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Documents  Danchin, Raphaël | enregistrements trouvés : 13

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Research talks;Partial Differential Equations;Mathematical Physics

In the fifties John Nash astonished the geometers with his celebrated isometric embedding theorems. A folkloristic explanation of his first theorem is that you should be able to put any piece of paper in your pocket without crumpling or folding it, no matter how large it is.
Ten years ago László Székelyhidi and I discovered unexpected similarities with the behavior of some classical equations in fluid dynamics. Our remark sparked a series of discoveries and works which have gone in several directions. Among them the most notable is the recent proof of Phil Isett of a long-standing conjecture of Lars Onsager in the theory of turbulent flows. In a joint work with László, Tristan Buckmaster and Vlad Vicol we improve Isett's theorem to show the existence of dissipative solutions of the incompressible Euler equations below the Onsager's threshold.
In the fifties John Nash astonished the geometers with his celebrated isometric embedding theorems. A folkloristic explanation of his first theorem is that you should be able to put any piece of paper in your pocket without crumpling or folding it, no matter how large it is.
Ten years ago László Székelyhidi and I discovered unexpected similarities with the behavior of some classical equations in fluid dynamics. Our remark sparked a series of ...

35Q31 ; 35D30 ; 76B03

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- viii; 242 p.
ISBN 978-1-4704-3646-9

Contemporary mathematics , 0710

Localisation : Collection 1er étage

mécanique des fluides # équation de Navier-Stokes # fluide non-newtonien # magnétohydrodynamique (MHD) # mélange de fluides

35B40 ; 35B65 ; 35Q30 ; 35Q35 ; 76D05 ; 76D07 ; 76N10 ; 76W05 ; 35-06

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Research talks

In this talk I will present a couple of results for the existence of solutions to the one-dimensional Euler, Navier-Stokes and multi-dimensional Navier-Stokes systems. The purpose of the talk is to focus on the role of the pressure in the compressible fluid equations, and to understand whether or not it can be replaced by the nonlocal attraction-repulsion terms arising in the models of collective behaviour.

76N10 ; 35Q35

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Research talks

A common way to prove global well-posedness of free boundary problems for incompressible viscous fluids is to transform the equations governing the fluid motion to a fixed domain with respect to the time variable. An elegant and physically reasonable way to do this is to introduce Lagrangian coordinates. These coordinates are given by the transformation rule

$x(t)=\xi +\int_{0}^{t}u(\tau ,\xi ) d\tau $

where $u(\tau ,\xi )$ is the velocity vector of the fluid particle at time $\tau$ that initially started at position $\xi$. The variable $x(t)$ is then the so-called Eulerian variable and belongs to the coordinate frame where the domain that is occupied by the fluid moves with time.The variable $\xi$ is the Lagrangian variable that belongs to time fixed variables. In these coordinates the fluid only occupies the domain $\Omega_{0}$ that is occupied at initial time t = 0.
To prove a global existence result for such a problem, it is important to guarantee the invertibility of this coordinate transform globally in time. By virtue of the inverse function theorem, this is the case if

$\nabla_{\xi }x(t)=Id+\int_{0}^{t}\nabla_{\xi }u(\tau ,\xi )d\tau $

is invertible. By using a Neumann series argument, this is invertible, if the integral termon the right-hand side is small in $L^{\infty }(\Omega _{0})$. Thus, it is important to have a global control of this $L^{1}$-time integral with values in $L^{\infty }(\Omega _{0})$. If the domain is bounded, this can be controlled by exponential decay properties of the corresponding semigroup operators that describe the motion of the linearized fluid equation. On unbounded domains, however, these decay properties are not true anymore. While there are technical possibilities to fix these problems if the boundary is compact, these fixes cease to work if the boundary is non-compact.
As a model problem, we consider the case where $\Omega _{0}$ is the upper half-space. To obtain estimates of the $L^{1}$-time integral we use the theorem of Da Prato and Grisvard of 1975 about maximal regularity in real interpolation spaces. The need of global in timecontrol, however, makes it necessary to work out a version of this theorem that involves “homogeneous” estimates only (this was also done in the book of Markus Haase). In the talk, we show how to obtain this global Lagrangian coordinate transform from this theorem of Da Prato and Grisvard.
A common way to prove global well-posedness of free boundary problems for incompressible viscous fluids is to transform the equations governing the fluid motion to a fixed domain with respect to the time variable. An elegant and physically reasonable way to do this is to introduce Lagrangian coordinates. These coordinates are given by the transformation rule

