London PDE Seminar

Past Seminars

A list of past talks at the London PDE Seminar can be found here.

Date/Time: 11:00 (UK time), Friday, 23 June 2023

Speaker: Warren Li (Princeton)

Title: Kasner-like description of spacelike singularities in spherical symmetry

Abstract: Following the investigations of Christodoulou, the spherically symmetric Einstein-(Maxwell)-scalar field equations have been popularised as a toy model for the Einstein equations, particularly regarding singularity formation and the cosmic censorship conjectures. Focusing on the so-called spacelike part of the singularity, we present new results describing precise leading order blow-up rates of the scalar field, the Hawking mass, and the Kretschmann scalar near the singularity. We also introduce the BKL heuristics of relativistic cosmology, and explain how our results may be viewed in this cosmological context.

Date/Time: 11:00 (UK time), Monday, 19 June 2023 (special date)

Speaker: Wilhelm Schlag (Yale)

Title: On continuous time bubbling for the harmonic map heat flow in two dimensions

Abstract: I will describe recent work with Jacek Jendrej (CNRS, Paris Nord) and Andrew Lawrie (MIT) on harmonic maps of finite energy from the plane to the two sphere, without making any symmetry assumptions. While it has been known since the 1990s that bubbling occurs along a carefully chosen sequence of times via an elliptic Palais-Smale mechanism, we show that this continues to hold continuously in time. The key notion is that of the "minimal collision energy" which appears in the soliton resolution result by Jendrej and Lawrie on critical equivariant wave maps.

Date/Time: 11:00 (UK time), Friday, 16 June 2023

Speaker: Irfan Glogić (Vienna)

Title: Global-in-space stability of self-similar blowup for the wave maps equation

Abstract: We consider wave maps from the \((1+d)\)-dimensional Minkowski space into the \(d\)-sphere. Numerical simulations of this model indicate that in the energy supercritical case, \(d \geq 3\), generic large data lead to finite time blowup via an explicitly known self-similar solution. In the effort of rigorously proving these observations, many works have been produced over the last decade, starting with the pioneering work of Aichelburg-Donninger-Schörkhuber. In this talk, we outline a novel general framework for the analysis of spatially global stability of self-similar solutions to semilinear wave equations. We then implement this scheme in the aforementioned context of wave maps, thereby obtaining the first nonlinear stability result that is global-in-space. At the and, we discuss further open problems as well as the new mathematical challenges that our approach generates.

Date/Time: 10:00 (UK time), Tuesday, 30 May 2023 (special date and time)

Speaker: Yan Guo (Brown)

Title: Global smooth axisymmetric Euler flows with rotation

Abstract: Consider incompressible Euler equations in the presence of a constant rotation (e.g. Coriolis force). We construct global in time smooth axisymmetric flows with small amplitude.

Date/Time: 11:00 (UK time), Friday, 9 June 2023

Speaker: Toan Nguyen (Penn State)

Title: Landau damping and the survival threshold

Abstract: The talk is to precise the classical notion of Landau damping and to provide the survival threshold of spatial frequencies that dictates the transition from purely oscillatory modes, known as Langmuir waves, to the free dynamics of electrons near spatially homogeneous backgrounds, classically modeled by the Vlasov-Poisson system in plasma physics or the Hartree-Coulomb equations in quantum mechanics. The transition occurs due to the exact resonant interaction between excited electrons and the oscillatory waves, namely the classical Landau damping.

Date/Time: 11:00 (UK time), Friday, 26 May 2023

Speaker: Juhi Jang (University of Southern California)

Title: Gravitational collapse of gaseous stars

Abstract: A classical model of a self-gravitating Newtonian star is given by the gravitational Euler-Poisson system, while a relativistic star is modeled by the Einstein-Euler system. I will review some recent progress on the local and global dynamics of Newtonian star solutions, and discuss mathematical construction of self-similar gravitational collapse of Newtonian stars including Larson-Penston solution for the isothermal stars, Yahil solution for polytropic stars, which show the existence of smooth initial data that lead to finite time collapse, characterized by the blow-up of the star density. If time permits, I will also discuss the relativistic analogue of Larson-Penston solutions and formation of naked singularities for the Einstein-Euler system. The talk is based on joint works with Yan Guo, Mahir Hadzic, and Matthew Schrecker.

