| Talk_id | Date | Speaker | Title |
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16478
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Wednesday 2/6
4:10 PM
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Mihai Tohaneanu, University of Kentucky
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Local energy estimates on black hole backgrounds
- Mihai Tohaneanu, University of Kentucky
- Local energy estimates on black hole backgrounds
- 02/06/2019
- 4:10 PM - 5:00 PM
- C517 Wells Hall
Local energy estimates are a robust way to measure decay of solutions to linear wave equations. I will discuss several such results on black hole backgrounds, such as Schwarzschild, Kerr, and suitable perturbations converging at various rates, and briefly discuss applications to nonlinear problems. The most challenging geometric feature one needs to deal with is the presence of trapped null geodesics, whose presence yield unavoidable losses in the estimates. This is joint work with Lindblad, Marzuola, Metcalfe, and Tataru.
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17490
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Wednesday 2/13
4:10 PM
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Albert Ai, UC Berkeley
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Low Regularity Solutions for Gravity Water Waves
- Albert Ai, UC Berkeley
- Low Regularity Solutions for Gravity Water Waves
- 02/13/2019
- 4:10 PM - 5:00 PM
- C517 Wells Hall
We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness, below that which is attainable by energy estimates alone. This program was initiated for gravity water waves by Alazard-Burq-Zuily, by proving Strichartz estimates with loss. We discuss how these Strichartz estimates, and thus the low regularity threshold, can be sharpened by applying an integration along the Hamilton flow combined with local smoothing estimates.
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17491
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Wednesday 3/13
4:10 PM
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Jonas Lührmann, Johns Hopkins
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Local smoothing estimates for Schrödinger equations on hyperbolic space and applications
- Jonas Lührmann, Johns Hopkins
- Local smoothing estimates for Schrödinger equations on hyperbolic space and applications
- 03/13/2019
- 4:10 PM - 5:00 PM
- C517 Wells Hall
We establish frequency-localized local smoothing estimates for Schrödinger equations on hyperbolic space. The proof is based on the positive commutator method and a heat flow based Littlewood-Paley theory. Our results and techniques are motivated by applications to the problem of stability of solitary waves to nonlinear Schrödinger-type equations on hyperbolic space.
The talk is based on joint work with Andrew Lawrie, Sung-Jin Oh, and Sohrab Shahshahani.
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17492
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Wednesday 3/27
4:10 PM
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Yash Jhaveri, Institute for Advanced Study
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Higher Regularity of the Singular Set in the Thin Obstacle Problem
- Yash Jhaveri, Institute for Advanced Study
- Higher Regularity of the Singular Set in the Thin Obstacle Problem
- 03/27/2019
- 4:10 PM - 5:00 PM
- C517 Wells Hall
In this talk, I will give an overview of some of what is known about solutions to the thin obstacle problem, and then move on to a discussion of a higher regularity result on the singular part of the free boundary. This is joint work with Xavier Fernández-Real.
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17493
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Wednesday 4/24
4:10 PM
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Nestor Guillen, University of Massachusetts at Amherst
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Transportation methods for Lévy measures and applications
- Nestor Guillen, University of Massachusetts at Amherst
- Transportation methods for Lévy measures and applications
- 04/24/2019
- 4:10 PM - 5:00 PM
- C517 Wells Hall
The comparison principle for second order elliptic equations is one of the cornerstone of the theory of viscosity solutions. Works of Sayah and more recently by Jakobsen-Karlsen and Barles-Imbert have expanded it to many important subfamilies of nonlocal elliptic equations. In joint work with Chenchen Mou and Andrzej Święch we show how optimal transportation methods can be used to couple Lévy measures, which encode the integro-differential part of nonlocal operators, which allow us to obtain comparison principles for new families of nonlocal equations. Our method puts all previous subfamilies of nonlocal operators (Levy-Ito operators, operators of order less than 1) in a single framework, while also yielding results for new families.
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