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JeanClaude Saut  Tue, 23. Nov 21, 14:30  
New and old on the Intermediate Long Wave equation  
We survey new and old results on the Intermediate Long Wave (ILW) equation from modeling, PDE and integrability aspects.  

Patrick Gérard  Wed, 24. Nov 21, 9:15  
High frequency approximation of solutions of the BenjaminOno equation on the torus  
For solutions of the BenjaminOno equation with periodic boundary conditions, I will discuss the link in the high frequency regime between the nonlinear Fourier transform inherited from the integrable structure, and a gauge transform introduced by T. Tao in 2004 in the context of the low regularity initial value problem. As an application, we will get optimal high frequency approximations of solutions. This talk is based on a recent joint work with T. Kappeler and P. Topalov.  

Thomas Kappeler  Wed, 24. Nov 21, 10:45  
Normal form coordinates for the BenjaminOno equation having ex pansions in terms of pseudodifferential operators  
Using the Birkhoff map of the BenjaminOno equation as a starting point, we deform it near an arbitrary compact family of finite dimensional tori, invariant under the BenjaminOno flow, so that the following main properties hold: (i) When restricted to the family of finite dimensional tori, the transformation coincides with the Birkhoff map. (ii) Up to a remainder term, which is smoothing to any given order, it is a pseudodifferential operator of order 0, with principal part given by the Fourier transform, modified by a phase factor. (iii) The transformation is canonical and the pullback of the BenjaminOno Hamiltonian by it is in normal form up to order three. Such coordinates are a key ingredient for studying the stability of finite gap solutions of arbitrary size of the BenjaminOno equation under small, quasilinear, momentum preserving perturbations. This is joint work with Riccardo Montalto.  

Christian Klein  Wed, 24. Nov 21, 14:00  
Hybrid approaches to DaveyStewartson II systems  
We present a detailed numerical study of solutions to DaveyStewartson (DS) II systems, nonlocal nonlinear Schrödinger equations in two spatial dimensions. A possible blowup of solutions is studied, a conjecture for a selfsimilar blowup is formulated. In the integrable cases, numerical and hybrid approaches for the inverse scattering are presented.  

Goeksu Oruk  Wed, 24. Nov 21, 15:15  
A Numerical Approach for the Spectral Stability of Periodic Travelling Wave Solutions to the Fractional BenjaminBonaMahony Equation  
Currently, the studies on periodic travelling waves of the nonlinear dispersive equations are becoming very popular. In this study we investigate the spectral stability of the periodic waves for the fractional BenjaminBonaMahony (fBBM) equation, numerically. For the numerical generation of periodic travelling wave solutions we use an iteration method which is based on a modification of Petviashvili algorithm. This is a joint work with S. Amaral, H. Borluk, G.M. Muslu and F. Natali.  

Anton Arnold  Thu, 25. Nov 21, 9:15  
Optimal nonsymmetric FokkerPlanck equation for the convergence to a given equilibrium  
We are concerned with finding FokkerPlanck equations in whole space with the fastest exponential decay towards a given equilibrium. For a prescribed, anisotropic Gaussian we determine a nonsymmetric FokkerPlanck equation with linear drift that shows the highest exponential decay rate for the convergence of its solutions towards equilibrium. At the same time it has to allow for a decay estimate with a multiplicative constant arbitrary close to its infimum. This infimum is 1, corresponding to the highrotational limit in the FokkerPlanck drift. Such an optimal Fokkerplanck equation is constructed explicitly with a diffusion matrix of rank one, hence being hypocoercive. The proof is based on the recent result that the L2 projector norms of the FokkerPlanck equation and of its driftODE coincide. Finally we give an outlook onto using FokkerPlanck equation with tdependent coefficients. This talk is based on a joint work with Beatrice Signorello.  

Nikola Stoilov  Thu, 25. Nov 21, 10:45  
Numerical study of DaveyStewartson I I systems  
An efficient high precision hybrid numerical approach for integrable DaveyStewartson (DS) I equations for trivial boundary conditions at infinity is presented for Schwartz class initial data. The code is used for a detailed numerical study of DS I solutions in this class. Localized stationary solutions are constructed and shown to be unstable against dispersion and blowup. A finitetime blowup of initial data in the Schwartz class of smooth rapidly decreasing functions is discussed.  