Date/Time: 11:00 (UK time), Friday, 19 May 2023

Speaker: José Espinar (Universidad de Cádiz)

Title: Min-Oo conjecture for fully nonlinear conformally invariant equations

Abstract: In this talk we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard \(n\)-sphere \(\mathbb{S}^n\) under suitable conditions along the boundary. We emphasize that our results do not assume concavity assumption on the fully nonlinear equations we will work with.

This proves rigidity for compact connected locally conformally flat manifolds \( (M,g) \) with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere \( \partial D(r) \), where \( D(r) \) denotes a geodesic ball of radius \( r\in (0,\pi/2] \) in \( \mathbb{S}^n \), and totally umbilical with mean curvature bounded bellow by the mean curvature of this geodesic sphere. Under the above conditions, \( (M,g) \) must be isometric to the closed geodesic ball \( \overline{D(r)} \). In particular, we recover the solution by F.M. Spiegel to the Min-Oo conjecture for locally conformally flat manifolds.

As a side product, in dimension \(2\) our methods provide a new proof to Toponogov's Theorem about rigidity of compact surfaces carrying a shortest simple geodesic. Roughly speaking, Toponogov's Theorem is equivalent to a rigidity theorem for spherical caps in the Hyperbolic three-space \( \mathbb H^3 \). In fact, we extend it to obtain rigidity for super-solutions to certain Monge-Ampère equations.

Date/Time: 11:00 (UK time), Friday, 5 May 2023

Speaker: Alexander Cliffe (Oxford)

Title: Two-Dimensional Riemann Problems in Gas Dynamics

Abstract: The Riemann problem is the most simple initial value problem having discontinuous initial data which are piecewise constant and invariant under scaling. For systems of conservation laws in one space dimension, the solutions of Riemann problems form the building blocks for the general initial value problem, and the theory is well developed thanks to fundamental contributions from Lax (1957) and Glimm (1965). However, in two space dimensions, rigorous analytic results are few and far between, and many basic questions remain unanswered. We discuss recent progress in this area, focusing mainly on the two-dimensional shock reflection/diffraction problems, which can be reformulated as free boundary problems for a nonlinear PDE of mixed hyperbolic-elliptic type. Moreover, we present our latest result: the existence of a global four-shock interaction configuration in potential flow. Further details may be found in Chen-Feldman (2018) and Chen-AC-Huang-Liu-Wang (preprint: 2023).

Date/Time: 12:00 (UK time), Friday, 28 April 2023

Speaker: Shiwu Yang (Beijing)

Title: Asymptotic decay for defocusing semilinear wave equations

Abstract: In this talk, I will report recent progress on global behaviors for solutions of energy subcritical defocusing semilinear wave equations with pure power nonlinearity. We prove various type of decay estimates for the solutions in Minkowski space for all space dimensions. The proof is based on vector field method with new multipliers. We also extend similar results to the Schwarzschild black hole spacetimes. These are based on joint works with H. Mei and D. Wei.

The key mechanism to our approach is the construction of a pseudodifferential commutator, which, combined with a weighted Coifman-Meyer-type estimate, gives an integrated decay bound on certain quantities in frequency space without degeneration at trapping.

