Programme
Monday, 24 June
Tuesday, 25 June
Wednesday, 26 June
Thursday, 27 June
Titles and Abstracts (Mini-Courses)
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John Anderson
Title: Long time behavior of nonlinear wave equations
In these lectures, I hope to describe some of the common techniques and properties of waves that are involved in studying nonlinear wave equations for time scales which are large compared to what is covered by the proof of local well-posedness. The lectures will walk through two examples: the first example is a toy model for the Einstein equations of general relativity and the second is a toy model for the compressible Euler equations of gas dynamics. These seemingly very similar equations will display significantly different behavior over large time scales: the first will essentially behave like the linear wave equation, while the second will form a singularity (a shock) in finite time. In addition to describing the mathematical techniques that these examples display, I hope to describe how such questions arise naturally when studying physical models.
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Tarek Elgindi
Title: Finite-time singularity formation in incompressible fluids
We will review various results concerning singularity formation in incompressible fluids equations. We will discuss two techniques, scale-invariance and self-similarity, that were introduced to prove singularity formation in the Euler equation in different settings.
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Grigorios Fournodavlos
Title: On the nature of the Big Bang singularity
In this mini-course we will begin with a review of the classical BKL heuristics regarding solutions to the Einstein equations containing a Big Bang singularity. We will discuss the conjectural picture of the very rich dynamics of Big Bang singularities and comment on the existing works in the literature. Then, we will present some of the state of the art results concerning solutions that exhibit Kasner-like behavior.
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Titles and Abstracts (Talks)
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Daniel Ginsberg
Title: The stability of irrotational shocks and the Landau law of decay
It is well-known that in three space dimensions, smooth solutions to the equations describing a compressible gas can break down in finite time. One type of singularity which can arise is known as a "shock", which is a hypersurface of discontinuity across which the integral forms of conservation< of mass and momentum hold and through which there is nonzero mass flux. One can find approximate solutions to the equations of motion which describe expanding spherical shocks. We use these model solutions to construct global-in-time solutions to the irrotational compressible Euler equations with shocks. This is joint work with Igor Rodnianski.
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Cécile Huneau
Title: Burnett's conjecture in General Relativity
In this talk, I will present a work in collaboration with Jonathan Luk, on the behaviour of weak limits of solutions to Einstein equations. In 1989, Burnett conjectured that a metric, obtained as a uniform limit of solutions to Einstein vacuum equation, whose derivatives converge only weakly, is a solution to Einstein-massless Vlasov equations. This conjecture has been back in the spotlight with the work of Green and Wald on perturbations in cosmology. In a recent work with Jonathan Luk, we prove this conjecture, without symmetry assumption, in generalized wave coordinates.
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Hyunju Kwon
Title: Strong Onsager conjecture
Smooth solutions to the incompressible 3D Euler equations conserve kinetic energy in every local region of a periodic spatial domain. In particular, the total kinetic energy remains conserved. When the regularity of an Euler flow falls below a certain threshold, a violation of total kinetic energy conservation has been predicted due to anomalous dissipation in turbulence, leading to Onsager's theorem. Subsequently, the \( L^3 \)-based strong Onsager conjecture has been proposed to reflect the intermittent nature of turbulence and the local evolution of kinetic energy. This conjecture states the existence of Euler flows with regularity below the threshold of \( B^{1/3}_{3,\infty} \) which not only dissipate total kinetic energy but also exhibit intermittency and satisfy the local energy inequality. In this talk, I will discuss the resolution of this conjecture based on recent collaboration with Matthew Novack and Vikram Giri.
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Jonathan Luk
Title: Late time tails of linear and nonlinear waves
I will present a recent joint work with Sung-Jin Oh (Berkeley), where we develop a general method for understanding the precise late time asymptotic behavior of solutions to linear and nonlinear wave equations in odd spatial dimensions. In particular, we prove that in the presence of a nonlinearity and/or a dynamical background, the late time tails are in general different from the better understood case of linear equations on stationary backgrounds. I will explain how the late time tails are related to the problem of the singularity structure in the interior of generic dynamical vacuum black holes in general relativity.
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Georgios Moschidis
Title: Weak turbulence on Schwarzschild-AdS spacetime
In the presence of confinement, the Einstein field equations are expected to exhibit turbulent dynamics. The simplest example of such behaviour is described by the AdS instability conjecture, put forward by Dafermos and Holzegel in 2006; this conjecture suggests that generic small perturbations of the AdS initial data lead to the formation of trapped surfaces when reflecting boundary conditions are imposed at conformal infinity. However, whether a similar scenario also holds in the more complicated case of the exterior region of an asymptotically AdS black hole spacetime has been the subject of debate.
In this talk, we will show that weak turbulence does emerge in the dynamics of a quasilinear toy model for the vacuum Einstein equations on the Schwarzschild-AdS exterior spacetimes for an open and dense set of black hole mass parameters. This is joint work with Christoph Kehle.
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Fabio Pusateri
Title: Recent results on long-time existence for the free boundary Euler equations
We will present some recent results on the long-time regularity of solutions to the free boundary Euler equations.
First, we will discuss a result for the full problem with vorticity and gravity in three space dimensions. For small and localized initial data, we show regularity up to a time that is (almost) of the order of one over the size of the initial vorticity. This is a natural time scale for the evolution of the vorticity and, importantly, it is independent of the size of the irrotational components. This is joint work with Daniel Ginsberg (CUNY).
We will then discuss recent and ongoing joint works with Yu Deng (USC) and Alexandru Ionescu (Princeton), motivated by the theory of weak wave turbulence, about irrotational solutions on large tori. We obtain new deterministic results for gravity waves in one space dimension and a long-time result for random solutions.
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