Johannes von Lindheim, Technische Universität Berlin

"What is... a coherent structure?"

Although a formal definition of a coherent (or turbulent) structure has so far been elusive, it is generally thought to be a region of space and time within which the flow field exhibits a characteristic coherent pattern. With the contribution from flow visualization experiments and DNS simulations, Pope (2001) classifies these structures into eight categories. In this talk, I will give a brief overview on turbulent structures and then shift the focus to identifying one category of such structures, namely vortices, both in the Eulerian and Lagrangian specification of the flow field.

Abhishek Harikrishnan, Freie Universität Berlin

"What is... a shearlet?"

Shortcomings in Fourier or Wavelet representations of multivariate data, most commonly images, have lead to the development of shearlets, originally introduced for the sparse approximation of functions from  L2(R2).  They are a multiscale geometric framework tuned to efficiently encode anisotropic features of such functions using parabolic scalings and shearings of some mother shearlet psi. In this talk, I will show the construction of shearlet frames, as well as a theorem showing almost-optimal sparse approximation behavior of shearlets with respect to a model class of images with anisotropic features (so-called cartoon-like functions). Moreover, I will present a few examples of applications of shearlets from image processing (denoising, inpainting, super-resolution, medical imaging...) by means of sparse regularization, which has been proven to be a useful prior in such imaging problems.

Peter Friz, Technische Universität Berlin

Multiscale Systems, Homogenization and Rough Paths

Rough paths (and its recent generalisations: paracontrolled calculus, regularity structures) provide a powerful framework to the analysis of (partial) differential equations equations. Typical applications include highly-oscillatory systems (a priori well-posed, but with unclear limiting behaviour) and stochastic equations (analytically ill-posed, though sometimes within reach of Ito's stochastic analysis). After presenting some general ideas, I will explain how rough paths have been used to solve a concrete fast-slow homogenisation problem originally posed in [5].

 

References:
[1] Ilya Chevyrev, Peter K. Friz, Alexey Korepanov, Ian Melbourne, Huilin Zhang; Deterministic homogenization for discrete-time fast-slow systems under optimal moment assumptions. arXiv 2019
[2] Ilya Chevyrev, Peter K. Friz, Alexey Korepanov, Ian Melbourne, Huilin Zhang; Multiscale systems, homogenization, and rough paths; arXiv 2017 and 2019 Springer Volume Varadhan 75
[3] Peter Friz, Paul Gassiat and Terry Lyons, Physical Brownian motion in a magnetic field as a rough path. Trans. AMS 2015
[4] David Kelly, Ian Melbourne, Deterministic homogenization for fast-slow systems with chaotic noise, Journal of Functional Analysis 2017
[5] Melbourne and A. M. Stuart; Diffusion limits of chaotic skew-product flows, Nonlinearity 2011.

Upanshu Sharma, École des Ponts Paris Tech

Estimating coarse-graining error beyond reversibility

Coarse-graining is the procedure of approximating a complex and high-dimensional system by a simpler and lower dimensional one. Typically, such an approximation is achieved by using a coarse-graining map  F, which projects the full state of a system  X  (representing for instance the position of particles in the system), onto a lower-dimensional state space. Assuming that the state X evolves according to a stochastic differential equation (SDE), it is easy to identify the evolution of  F(X)  — however, this cannot be used in practice since the evolution still depends on the original state space. Legoll and Lelièvre (2010) addressed this by introducing a natural approximation, called the effective dynamics, and quantified the error of this approximation when  X  solves the overdamped Langevin equation. Since then a variety of results have been discussed in this direction, in the setting when  X  evolves according to a reversible SDE. In this talk, I will show that reversibility is not a limitation, and both the construction of the effective dynamics and the error estimates can be generalised to the setting of non-reversible SDEs. This is joint work with C. Hartmann and L. Neureither.

Claude Bardos Emeritus-Professor, Laboratoire Jacques Louis Lions,  Université Paris–Diderot

From the d'Alembert paradox to the 1984 Kato criteria via the 1941  $1/3$ Kolmogorov law and the 1949 Onsager conjecture

Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann , Piotr and Agneska Gwiadza , were motivated by the following issues:
The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence.

As a consequence.

I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.
Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is equivalent to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.

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