Date/Time: 11:00 (UK time), Friday, 28 April 2023

Speaker: Pin Yu (Tsinghua)

Title: On the stability of multi-dimensional rarefaction waves

Abstract: In his pioneering work in 1860, Riemann proposed the Riemann problem and solved it for isentropic gas in terms of shocks and rarefaction waves. It eventually became the foundation of the theory of one-dimension conservation laws developed in the 20th century. We prove the non-nonlinear structural stability of the Riemann problem for multi-dimensional isentropic Euler equations in the regime of rarefaction waves. This is a joint work with Tian-Wen Luo.

Date/Time: 12:00 (UK time), Friday, 24 March 2023

Speaker: Christopher Kauffman (Münster)

Title: The wave equation with small first-order linear perturbations on subextremal Kerr spacetimes

Abstract: I will discuss a recent work with Gustav Holzegel, in which we prove integrated decay bounds for solutions of the wave equation with small linear perturbations on subextremal Kerr spacetimes. Our proof adapts the framework introduced by Dafermos–Rodnianski–Shlapentokh-Rothman for the homogeneous wave equation. We additionally introduce a method to compensate for certain obstructions caused by the necessary degeneration of Morawetz-type estimates in these spacetimes, which in turn is due to the presence of trapped null geodesics.

The key mechanism to our approach is the construction of a pseudodifferential commutator, which, combined with a weighted Coifman-Meyer-type estimate, gives an integrated decay bound on certain quantities in frequency space without degeneration at trapping.

Date/Time: 11:00 (UK time), Friday, 24 March 2023

Speaker: Grigalius Taujanskas (Cambridge)

Title: Scattering of Maxwell Potentials on Curved Spacetimes

Abstract: The conformal approach to scattering is a combination of the ideas of Penrose's conformal compactification, the classical scattering theory of Lax and Phillips, and Friedlander's work on radiation fields, all developed in the 1960s. Recently there has been a resurgence of interest in the development of precise scattering theories, in particular on curved spacetimes, due to their importance for asymptotics, stability of spacetimes, and potential applications to quantum gravity. In this talk I will review the general setup of these ideas and show how to construct a scattering theory for Maxwell potentials on a non-trivial class of curved spacetimes, called Corvino–Schoen–Chrusciel–Delay spacetimes, where the combination of spacetime curvature and gauge freedom in the Maxwell potential have implications for the regularity of the initial and scattering data. This is based on joint work with J.-P. Nicolas (Brest).

Date/Time: 11:15 (UK time), Friday, 10 March 2023

Speaker: Tobias Barker (Bath)

Title: On symmetry breaking for the Navier-Stokes equations

Abstract: Motivated by an open question posed by Chemin, Zhang and Zhang, we investigate quantitative symmetry breaking for the 3D incompressible Navier-Stokes equations with initial third component of the velocity equal to zero. Specifically I will discuss: (i) isotropic norm inflation with unfavourable initial pressure gradient, (ii) quantitative symmetry breaking with favourable initial pressure gradient. Joint work with Christophe Prange (CNRS, Cergy) and Jin Tan (Cergy).

Date/Time: 11:15 (UK time), Friday, 24 February 2023

Speaker: Hamed Masaood (Imperial)

Title: A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole

Abstract: I will talk about the scattering problem in general relativity, and present a construction of a scattering theory for the linearised Einstein equations in a double null gauge against a Schwarzschild background. The construction begins by dealing with the gauge invariant components of the linearised system via the spin \(\pm2\) Teukolsky equations and the Teukolsky-Starobinsky identities; this was the subject of the paper arXiv:2007.13658. In this talk I will discuss how this theory can then be extended to the full system. Key to this step is the identification of suitable asymptotic gauge conditions on scattering data. Here, a Bondi-adapted double null gauge is shown to provide the necessary gauge rigidity, in a manner that enables the identification of a Hilbert-space isomorphism between finite energy scattering data and a suitable space of finite energy Cauchy data. In particular, the scattering theory we construct enables us to show that, in the global scattering problem, the linear memory effects at past and future null infinity are related by an antipodal map on the \(2\)-sphere.

