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Documents  35Q35 | enregistrements trouvés : 16

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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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We investigate the gyrokinetic limit for the two-dimensional Vlasov-Poisson system in a regime studied by F. Golse and L. Saint-Raymond [1, 3]. 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 analyze the asymptotics for a Vlasov-Poisson system describing the interaction of a bounded density of particles with a moving point charge, characterized by a Dirac mass in the phase-space.
We investigate the gyrokinetic limit for the two-dimensional Vlasov-Poisson system in a regime studied by F. Golse and L. Saint-Raymond [1, 3]. 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 analyze the asymptotics for a Vlasov-Poisson system describing the interaction of a bounded density of particles with a moving point charge, characterized by ...

76X05 ; 82C21 ; 35Q35 ; 35Q83 ; 35Q60 ; 82D10

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We construct a hierarchy of hybrid numerical methods for multi-scale kinetic equations based on moment realizability matrices, a concept introduced by Levermore, Morokoff and Nadiga. Following such a criterion, one can consider hybrid scheme where the hydrodynamic part is given either by the compressible Euler or Navier-Stokes equations, or even with more general models, such as the Burnett or super-Burnett systems.
PDE - numerical methods - Boltzmann equation - fluid models - hybrid methods
We construct a hierarchy of hybrid numerical methods for multi-scale kinetic equations based on moment realizability matrices, a concept introduced by Levermore, Morokoff and Nadiga. Following such a criterion, one can consider hybrid scheme where the hydrodynamic part is given either by the compressible Euler or Navier-Stokes equations, or even with more general models, such as the Burnett or super-Burnett systems.
PDE - numerical methods - ...

35Q35 ; 65N08 ; 65N22

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Multi angle  On the Hall-MHD equations
Chae, Dongho (Auteur de la Conférence) | CIRM (Editeur )

In this talk we present recent results on the Hall-MHD system. We consider the incompressible MHD-Hall equations in $\mathbb{R}^3$.

$\partial_tu +u \cdot u + \nabla u+\nabla p = \left ( \nabla \times B \right )\times B +\nu \nabla u,$
$\nabla \cdot u =0, \nabla \cdot B =0, $
$\partial_tB - \nabla \times \left (u \times B\right ) + \nabla \times \left (\left (\nabla \times B\right )\times B \right ) = \mu \nabla B,$
$u\left (x,0 \right )=u_0\left (x\right ) ; B\left (x,0 \right )=B_0\left (x\right ).$

Here $u=\left (u_1, u_2, u_3 \right ) = u \left (x,t \right ) $ is the velocity of the charged fluid, $B=\left (B_1, B_2, B_3 \right ) $ the magnetic field induced by the motion of the charged fluid, $p=p \left (x,t \right )$ the pressure of the fluid. The positive constants $\nu$ and $\mu$ are the viscosity and the resistivity coefficients. Compared with the usual viscous incompressible MHD system, the above system contains the extra term $\nabla \times \left (\left (\nabla \times B\right )\times B \right ) $ , which is the so called Hall term. This term is important when the magnetic shear is large, where the magnetic reconnection happens. On the other hand, in the case of laminar ows where the shear is weak, one ignores the Hall term, and the system reduces to the usual MHD. Compared to the case of the usual MHD the history of the fully rigorous mathematical study of the Cauchy problem for the Hall-MHD system is very short. The global existence of weak solutions in the periodic domain is done in [1] by a Galerkin approximation. The global existence in the whole domain in $\mathbb{R}^3$ as well as the local well-posedness of smooth solution is proved in [2], where the global existence of smooth solution for small initial data is also established. A refined form of the blow-up criteria and small data global existence is obtained in [3]. Temporal decay estimateof the global small solutions is deduced in [4]. In the case of zero resistivity we present finite time blow-up result for the solutions obtained in [5]. We note that this is quite rare case, as far as the authors know, where the blow-up result for the incompressible flows is proved.
In this talk we present recent results on the Hall-MHD system. We consider the incompressible MHD-Hall equations in $\mathbb{R}^3$.

$\partial_tu +u \cdot u + \nabla u+\nabla p = \left ( \nabla \times B \right )\times B +\nu \nabla u,$
$\nabla \cdot u =0, \nabla \cdot B =0, $
$\partial_tB - \nabla \times \left (u \times B\right ) + \nabla \times \left (\left (\nabla \times B\right )\times B \right ) = \mu \nabla B,$
$u\left (x,0 \right ...