$x(t)=\xi +\int_{0}^{t}u(\tau ,\xi ) d\tau $

where $u(\tau ,\xi )$ is ...

35Q35 ; 76D05

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Research talks

In this talk, I will present a recent study on traveling waves solutions to a 1D biphasic Navier-Stokes system coupling compressible and incompressible phases. With this original fluid equations, we intend to model congestion (or saturation) phenomena in heterogeneous flows (mixtures, collective motion, etc.). I will first exhibit explicit partially congested propagation fronts and show that these solutions can be approached by profiles which are solutions to a singular compressible Navier-Stokes system. The last part of the talk will be dedicated to the analysis of the stability of the approximate profiles. This is a joint work with Anne-Laure Dalibard. In this talk, I will present a recent study on traveling waves solutions to a 1D biphasic Navier-Stokes system coupling compressible and incompressible phases. With this original fluid equations, we intend to model congestion (or saturation) phenomena in heterogeneous flows (mixtures, collective motion, etc.). I will first exhibit explicit partially congested propagation fronts and show that these solutions can be approached by profiles which ...

35Q35 ; 35L67

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Research talks

In this joint work with Athanasios Tzavaras (KAUST) and Corrado Lattanzio (L’Aquila) we develop a relative entropy framework for Hamiltonian flows that in particular covers the Euler-Korteweg system, a well-known diffuse interface model for compressible multiphase flows. We put a particular emphasis on extending the relative entropy framework to the case of non-monotone pressure laws which make the energy functional non-convex.The relative entropy computation directly implies weak (entropic)-strong uniqueness, but we will also outline how it can be used in other contexts. Firstly, we describe how it can be used to rigorously show that in the large friction limit solutions of Euler-Korteweg converge to solutions of the Cahn-Hilliard equation. Secondly, we explain how the relative entropy can be used for obtaining a posteriori error estimates for numerical approximation schemes. In this joint work with Athanasios Tzavaras (KAUST) and Corrado Lattanzio (L’Aquila) we develop a relative entropy framework for Hamiltonian flows that in particular covers the Euler-Korteweg system, a well-known diffuse interface model for compressible multiphase flows. We put a particular emphasis on extending the relative entropy framework to the case of non-monotone pressure laws which make the energy functional non-convex.The relative ...

35Q31 ; 76D45 ; 76T10

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Research talks

The Euler-Korteweg system corresponds to compressible, inviscid fluids with capillary forces. It can be used to model diffuse interfaces. Mathematically it reads as the Euler equations with a third order dispersive perturbation corresponding to the capillary tensor.

In dimension one there exists traveling waves with equal or different limit at infinity, respectively solitons and kinks. Their stability is ruled by a simple criterion a la Grillakis-Shatah-Strauss. This talk is devoted to the construction of multiple traveling waves, namely global solutions that converge as $t\rightarrow \infty $ to a profile made of several (stable) traveling waves. The waves constructed have both solitons and kinks. Multiple traveling waves play a peculiar role in the dynamics of dispersive equations, as they correspond to solutions that follow in some sense a purely nonlinear evolution.
The Euler-Korteweg system corresponds to compressible, inviscid fluids with capillary forces. It can be used to model diffuse interfaces. Mathematically it reads as the Euler equations with a third order dispersive perturbation corresponding to the capillary tensor.

In dimension one there exists traveling waves with equal or different limit at infinity, respectively solitons and kinks. Their stability is ruled by a simple criterion a la ...

