# Turb1d 2020

The Universidad de Cantabria (UC) and the Institute Camille Jordan (ICJ) are organizing an online conference on the study of mathematical models of turbulence and related phenomena.

**Schedule**

Monday 23rd November, 14h00 - 15h00 (central EU time): Adam Kubica (Warsaw University of Technology);

Title:**Kolmogorov's two-equation model of turbulence.**

Abstract: It will be considered the Kolmogorov's two-equation model of turbulence in three dimensional domain with periodic boundary conditions. First, I will present the result concerning the existence of local in time solution. The proof is based on Galerkin method applied for an appropriate truncated problem. Next, it will be shown the existence of global in time solution under some smallness assumption imposed on initial data. This is a joint work with Przemysław Kosewski.Tuesday 24th November, 14h00 - 15h00 (central EU time): Alexei Mailybaev (IMPA).

Title:**Spontaneously stochastic solutions**

Abstract: We discuss turbulence models that lack uniqueness of solutions. We consider such models within a broader context of differential equations that lack Lipschitz continuity and, therefore, require a regularization procedure or a selection criterion for defining relevant solutions. Starting with simplified models, we show how spontaneously stochastic solutions arise in these formally deterministic systems. This phenomenon occurs when one takes into account an infinitesimal noise in the regularization process. Then we present numerical results demonstrating spontaneous stochasticity in realistic models of turbulence.Wednesday 25th November, 14h00 - 15h00 (central EU time): Josef Málek (Charles University);

Title:**Large data analysis for Kolmogorov's two-equation model of turbulence and for activated fluids**

Abstract: The talk will be split into two parts. In both parts, we will present recently published results concerning long-time and large-data existence of a (suitable) weak solution to three-dimensional internal unsteady flows of incompressible fluids subject to slipping mechanisms (including Navier's slip and stick-slip) on the impermeable wall.

In the first part, we focus on the Kolmogorov's two equation model of turbulence. Kolmogorov seems to have been the first to recognize that a two-equation model of turbulence might be used as the basis of turbulent flow prediction. Nowadays, a whole hierarchy of phenomenological two-equation models of turbulence is in place. The structure of their governing equations is similar to the Navier–Stokes equations for incompressible fluids, the difference is that the viscosity is not constant but depends on two scalar quantities that measure the effect of turbulence: the average of the kinetic energy of velocity fluctuations (i.e. the turbulent energy) and the measure related to the length scales of turbulence. For these two scalar quantities two additional evolutionary convection–diffusion equations are added to the generalized Navier–Stokes system. Although Kolmogorov’s model has so far been almost unnoticed, it exhibits interesting features. First of all, in contrast to other two-equation models of turbulence, there is no source term in the equation for the frequency. Consequently, nonhomogeneous Dirichlet boundary conditions for the quantities measuring the effect of turbulence are assigned to a part of the boundary. Second, the structure of the governing equations is such that one can find an “equivalent" reformulation of the equation for turbulent energy that eliminates the presence of the energy dissipation acting as the source in the original equation for turbulent energy and which is merely an L1 quantity. Third, the material coefficients such as the viscosity and turbulent diffusivities may degenerate, and thus the a priori control of the derivatives of the quantities involved is unclear. The results presented in this part carries on the articles:

* Miroslav Bulíček, Josef Málek: Large data analysis for Kolmogorov's two-equation model of turbulence. Nonlinear Anal. Real World Appl. 50 (2019), 104–143.

* Miroslav Bulíček, Roger Lewandowski, Josef Málek: On evolutionary Navier-Stokes-Fourier type systems in three spatial dimensions. Comment. Math. Univ. Carolin. 52 (2011), no. 1, 89–114.