Date/Time: 11:15 (UK time), Friday, 10 February 2023

Speaker: Arthur Touati (IHES)

Title: Geometric optics approximation for the Einstein vacuum equations

Abstract: In this talk I will present recent work on the rigorous justification of the geometric optics approximation for the Einstein vacuum equations, and its link with the Burnett conjecture in general relativity. I will start by presenting the initial value problem for the Einstein vacuum equations formulated in wave coordinates. Then I will give the state of the art on the Burnett conjecture, focusing on the approaches in U(1) symmetry and double null gauge. I will then present my main result and sketch its proof, highlighting the quasi- and semi-linear challenges. I will conclude my talk by discussing other quadratic wave equations.

Date/Time: 11:15 (UK time), Friday, 27 January 2023

Speaker: Christopher Alexander (UCL)

Title: General Relativistic Shock Waves that Exhibit Cosmic Acceleration

Abstract: This talk concerns the construction and analysis of a new family of exact general relativistic shock waves. The construction resolves the open problem of determining the expanding waves created behind a shock-wave explosion into a static isothermal sphere with an inverse square density and pressure profile. The construction involves matching two self-similar families of solutions to the perfect fluid Einstein field equations across a spherical shock surface. The matching is accomplished in Schwarzschild coordinates where the shock waves appear one derivative less regular than they actually are. Separately, both families contain singularities, but as matched shock-wave solutions, they are singularity free. This construction is also accompanied by a novel existence proof in the pure radiation case.

These shock-wave solutions represent an intriguing new mechanism in General Relativity for exhibiting accelerations in asymptotically Friedmann spacetimes, analogous to the accelerations modelled by the cosmological constant in the Standard Model of Cosmology. However, unlike in the Standard Model, these shock-wave solutions solve the Einstein field equations in the absence of a cosmological constant, opening up the question of whether a purely mathematical mechanism could account for the cosmic acceleration observed today, rather than dark energy.

Date/Time: 15:00 (UK time), Friday, 25 November 2022

Speaker: Stefan Czimek (Leipzig)

Title: Obstruction-free gluing for the Einstein equations

Abstract: We present a new approach to the gluing problem in General Relativity, that is, the problem of matching two solutions of the Einstein equations along a spacelike or characteristic (null) hypersurface. In contrast to previous constructions, the new perspective actively utilizes the nonlinearity of the constraint equations. As a result, we are able to remove the 10-dimensional spaces of obstructions to gluing present in the literature. As application, we show that any asymptotically flat spacelike initial data set can be glued to Schwarzschild initial data of sufficiently large mass. This is joint work with I. Rodnianski.

Date/Time: 15:00 (UK time), Friday, 18 November 2022

Speaker: Joonhyun La (Imperial College London)

Title: Vorticity Convergence from Boltzmann to 2D incompressible Euler equations below Yudovich class

Abstract: It is challenging to perform a multiscale analysis of mesoscopic systems exhibiting singularities at the macroscopic scale. In this paper, we study the hydrodynamic limit of the Boltzmann equations \[ \mathrm{St} \partial_t F + v \cdot \nabla_x F = \frac{1}{\mathrm{Kn} } Q(F, F) \] toward the singular solutions of 2D incompressible Euler equations whose vorticity is unbounded \[ \partial_t u + u \cdot \nabla_x u + \nabla_x p = 0, \quad \mathrm{div} u = 0. \] We obtain a microscopic description of the singularity through the so-called kinetic vorticity and understand its behavior in the vicinity of the macroscopic singularity. As a consequence of our new analysis, we settle affirmatively an open problem of convergence toward Lagrangian solutions of the 2D incompressible Euler equation whose vorticity is unbounded (\(\omega \in L^{\mathfrak{p} }\) for any fixed \(1 \le \mathfrak{p} < \infty\)). Moreover, we prove the convergence of kinetic vorticities toward the vorticity of the Lagrangian solution of the Euler equation. In particular, we obtain the rate of convergence when the vorticity blows up moderately in \(L^{\mathfrak{p} }\) as \(\mathfrak{p} \rightarrow \infty\) (localized Yudovich class).

