Program

Time: 4:30 - 5:30pm

Speaker: Wilhelm Schlag, Professor of Mathematics, Yale University

Abstract
Wilhelm will present a sample of results on the long-term dynamics and blowup of critical wave maps. In particular, he will discuss an ongoing project with Krieger and Miao on the rigidity of the blowup solutions from 2006 constructed by Krieger, Tataru, and Wilhelm under sufficiently regular perturbations.

Time: 6 - 7pm

Speaker: Jacob Bedrossian, Professor, Department of Mathematics

Abstract
In 1959, Batchelor predicted that passive scalars advected in fluids at finite Reynolds number with small diffusivity κ should display a |k|−1 power spectrum over a small-scale inertial range in a statistically stationary experiment. This prediction has been experimentally and numerically tested extensively in the physics and engineering literature and is a core prediction of passive scalar turbulence. Together with Alex Blumenthal and Sam Punshon-Smith, we have provided the first mathematically rigorous proof of this prediction for a scalar field evolving by advection-diffusion in a fluid governed by the 2D Navier-Stokes equations and 3D hyperviscous Navier-Stokes equations in a periodic box subjected to stochastic forcing at arbitrary Reynolds number. As conjectured by physicists, we also show the results in fact hold for a variety of toy models, though Navier-Stokes at high Reynolds number is the most physically relevant and the most difficult mathematically that we have considered thus far. These results are proved by studying the Lagrangian flow map using extensions of ideas from random dynamical systems. We prove that the Lagrangian flow has a positive Lyapunov exponent (Lagrangian chaos) and show how this can be upgraded to almost sure exponential mixing of passive scalars at zero diffusivity and further to uniform-in-diffusivity mixing. This in turn is a sufficiently precise understanding of the low-to-high frequency cascade to deduce Batchelor's prediction.

Time: 3 - 4pm

Speaker: Monica Musso, Professor, University of Bath

Abstract
The Euler equations (1755) define a system of non-linear PDEs that models the dynamics of an inviscid, incompressible fluid.  In dimension 2, a classical problem is the desingularized  N-vortex problem, namely the existence of true smooth solutions of  Euler equations  with  highly concentrated vorticities around N points.  We construct smooth solutions with concentrated vorticities around points which evolve according to the Hamiltonian system for the Kirkhoff-Routh energy. The profile around each point resembles a scaled finite mass solution of Liouville's equation. We discuss extensions of this analysis to the case of vortex filaments in 3-dimensional space, along the lines of Da Rios 1904 vortex filament conjecture in connection with the binormal flow of curves.  The results are in collaboration with Juan Dávila, Manuel del Pino and Juncheng Wei.

Time: 4:30 - 5:30pm

Speaker: Joachim Krieger, Professor, and Chair of Partial Differential Equations

Abstract
Joachim will discuss ongoing work with S. Miao and W. Schlag on establishing stability and in fact a kind of rigidity property of the blow-up solutions constructed by K.-Schlag-Tataru in 2006 under perturbations outside the equivariant setting.

Time: 8 - 9pm

Speaker: Slim Ibrahim,  Professor of Mathematics, University of Victoria

Abstract
In recent work with Y. Deng, we used the concept of ground and excited states in nonlinear dispersive equations (e.g. Klein-Gordon and Schrödinger equations) to characterize solutions in the N-body problem with strong force under some energy constraints. In this talk, I will explore this method to a restricted 3-body problem (Hill’s type lunar problem). It turns out to have very nice analogies to the nine-set theorem studied by Nakanishi-Schlag.

Time: 2 - 3pm

Speaker:  Mahir Hadzic, Associate Professor,  University College London

Abstract
Radial finite mass and compactly supported steady states of the asymptotically flat Einstein-Vlasov and Einstein-Euler systems represent isolated self-gravitating stationary galaxies and stars respectively. Upon the specification of the equation of state, such steady states are naturally embedded in 1-parameter families of solutions parametrised by the size of their central redshift.