In the second part, our attention concentrates on a special class of the activated fluids, namely the fluids that behave as an Euler fluid prior activation and behave as a viscous fluid once activation takes place. Such fluids appear naturally in a new classification of incompressible fluids characterized by a continuous monotone relation between the velocity gradient and the Cauchy stress. Similar classification can be made for boundary conditions that we view as the constitutive relations on the boundary. The presentation of this part is based on the article:

*Jan Blechta, Josef Málek, KR Rajagopal: On the classification of incompressible fluids and a mathematical analysis of the equations that govern their motion. SIAM J. Math. Anal. 52 (2020), no. 2, 1232–1289.Thursday 26th November, 14h00 - 15h00 (central EU time): Roberta Bianchini (Italian National Research Council).

Title:**Linear inviscid damping for shear flows in the 2D stably stratified regime**

Abstract: In this talk, we discuss the linear asymptotic stability of shear flows near Couette for a class of 2D incompressible stratified fluids in the inviscid regime. An inviscid damping result, with optimal decay rates, will be provided for the velocity field in the Couette case. These results are obtained by means of a suitable point-wise approach in frequency space, which allows to confirm the predictions due to Hartman in 1975 and improve existing results. For the previously unexplored case of general shear flows close to Couette, stability in any Sobolev space H^s is obtained. Remarkably, our results hold under the celebrated Miles-Howard criterion for stratified fluids. This is a joint work with Michele Coti Zelati and Michele Dolce (Imperial College, London, UK).Friday 27th November, 14h00 - 15h00 (central EU time): Sarka Necasová (Institute of Mathematics AS CR);

Title:**Singular limits as a tool for rigorous derivation of models : Compressible Primitive Equations, the quasi-geostrophic system**

Abstract: The lecture addresses to the singular limits as a tools for rigorous derivation of models.Firstly, we obtain the hydrostatic approximation by taking the small aspect ratio limit to the Navier-Stokes equations. The aspect ratio (the ratio of the depth to horizontal width) is a geometrical constraintin general large scale geophysical motions that the vertical scale is significantly smaller than horizontal.We use theversatilerelative entropy inequality to prove rigorously the limit from the compressible Navier-Stokes equations to the compressible Primitive Equations, see [3].Secondly, we study the ”‘classical” weak-strong uniqueness property and singular limit for the com-pressible Primitive Equations (PE). We show that a weak solution coincides with the strong solutionemanating from the same initial data. On the other hand, we prove compressible PE will approach tothe incompressible inviscid PE equations in the regime of low Mach number and large Reynolds numberin the case of well-prepared initial data, see [2].Thirdly, we will focus on the low Mach number and low Rossby number regime in the case of rotatingcompressible system. Based on the concept ofdissipative measure-valued solution, the quasi-geostrophic system is identified as the limit problem in the case of ill-prepared initial data, see [3]. Joint work with Hongjun Gao and Tong Tang (University of Nanjing, China)

[1] Sarka Necasová, Tong Tang:On a singular limit for the compressible rotating Euler system, J. Math.Fluid Mech. 22 (2020), no. 3, Art. 43, 14 pp.

[2] Hongjun Gao, Sarka Necasová, Tong Tang:On weak-strong uniqueness and singular limit for thecompressible Primitive EquationsDiscrete Contin. Dyn. Syst. Journal 40, 7, 4287–4305 (2020).

[3] Hongjun Gao, Sarka Necasová, Tong Tang:On the hydrostatic approximation of compressible anisotropicNavier-Stokes equations - rigorous justification, arXiv:2011.04810

The talks are hosted by the Faculty of Science of University of Cantabria using Microsoft TEAMS.

**Instructions to attend the seminar**:

1) join the mailing list by writing an email to **rafael(dot)granero(at)unican.es**. You only need to do this once.

2) download Microsoft TEAMS and register yourself. You will not need a license.

3) you will receive an email with the link to join the talk a couple of hours before the starting time.

We also ask for your patience with any technical difficulties.

**Organizers:** Francesco Fanelli (ICJ) and Rafael Granero-Belinchón (UC).