Date/Time: 15:00 (UK time), Friday, 4 November 2022

Speaker: Alexander Pushnitski (King's College London)

Title: Turbulent solutions to the cubic Szegö equation on the real line

Abstract: The cubic Szegö equation (CSE) on the real line is a model evolution equation with a cubic nonlinearit. It is completely integrable and possesses a Lax pair. The CSE on the real line was introduced by Pocovnicu (following the work of Gérard-Grellier on a similar equation on the unit circle), who exhibited turbulent solutions, i.e. solutions with growing Sobolev norms. In this talk, I will discuss a recent joint work with Patrick Gérard, where we prove that initial conditions of turbulent solutions to the CSE form a dense st. Our analysis relies on a detailed study of an inverse spectral problem for Hankel operators; this problem arises from the Lax pair for the CSE.

Date/Time: 15:00 (UK time), Friday, 28 October 2022

Speaker: Shengwen Wang (Queen Mary University of London)

Title: A Brakke type regularity theorem for the Allen-Cahn flow

Abstract: We will talk about an analogue of the Brakke's local regularity theorem for the \(\epsilon\) parabolic Allen-Cahn equation. In particular, we show uniform \(C_{2,\alpha}\) regularity for the transition layers converging to smooth mean curvature flows as \(\epsilon\) tend to \(0\) under the almost unit-density assumption. This can be viewed as a diffused version of the Brakke regularity for the limit mean curvature flow. This talk is based on joint work with Huy Nguyen.

Date/Time: 15:00 (UK time), Friday, 14 October 2022

Speaker: Stefanos Aretakis (University of Toronto)

Title: Observational signatures for extremal black holes

Abstract: I will present (isotropic and anisotropic) observational signatures for extremal black holes. These signatures rely on the precise late time asymptotics for solutions to the wave equation on such backgrounds. I will also present asymptotics for subextremal backgrounds.

Date/Time: 14:00 (UK time), Friday, 7 October 2022

Speaker: Rita Teixeira da Costa (Princeton University)

Title: Decay for the Teukolsky equation on subextremal Kerr black holes

Abstract: The Teukolsky equation is one of the fundamental equations governing linear gravitational perturbations of the Kerr black hole family. We study fixed frequency solutions and obtain estimates which are uniform in the frequency parameters. Due to the separability of the Teukolsky equation, as a corollary we show that solutions are bounded and decay in time; this is a key first step in establishing linear stability of Kerr. Moreover, our estimates can also shed light on more delicate features of the Teukolsky equation, such as their scattering properties. This is joint work with Yakov Shlapentokh-Rothman (Toronto).

Date/Time: 14:00 (UK time), Friday, 17 June 2022

Speaker: Sameer Iyer (University of California, Davis)

Title: Reversal in the stationary Prandtl equations

Abstract: We discuss a recent result in which we investigate reversal and recirculation for the stationary Prandtl equations. Reversal describes the solution after the Goldstein singularity, and is characterized by spatio-temporal regions in which \(u > 0\) and \(u < 0\). The classical point of view of regarding the Prandtl equations as an evolution \(x\) completely breaks down. Instead, we view the problem as a quasilinear, mixed-type, free-boundary problem. Joint work with Nader Masmoudi.

Date/Time: 11:00 (UK time), Friday, 17 June 2022

Speaker: Renato Velozo (University of Cambridge)

Title: Stability of Schwarzschild for the spherically symmetric Einstein-massless Vlasov system

Abstract: The Einstein-massless Vlasov system is a relevant model in the study of collisionless many-particle systems in general relativity. In this talk, I will present a stability result for the exterior of Schwarzschild spacetime as a solution of this system assuming spherical symmetry. We exploit the normal hyperbolicity of the null geodesic flow in a neighborhood of the trapped set, to obtain decay estimates for the stress energy momentum tensor. The main result requires a precise understanding of radial derivatives of the energy momentum tensor, which we estimate using Jacobi fields on the tangent bundle in terms of the Sasaki metric.