In the first part of the talk we prove that highly relativistic galaxies/stars (the ones with high central redshift) are linearly unstable. This confirms an instability scenario suggested in 1960s by Zeldovich et al. in the Vlasov case, and Wheeler et al. in the Euler case. We also highlight the differences to the Newtonian case, explain the Hamiltonian structure of the linearised flow and prove an exponential trichotomy result. This is a joint work with Gerhard Rein and Zhiwu Lin.

In the second part of the talk we explain and rigorously prove the Turning Point Principle for the Einstein-Euler system, proposed by Wheeler et al. We thereby obtain very precise spectral information about the linearised star dynamics, which can be read off solely from the so-called mass-radius curve parametrised by the central redshift. This is a joint work with Zhiwu Lin.

Time: 4:30 - 5:30pm

Speaker: Toan T. Nguyen, Associate Professor, Penn State University

Abstract
The talk presents a recent joint work with Emmanuel Grenier (ENS Lyon) and Igor Rodnianski (Princeton) on Landau damping and Echoes in a plasma modeled by the classical Vlasov-Poisson system.

Time: 6 - 7pm

Speaker: Manuel Del Pino, Center for Nonlinear Analysis & Partial Differential Equations

Abstract
In this talk I will review some recent results on blow-up for the classical Patlak-Keller-Segel model of chemotaxis in two-dimensions.  We provide a construction with a proof of nonradial stability for a solution with infinite time blow  in the so-called critical mass case. In the supercritical mass scenario we find solutions with multiple blow-up at a single point. The method corresponds to a careful distinction and gluing between inner and outer regimes.  These are joint works  with Juan Davila, Jean Dolbeault, Monica Musso and Juncheng We

Time: 8 - 9pm

Speaker: Hao Jia, Professor, University of Minnesota

Abstract
Stability of coherent structures in two dimensional Euler equations, such as shear flows and vortices, is an important problem and a classical topic in fluid dynamics. Full nonlinear asymptotic stability results are difficult to obtain since the rate of stabilization is slow, the convergence of vorticity occurs only in a weak, distributional sense, and the nonlinearity is strong. In a breakthrough work, Bedrossian and Masmoudi proved the first nonlinear asymptotic stability result, near the Couette flow (linear shear). In this talk, we will explain the physical relevance of the problem, survey recent progress, and in particular, discuss our results on the nonlinear asymptotic stability of general monotonic shear flows. If time permits, further open problems in the area will also be mentioned. This is joint work with Alexandru Ionescu.

Time: 2 - 3pm

Speaker: Birgit Schörkhuber, Professor, Karlsruher Institute for Technology

Abstract
Birgit considers the heat flow for Yang-Mills connections on $\R^d$ in supercritical dimensions $5 \leq d \leq 9$, where the model admits self-similar blowup solutions, with an explicit example given by Weinkove. She will discuss the stability of the Weinkove solution and show that it is nonlinearly asymptotically stable under small equivariant perturbations. The talk is based on joint works with Irfan Glogić and Roland Donninger (Vienna).

Time: 3 - 4pm

Speaker: Slim Tayachi. Professor of Mathematics, University of Tunis El Manar

Abstract
In this talk, we study global well-posedness and long time asymptotic behavior of solutions to the nonlinear heat equation with absorption. We focus particularly on highly singular initial values which are antisymmetric with respect to some variables. Our approach is to study well-posedness and large time behavior on sectorial domains, and then to extend the results by reflection to solutions on the whole space which are antisymmetric. Our results include, among others, new examples of self-similar and asymptotically self-similar solutions. This is a joint work with Hattab Mouajria and Fred B. Weissler.

Time: 4:30 - 5:30pm

Speaker: Hatem Zaag, CNRS Senior Researcher, in the National institute for mathematical sciences and their interaction (INSMI).

Abstract
We consider a nonlocal parabolic equation, modelling MicroElectro-Mechanical Systems (MEMS), such as microphones, etc. In such a device, we should avoid to have "touch-down", when a membrane touches a metallic plate, damaging the device. In this work, we construct a "touch-down" solution and describe its profile.