Ola Maehlen  Fri, 26. Nov 21, 9:15  
Onesided Hölder regularity of global weak solutions of negative order dis persive equations  
The majority of dispersive equations in one spacedimension can be realized as dispersive perturbations of the Burgers equation ut + uux = Lux, where L is a local or nonlocal symmetric operator. For negative order dispersion, the Burg ers’ nonlinearity dominates and classical solutions break down due to shockformation/wave breaking. Using hyperbolic techniques we establish global existence and uniqueness of entropy solutions, with L2 initial data, for a family of negative order dispersive equations, but our main focus will be on a new generalization of the classical Oleïnik estimate for Burgers equation. We obtain one sided Hölder regularity for the solutions, which in turn controls their height and provides a novel bound of the lifespan of classical solutions based on their initial skewness. This is joint work with Jun Xue (NTNU).  

Didier Pilod  Fri, 26. Nov 21, 10:30  
Unconditional uniqueness for the BenjaminOno equation POSTPONED  
We study the unconditional uniqueness of solutions to the BenjaminOno equation with initial data in Hs, both on the real line and on the torus. We use the gauge transformation of Tao and two iterations of normal form reductions via integration by parts in time. By employing a refined Strichartz estimate we establish the result below the regularity threshold s = 1/6. As a byproduct of our proof, we also obtain a nonlinear smoothing property on the gauge variable at the same level of regularity. This talk is based on a joint work with Razvan Mosincat (University of Bergen).  

Francois Golse  Mon, 20. Dec 21, 12:00  
From NBody Schrödinger to EulerPoisson  
This talk presents a joint meanfield and classical limit by which the EulerPoisson system is rigorously derived from the Nbody Schrödinger equation with Coulomb interaction in space dimension 3. One of the key ingredients in this derivation is a remarkable inequality for the Coulomb potential which has been obtained by S. Serfaty in 2020 (Duke Math. J.). 2)  

Jakob Möller  Mon, 20. Dec 21, 12:30  
The PauliPoisson equation and its cassical limit  
The PauliPoisson equation is a semirelativistic description of electrons under the influence of a given linear (strong) magnetic field and a selfconsistent electric potential computed from the Poisson equation in 3 space dimensions. It is a system of two magnetic Schrödinger type equations for the two components of the spinor, coupled by the additional SternGerlach term of magnetic field and spin represented by the Pauli matrices. On the other hand the PauliPoiswell equation includes the selfconsistent description of the magnetic field by coupling it via a three Poisson equations with the Pauli current as source term to the Pauli equation. The PauliPoiswell equation offers a fully selfconsistent description of spin1/2particles in the semirelativistic regime. We introduce the equations and study the semiclassical limit of PauliPoisson towards a semirelativistic Vlasov equation with Lorentz force coupled to the Poisson equation. We use Wigner transform methods and a mixed state formulation, extending the work of LionsPaul and MarkowichMauser on the semiclassical limit of the SchrödingerPoisson equation. We also present a result on global weak solutions of the PauliPoiswell equation.  

Ivan Moyano  Mon, 20. Dec 21, 15:00  
Unique continuation, Carleman estimates and propagation of smallness with applications in observability  
Based on a series of works in collaboration with Gilles Lebeau and Nicolas Burq Propagation of smallness and control for heat equations (with Nicolas Burq, to appear in JEMS), Spectral Inequalities for the Schrödinger operator (with Gilles Lebeau). Propagation of smallness and spectral estimates (with Nicolas Burq) And the recent advances in propagation of smallness introduced by Logonuv and Malinnikova. A. Logunov and E. Malinnikova. Quantitative propagation of smallness for solutions of elliptic equations. Preprint, Arxiv, (arXiv:1711.10076), 2017 A. Logunov. Nodal sets of Laplace eigenfunctions : polynomial upper estimates of the Hausdorff measure. Ann. of Math. (2), 187(1):221–239, 2018.  

Nicolas Besse  Mon, 20. Dec 21, 15:30  
Trying to prove quasilinear theory in plasma physics  
The aim of quasilinear theory is to explain relaxation or saturation of kinetic instabilities governed by the VlasovPoisson (VP) equation, by showing that in fact the Hamiltonian dynamics of VP can be approximated by a diffusion equation in velocity for the spaceaverage distribution function.  

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