the novel jump condition.
Lateral boundary conditions for a Kolmogorov-type PDEread_more
HG G 43

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