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Geometric structures naturally appear in fluid motions. One of the best known examples is Saturn's Hexagon, the huge cloud pattern at the level of Saturn's north pole, remarkable both for the regularity of its shape and its stability during the past decades. In this paper we will address the spontaneous formation of hexagonal structures in planar viscous flows, in the classical setting of Leray's solutions of the Navier–Stokes equations. Our analysis also makes evidence of the isotropic character of the energy density of the fluid for sufficiently localized 2D flows in the far field: it implies, in particular, that fluid particles of such flows are nowhere at rest at large distances.[-]

Geometric structures naturally appear in fluid motions. One of the best known examples is Saturn's Hexagon, the huge cloud pattern at the level of Saturn's north pole, remarkable both for the regularity of its shape and its stability during the past decades. In this paper we will address the spontaneous formation of hexagonal structures in planar viscous flows, in the classical setting of Leray's solutions of the Navier–Stokes equations. Our ...[+]

35Q30 ; 76D05

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As a toy model for the viscous interaction of planar vortices, we consider the solution of the two-dimensional Navier-Stokes equation with singular initial data corresponding to a pair of point vortices with opposite circulations. In the large Reynolds number regime, we construct an approximate solution which takes into account the deformation of the stream lines due to vortex interactions, as well as the corrections to the translation speed of the dipole due to finite size effects. Using energy estimates based on Arnold's variational characterization of equilibria for the Euler equation, we then show that our approximation remains valid over a very long time interval, if the viscosity is sufficiently small. This is a joint work with Michele Dolce (Lausanne), which relies on previous studies in collaboration with Vladimir Sverak (Minneapolis).[-]

As a toy model for the viscous interaction of planar vortices, we consider the solution of the two-dimensional Navier-Stokes equation with singular initial data corresponding to a pair of point vortices with opposite circulations. In the large Reynolds number regime, we construct an approximate solution which takes into account the deformation of the stream lines due to vortex interactions, as well as the corrections to the translation speed of ...[+]

35Q30 ; 76D05 ; 76D17 ; 35C20 ; 35B35

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In this talk, I will introduce nonlinear stability results of 2-D Couette flow for the NavierStokes equations at high Reynolds number. To achieve the optimal stability threshold, two kinds of physical effects including inviscid damping and enhanced dissipation play a crucial role. This talk is based on three joint papers

35Q30 ; 76E30 ; 76F06

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Reduced MHD models in Tokamak geometry are convenient simplifications of full MHD and are fundamental for the numerical simulation of MHD stability in Tokamaks. This presentation will address the mathematical well-posedness and the justification of the such models.
The first result is a systematic design of hierachies of well-posed reduced MHD models. Here well-posed means that the system is endowed with a physically sound energy identity and that existence of a weak solution can be proved. Some of these models will be detailed.
The second result is perhaps more important for applications. It provides understanding on the fact the the growth rate of linear instabilities of the initial (non reduced) model is lower bounded by the growth rate of linear instabilities of the reduced model.
This work has been done with Rémy Sart.[-]

Reduced MHD models in Tokamak geometry are convenient simplifications of full MHD and are fundamental for the numerical simulation of MHD stability in Tokamaks. This presentation will address the mathematical well-posedness and the justification of the such models.
The first result is a systematic design of hierachies of well-posed reduced MHD models. Here well-posed means that the system is endowed with a physically sound energy ...[+]

76W05 ; 35L65 ; 65M60 ; 35Q30

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The inhomogeneous incompressible Navier-Stokes equations that govern the evolution of viscous incompressible flows with non-constant density have received a lot of attention lately. In this talk, we shall mainly focus on the singular situation where the density is discontinuous, which is in particular relevant for describing the evolution of a mixture of two incompressible and non reacting fluids with constant density, or of a drop of liquid in vacuum. We shall highlight the places where tools in harmonic analysis play a key role, and present a few open problems.[-]

The inhomogeneous incompressible Navier-Stokes equations that govern the evolution of viscous incompressible flows with non-constant density have received a lot of attention lately. In this talk, we shall mainly focus on the singular situation where the density is discontinuous, which is in particular relevant for describing the evolution of a mixture of two incompressible and non reacting fluids with constant density, or of a drop of liquid in ...[+]

35Q30 ; 76D05 ; 35Q35 ; 76D03

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We revise recent contributions to 2D Euler and Navier-Stokes equations with and without noise, but always in the case of stochastic solutions. The role of white noise initial conditions will be stressed and related to some questions about turbulence.

35Q30 ; 35Q31 ; 60H15 ; 60H40

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I will discuss recent developments concerning the non-uniqueness of distributional solutions to the Navier-Stokes equation.

35Q30 ; 76D05 ; 35Q35

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In these lectures, we will focus on the analysis of oceanographic models. These models involve several small parameters: Mach number, Froude number, Rossby number... We will present a hierarchy of models, and explain how they can formally be derived from one another. We will also present different mathematical tools to address the asymptotic analysis of these models (filtering methods, boundary layer techniques).

86A05 ; 34E13 ; 35Q30 ; 35Q86 ; 35Jxx

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In these lectures, we will focus on the analysis of oceanographic models. These models involve several small parameters: Mach number, Froude number, Rossby number... We will present a hierarchy of models, and explain how they can formally be derived from one another. We will also present different mathematical tools to address the asymptotic analysis of these models (filtering methods, boundary layer techniques).

86A05 ; 34E13 ; 35Q30 ; 35Q86 ; 35Jxx

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In these lectures, we will focus on the analysis of oceanographic models. These models involve several small parameters: Mach number, Froude number, Rossby number... We will present a hierarchy of models, and explain how they can formally be derived from one another. We will also present different mathematical tools to address the asymptotic analysis of these models (filtering methods, boundary layer techniques).

86A05 ; 34E13 ; 35Q30 ; 35Q86 ; 35Jxx

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Documents 35Q30 23 résultats

Geometric structures in 2D Navier-Stokes flows - Brandolese, Lorenzo (Auteur de la Conférence) | CIRM H Nouveau

The long way of a viscous vortex dipole - Gallay, Thierry (Auteur de la Conférence) | CIRM H Nouveau

Stability threshold of 2-D Couette flow for the Navier-Stokes equations - Zhang, Zhifei (Auteur de la Conférence) | CIRM H Nouveau

Mathematical properties of hierarchies of reduced MHD models - Després, Bruno (Auteur de la Conférence) | CIRM H Nouveau

Recent approaches based on harmonic analysis for the study of non regular solutions to the Navier-Stokes equations with variable density - Danchin, Raphaël (Auteur de la Conférence) | CIRM H Nouveau

Stochastic solutions of 2D fluids​ - Flandoli, Franco (Auteur de la Conférence) | CIRM H Nouveau

Intermittent weak solutions of the 3D Navier-Stokes equations - Vicol, Vlad (Auteur de la Conférence) | CIRM H Nouveau

Asymptotic methods for the study of oceanographic models - Lecture 1: model hierarchy - Dalibard, Anne-Laure (Auteur de la Conférence) | CIRM H Nouveau

Asymptotic methods for the study of oceanographic models - Lecture 2: filtering methods - Dalibard, Anne-Laure (Auteur de la Conférence) | CIRM H Nouveau

Asymptotic methods for the study of oceanographic models - Lecture 3: boundary layer methods - Dalibard, Anne-Laure (Auteur de la Conférence) | CIRM H Nouveau

Stochastic solutions of 2D fluids - Flandoli, Franco (Auteur de la Conférence) | CIRM H Nouveau