Time: 6 - 7pm

Speaker: Pierre Raphael, Professor, University of Cambridge

Abstract
Global existence and scattering for the defocusing nonlinear Schrodinger equation is a celebrated result by Ginibre-Velo in the early 80’s in the strictly energy sub critical case, and Bourgain in 94 in the energy critical case. In the energy super critical setting, the defocusing energy is conserved and controls the energy norm, but this is too weak to conclude to global existence which yet had been conjectured by many and confirmed by numerical computations. This is a canonical super critical problem which typically arises similarily in fluid mechanics, and there global existence is either completely open or a direct consequence of the existence of additional conservation laws. In this talk based on recent joint works with Merle (IHES), Rodnianski (Princeton) and Szeftel (Paris Sorbonne), I will describe the construction of new smooth and finite energy highly oscillatory blow up solutions for the defocusing NLS in suitable energy super critical regimes, and explain how these new bubbles are connected to the also new description of implosion mechanisms for viscous three dimensional compressible fluids.

Time: 2 - 3pm

Speaker:  Charles Collot

Abstract
The kinetic wave equation arises in many physical situations: the description of small random surface waves, or out of equilibria dynamics for large quantum systems for example. In this talk we are interested in its derivation as an effective equation from the nonlinear Schrodinger equation (NLS) for the microscopic description of a system. More precisely, we will consider (NLS) in a weakly nonlinear regime on the torus in any dimension greater than two, and for highly oscillatory random Gaussian fields as initial data. A conjecture in statistical physics is that there exists a kinetic time scale on which, statistically, the Fourier modes evolve according to the kinetic wave equation. We prove this conjecture up to an arbitrarily small polynomial loss in a particular regime, and obtain a more restricted time scale in other regimes. The main difficulty, that I will comment on, is that one needs to identify the leading order statistically observable nonlinear effects. This means understanding correlation between Fourier modes, and relating randomness with stability and local well-posedness. The key idea of the analysis is the use of Feynman interaction diagrams to understand the solution as colliding linear waves. We use this framework to construct an approximate solution as a truncated series expansion, and use in addition random matrices tools to obtain its nonlinear stability in Bourgain spaces. This is joint work with P. Germain.

Time: 4:30 - 5:30pm

Speaker: Taoufik Hmidi, University of Rennes

Abstract
We shall discuss the existence of time periodic solutions for 3d QG equations in the patch form. We look for those structures around any generic revolution shape stationary solution. We show the bifurcation of non trivial solutions from the largest eigenvalues of the associated linearized operator. This is a joint work with Claudia Garcia and Joan Mateu.

Time: 6 - 7pm

Speaker:
Massimiliano Berti, Professor of Mathematical Analysis at SISSA

Abstract
We prove the first bifurcation result of time quasi-periodic traveling waves solutions for space periodic water waves with vorticity. In particular we prove existence of small amplitude time quasi-periodic solutions of the gravity-capillary water waves equations with constant vorticity , for a bidimensional fluid over a at bottom delimited by a space-periodic free interface. Joint work with L. Franzoi and A. Maspero.

Time: 8 - 9pm

Speaker: Thomas Y. Hou,  Charles Lee Powell Professor of Applied and Computational Mathematics, CALTech

Abstract
Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. We first review the numerical evidence of finite time singularity for 3D axisymmetric Euler equations by Luo and Hou. The singularity is a ring like singularity that occurs at a stagnation point in the symmetry plane located at the boundary of the cylinder. We then present a novel method of analysis and prove that the 1D HL model develops finite time self-similar singularity. We also apply this method of analysis to prove finite time self-similar blowup of the original De Gregorio model for some smooth initial data on the real line with compact support. Self-similar blowup results for the generalized De Gregorio model for the entire range of parameter on the real line or on a circle have been obtained for Holder continuous initial data with compact support. Finally, we report our recent progress in analyzing the finite time singularity of the axisymmetric 3D Euler equations with initial data considered by Luo and Hou.