35Q35 ; 35C07 ; 35Q53 ; 35Q31 ; 35B35

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Research talks;Analysis and its Applications;Partial Differential Equations

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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Research talks;Partial Differential Equations;Mathematical Physics

Consider the motion of a viscous incompressible fluid in a 3D exterior domain $D$ when a rigid body $\mathbb R^3\setminus D$ moves with prescribed time-dependent translational and angular velocities. For the linearized non-autonomous system, $L^q$-$L^r$ smoothing action near $t=s$ as well as generation of the evolution operator $\{T(t,s)\}_{t\geq s\geq 0}$ was shown by Hansel and Rhandi [1] under reasonable conditions. In this presentation we develop the $L^q$-$L^r$ decay estimates of the evolution operator $T(t,s)$ as $(t-s)\to\infty$ and then apply them to the Navier-Stokes initial value problem. Consider the motion of a viscous incompressible fluid in a 3D exterior domain $D$ when a rigid body $\mathbb R^3\setminus D$ moves with prescribed time-dependent translational and angular velocities. For the linearized non-autonomous system, $L^q$-$L^r$ smoothing action near $t=s$ as well as generation of the evolution operator $\{T(t,s)\}_{t\geq s\geq 0}$ was shown by Hansel and Rhandi [1] under reasonable conditions. In this presentation we ...

35Q30 ; 76D05 ; 76D07

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Research talks;Partial Differential Equations;Mathematical Physics

Given initial data $(b_0, u_0)$ close enough to the equilibrium state $(e_3, 0)$, we prove that the 3-D incompressible MHD system without magnetic diffusion has a unique global solution $(b, u)$. Moreover, we prove that $(b(t) - e_3, u(t))$ decay to zero with rates in both $L^\infty$ and $L^2$ norm. (This is a joint work with Wen Deng).

35Q30 ; 76D03

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Research talks;Partial Differential Equations;Mathematical Physics

We investigate the gyrokinetic limit for the two-dimensional Vlasov-Poisson system in a regime studied by F. Golse and L. Saint-Raymond. First we establish the convergence towards the Euler equation under several assumptions on the energy and on the norms of the initial data. Then we provide a first analysis of the asymptotics for a Vlasov-Poisson system describing the interaction of a bounded density with a moving point charge.

82D10 ; 82C40 ; 35Q35 ; 35Q83 ; 35Q31

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Research talks;Partial Differential Equations;Mathematical Physics

We first summarize the derivation of viscoelastic (rate-type) fluids with stress diffusion that generates the models that are compatible with the second law of thermodynamics and where no approximation/reduction takes place. The approach is based on the concept of natural configuration that splits the total response between the current and initial configuration into the purely elastic and dissipative part. Then we restrict ourselves to the class of fluids where elastic response is purely spherical. For such class of fluids we then provide a mathematical theory that, in particular, includes the long-time and large-data existence of weak solution for suitable initial and boundary value problems. This is a joint work with Miroslav Bulicek, Vit Prusa and Endre Suli. We first summarize the derivation of viscoelastic (rate-type) fluids with stress diffusion that generates the models that are compatible with the second law of thermodynamics and where no approximation/reduction takes place. The approach is based on the concept of natural configuration that splits the total response between the current and initial configuration into the purely elastic and dissipative part. Then we restrict ourselves to the class ...

76A10 ; 80A10 ; 35D30 ; 35Q35

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- xv; 523 p.
ISBN 978-3-642-16829-1

Grundlehren der mathematischen wissenschaften , 0343

Localisation : Collection 1er étage

décomposition de Littlewood-Paley # équation aux dérivées partielles non linéaire # équation de transport # équation de la chaleur # équation d'onde # équation de Schroedinger # mécanique des fluides # relativité générale

35Q35 ; 76N10 ; 76D05 ; 35Q30 ; 35-02 ; 35Q55 ; 42B25 ; 42B37 ; 76B03 ; 76D03 ; 42-02 ; 35L60

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