GSSI Analysis & PDE Seminar
Organizers: Paolo Antonelli, Gabriele Benomio, Yuri Cacchiò, Sara Daneri, Luigi De Rosa, Francesco Paolo Maiale, Pierangelo Marcati, Stefano Modena, Eris Runa
Location & time: Gran Sasso Science Institute, weekly on Tuesdays, 2:30-3:30pm (map)
Upcoming Seminars: (see also GSSI Academic Calendar)
14 January 2025, 2:30-3:30pm, GSSI Main Lecture Hall
Speaker: Jules Pitcho (Gran Sasso Science Institute)
Title: Non-uniqueness and recovery of uniqueness: the transport equation
Abstract: In this talk, I will discuss the history of the solution theory for the transport equation along a rough vector field. In the late twentieth century, DiPerna and Lions established well-posedness for bounded weak solutions when the vector field is Sobolev, and a measure-theoretic uniqueness for the flow of Sobolev vector fields. Recent works have used convex integration to construct non-unique trajectories of Sobolev vector fields for almost every initial datum thereby showing that the measure-theoretic notion of uniqueness for the flow put forward by DiPerna and Lions is strictly weaker than point wise uniqueness of trajectories. Twenty years ago, Ambrosio proved well-posedness for bounded weak solutions when the vector field is of bounded variation. An example of Depauw then showed that this result is sharp: uniqueness of bounded weak solution is not to be expected for a general bounded vector field. But a weaker notion of uniqueness is still within reach for the Depauw vector field, and in fact for a whole class of vector field comprising it. A corresponding unique flow, stochastic in nature, also exists for such vector field. However for merely bounded vector fields, this weaker notion of uniqueness fails over a dense set.
21 January 2025, 2:30-3:30pm, GSSI Main Lecture Hall
Speaker: Raffaele Scandone (University of Naples Federico II)
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28 January 2025, 2:30-3:30pm, GSSI Main Lecture Hall
Speaker: Federico Cacciafesta (University of Padua)
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4 February 2025, 2:30-3:30pm, GSSI Main Lecture Hall
Speaker: Flavio Rossetti (Instituto Superior Técnico)
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18 February 2025, 2:30-3:30pm, GSSI Main Lecture Hall
Speaker: Martin Taylor (Imperial College London)
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25 February 2025, 2:30-3:30pm, GSSI Main Lecture Hall
Speaker: Michael Goldman (CNRS/École Polytechnique)
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4 March 2025, 2:30-3:30pm, GSSI Main Lecture Hall
Speaker: Massimo Sorella (Imperial College London)
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13 March 2025, 2:30-3:30pm, GSSI Main Lecture Hall
Speaker: Matthew Schrecker (University College London)
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18 March 2025, 2:30-3:30pm, GSSI Main Lecture Hall
Speaker: Lucio Galeati (University of L'Aquila)
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25 March 2025, 2:30-3:30pm, GSSI Main Lecture Hall
Speaker: Rita Teixeira da Costa (University of Cambridge)
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