Date/Time: 11:00 (UK time), Friday, 10 June 2022

Speaker: Xin Liu (Weierstrass Institute)

Title: Second law of thermodynamics and bounded entropy solutions in the compressible Navier-Stokes system

Abstract: In this talk, I will explain the issue of solutions with bounded entropy for the compressible Navier-Stokes equations. In particular, near the gas-vacuum interface, the entropy production rate is highly singular. This is related to the second law of thermodynamics. I will discuss
—the existence of equilibrium with bounded entropy to radiation gaseous star problem,
—a class of self-similar solutions to the compressible Navier-Stokes equations without heat conductivity and shear viscosity, and
—non-existence of entropy bounded solutions to the compressible Navier-Stokes equations without the degeneracy assumption in 2.

Date/Time: 14:00 (UK time), Friday, 27 May 2022

Speaker: Gabriele Benomio (Princeton University)

Title: A new system of linearised gravity on Kerr exterior spacetimes

Abstract: I will present the derivation of a new system of linearised gravity on Kerr exterior spacetimes and discuss some of its structural properties and applications.

Date/Time: 11:00 (UK time), Friday, 13 May 2022

Speaker: Maria Pia Gualdani (University of Texas at Austin)

Title: The Landau equation, global regularity versus blow-up in finite time

Abstract: Kinetic equations are used to describe evolution of interacting particles. The most famous kinetic equation is the Boltzmann equation: formulated by Ludwig Boltzmann in 1872, this equation describes motion of a large class of gases. Later, in 1936, Lev Landau derived a new mathematical model for motion of plasma. This latter equation was named the Landau equation. One of the main features of the Landau equation is nonlocality, meaning that particles interact at large, non-infinitesimal length scales. Moreover, the coefficients are singular and degenerate for large velocities. Many important questions, such as whether or not solutions become unbounded after a finite time, are still unanswered due to their mathematical complexity.

In this talk we concentrate on the mathematical results of the homogeneous Landau equation. We will first review existing results and open problems on global regularity versus blow-up in finite time. In the second part of the talk we will focus on recent developments of regularity theory for an isotropic version of the Landau equation. This is a joint work with J. Bedrossian, N. Guillen, and S. Snelson.

Date/Time: 14:00 (UK time), Friday, 1 April 2022

Speaker: Lauri Oksanen (University of Helsinki)

Title: Lorentzian Calderón problem under curvature bounds

Abstract: We introduce a method of solving inverse boundary value problems for wave equations on Lorentzian manifolds, and show that zeroth order coefficients can be recovered under certain curvature bounds. The set of Lorentzian metrics satisfying the curvature bunds has a non-empty interior in the sense of sufficiently smooth perturbations of the metric, and the interior contains the Minkowski metric. On the contrary, all previous results on this problem impose conditions on the metric that force it to be real analytic with respect to a suitably defined time variable. The analogous problem on Riemannian manifolds is called the Calderón problem, and in this case the known results require the metric to be independent of one of the variables. In particular, the Riemannian version of the problem is open near the Euclidean metric.

Our approach is based on a new unique continuation result in the exterior of the double null cone emanating from a point. The approach shares features with the classical Boundary Control method, and can be viewed as a generalization of this method to cases where no real analyticity is assumed.

The talk is based on joint work with Spyros Alexakis (University of Toronto) and Ali Feizmohammadi (Fields Institute).

Date/Time: 14:00 (UK time), Friday, 18 March 2022

Speaker: Angkana Rüland (Heidelberg University)

Title: On Rigidity, Flexibility and Scaling Laws: The Tartar Square

Abstract: Highly non-(quasi)-convex, matrix-valued differential inclusions arise in numerous physical applications. One such example is the modelling of shape-memory alloys. In these settings, often the exact differential inclusions display a striking dichotomy between rigidity and flexibility in that

* solutions of sufficiently high regularity obey the "characteristic equations" determined by the differential inclusion, the solutions are rigid in this sense,
* low regularity solutions are highly non-unique and hence extremely flexible.

In order to investigate this dichotomy further, in this talk, I explore the effects of regularizations in the form of singular perturbation problems with vanishing regularization strength for these differential inclusions. Motivated by applications in shape-memory alloys, I discuss the role of scaling properties in the singular perturbation strength. In particular, I discuss the scaling behaviour of a singular perturbation problem for the Tartar square. This is based on joint work with Jamie Taylor, Antonio Tribuzio, Christian Zillinger and Barbara Zwicknagl.

Date/Time: 14:00 (UK time), Friday, 4 March 2022

Speaker: Qian Wang (University of Oxford)

Title: Rough solutions of the 3-D compressible Euler equations

Abstract: I will talk about my work on the compressible Euler equations. We prove the local-in-time existence the solution of the compressible Euler equations in 3-D, for the Cauchy data of the velocity, density and vorticity \( (v,\varrho, \omega) \in H^s \times H^s\times H^{s'} \), \( 2 < s' < s \). The result extends the sharp result of Smith-Tataru and Wang, established in the irrotational case, i.e \( \omega=0 \), which is known to be optimal for \( s>2 \). At the opposite extreme, in the incompressible case, i.e. with a constant density, the result is known to hold for \( \omega \in H^s \), \( s > 3/2 \) and fails for \( s \le 3/2 \), see the work of Bourgain-Li. It is thus natural to conjecture that the optimal result should be \( (v, \varrho, \omega) \in H^s \times H^s \times H^{s'} \), \( s > 2, \, s' > 3/2 \). We view our work as an important step in proving the conjecture. The main difficulty in establishing sharp well-posedness results for general compressible Euler flow is due to the highly nontrivial interaction between the sound waves, governed by quasilinear wave equations, and vorticity which is transported by the flow. To overcome this difficulty, we separate the dispersive part of sound wave from the transported part, and gain regularity significantly by exploiting the nonlinear structure of the system and the geometric structures of the acoustic spacetime.

Date/Time: 14:00 (UK time), Friday, 18 February 2022

Speaker: Theo Drivas (Stony Brook University)

Title: Simultaneous Development of Shocks and Weak Discontinuities from Smooth Data

Abstract: I will discuss some recent work on shock formation and propagation of singularities for compressible Euler. The aim is to describe precisely the structure of an entropy producing shock in its early phase (starting from smooth initial conditions), and also describe with some detail a collection of weaker singularities that are born with it. These singularities are in the derivatives of the solution fields and travel along different characteristics than the shock. This reports on joint work with T. Buckmaster, S. Shkoller, and V. Vicol.

Date/Time: 14:00 (UK time), Friday, 4 February 2022

Speaker: Michele Dolce (Imperial College London)

Title: On the 2D Euler-Boussinesq system around a stably stratified Couette flow

Abstract: The 2D Euler–Boussinesq system approximately describes the dynamics of incompressible, inviscid and inhomogeneous fluids. In the talk, I will describe quantitative results for small perturbations around a stably stratified Couette flow, i.e. a shear flow with density increasing with depth. We show that the density and velocity perturbation undergo inviscid damping. On the other hand, the vorticity and density gradient of the perturbation experience an \(L^2\) polynomial growth, a phenomenon that we call shear-buoyancy instability. This is first precisely quantified at the linear level. For the nonlinear problem, the result holds on the optimal time-scale on which a perturbative regime can be considered. This is based on joint works with J. Bedrossian, R. Bianchini and M. Coti Zelati.

Date/Time: 14:00 (UK time), Friday, 21 January 2022

Speaker: Carsten Gundlach (University of Southampton)

Title: Naked singularity formation in vacuum gravitational wave collapse

Abstract: I will explain in general terms how critical collapse provides a natural route to forming naked singularities in the time evolution of almost generic smooth initial data. Then I will review rigorous mathematical results on naked singularity formation in the spherical scalar field-Einstein system and in vacuum general relativity. Finally I will review numerical experiments suggesting that critical collapse does create naked singularites in vacuum gravity, and my own numerical approach.

Date/Time: 14:00 (UK time), Friday, 10 December 2021

Speaker: Fred Alford (Imperial College London)

Title: A mathematical study of Hawking radiation for Reissner Nordstrom black holes

Abstract: In the first part of this talk, we will (briefly) derive the original calculation by Hawking in 1974 to determine the radiation given off by a black hole, giving the result in the form of an integral of a classical solution to the linear wave equation. In the second part of the talk, we will take this integral as a starting point, and calculate the radiation given off by a forming spherically symmetric, charged black hole.

Date/Time: 14:00 (UK time), Friday, 26 November 2021

Speaker: Nastasia Grubic (ICMAT)

Title: On the 2D free boundary incompressible Euler equations with self-intersecting interface

Abstract: We show that free boundary incompressible Euler equations are locally well posed in a (highly symmetric) class of solutions in which the interface self-intersects and thus exhibits corner or cusp-like singularities. Contrary to what happens in all the previously known non-\(C^1\) water waves, the angle of these crests can change in time.

Date/Time: 14:00 (UK time), Friday, 12 November 2021

Speaker: Dejan Gajic (Radboud University)

Title: Instabilities of extremal black holes

Abstract: When Kerr black holes rotate at their maximally allowed angular velocity, they are said to be extremal. Extremal black holes display a variety of interesting phenomena that are not present in more slowly rotating black holes. I will introduce upcoming work on the existence of strong asymptotic instabilities of a non-axisymmetric nature for scalar waves propagating on extremal Kerr black hole backgrounds and I will discuss their connection with previous work on axisymmetric instabilities and the precise nature of late-time power law tails in the emitted radiation.

Date/Time: 14:00 (UK time), Friday, 29 October 2021

Speaker: Camille Laurent (Sorbonne Université)

Title: Concentration close to the cone for linear waves

Abstract: In this talk, we will be concerned with solutions to the linear wave equation. We study how the energy is asymptotically located with respect to the light cone and its translation. We derive some expressions of the exterior energy outside a shifted light cone. In particular, in odd dimension, we characterize the solutions that are asymptotically zero outside a shifted cone.

This is joint work with Raphaël Côte (Strasbourg)

Date/Time: 14:00 (UK time), Friday, 15 October 2021

Speaker: Charles Collot (Cergy Paris)

Title: On the derivation of the Kinetic Wave Equation

Abstract: The kinetic wave equation arises in weak wave turbulence theory. In this talk we are interested in its derivation as an effective equation from dispersive waves with quadratic and cubic nonlinearities for the microscopic description of a system (Nonlinear Schrodinger equations with random initial data). We will comment on space-homogeneous and inhomogeneous cases (equation on the torus and on the whole space). More precisely, we will consider such dispersive equations in a weakly nonlinear regime, and for highly oscillatory random Gaussian fields with localised enveloppes as initial data. A conjecture in statistical physics is that there exists a kinetic time scale on which, statistically, the energy spectrum of the solution evolves according to the kinetic wave equation.

I will present joint works with Ioakeim Ampatzoglou and Pierre Germain (Courant Institute, NYU) in which we approach the problem of the validity of this kinetic wave equation through the convergence and stability of the corresponding Dyson series. We are able to identify certain nonlinearities, dispersion relations, and regimes, and for which the convergence indeed holds almost up to the kinetic time (arbitrarily small polynomial loss).