35Q35 ; 76W05

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Vlasov-Poisson-Fokker-Planck equations provide a simplified model for a cloud of cold atoms in a Magneto Optical Trap. The strong field, or quasi-neutral regime, where the repulsive interaction dominates, is often relevant for experiments. Motivated by this example and more generally by trapped non neutral plasmas, we study this quasi-neutral limit, and show under certain conditions the convergence of the solution of Vlasov-Poisson-Fokker-Planck equations to the solution of incompressible Euler equation.
For an infinite or periodic system, this limit has already been studied by Y. Brenier and N. Masmoudi. New difficulties arise here from the Fokker-Planck operator, and especially from the boundary conditions (Joint work with D. Chiron, T. Goudon et N. Masmoudi).
Vlasov-Poisson-Fokker-Planck equations provide a simplified model for a cloud of cold atoms in a Magneto Optical Trap. The strong field, or quasi-neutral regime, where the repulsive interaction dominates, is often relevant for experiments. Motivated by this example and more generally by trapped non neutral plasmas, we study this quasi-neutral limit, and show under certain conditions the convergence of the solution of Vlasov-Poisson-Fokker-Planck ...

35Q35 ; 82D10

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Following the seminal work by Benamou and Brenier on the time continuous formulation of the optimal transport problem, we show how optimal transport techniques can be used in various areas, ranging from "the reconstruction problem" cosmology to a problem of volatility calibration in finance.

65K10 ; 85A30 ; 85A40 ; 35Q35

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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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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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We consider an acoustic waveguide modeled as follows:

$ \left \{\begin {matrix}
\Delta u+k^2(1+V)u=0& in & \Omega= \mathbb{R} \times]0,1[\\
\frac{\partial u}{\partial y}=0& on & \partial \Omega
\end{matrix}\right.$

where $u$ denotes the complex valued pressure, k is the frequency and $V \in L^\infty(\Omega)$ is a compactly supported potential.
It is well-known that they may exist non trivial solutions $u$ in $L^2(\Omega)$, called trapped modes. Associated eigenvalues $\lambda = k^2$ are embedded in the essential spectrum $\mathbb{R}^+$. They can be computed as the real part of the complex spectrum of a non-self-adjoint eigenvalue problem, defined by using the so-called Perfectly Matched Layers (which consist in a complex dilation in the infinite direction) [1].
We show here that it is possible, by modifying in particular the parameters of the Perfectly Matched Layers, to define new complex spectra which include, in addition to trapped modes, frequencies where the potential $V$ is, in some sense, invisible to one incident wave.
Our approach allows to extend to higher dimension the results obtained in [2] on a 1D model problem.
We consider an acoustic waveguide modeled as follows:

$ \left \{\begin {matrix}
\Delta u+k^2(1+V)u=0& in & \Omega= \mathbb{R} \times]0,1[\\
\frac{\partial u}{\partial y}=0& on & \partial \Omega
\end{matrix}\right.$

where $u$ denotes the complex valued pressure, k is the frequency and $V \in L^\infty(\Omega)$ is a compactly supported potential.
It is well-known that they may exist non trivial solutions $u$ in $L^2(\Omega)$, called trapped ...

35Q35 ; 35J05 ; 65N30 ; 41A60 ; 47H10 ; 76Q05 ; 35B40

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We consider the 1-D Schrödinger system with point vortex-type interactions that was derived by R. Klein, A. Majda and K. Damodaran and by V. Zakharov to modelize the dynamics of N nearly parallel vortex filaments in a 3-D incompressible fluid. We first prove a global in time result and display several classes of solutions. Then we consider the problem of collisions. In particular we establish rigorously the existence of a pair of almost parallel vortex filaments, with opposite circulation, colliding at some point in finite time. These results are joint works with E. Faou and E. Miot.
We consider the 1-D Schrödinger system with point vortex-type interactions that was derived by R. Klein, A. Majda and K. Damodaran and by V. Zakharov to modelize the dynamics of N nearly parallel vortex filaments in a 3-D incompressible fluid. We first prove a global in time result and display several classes of solutions. Then we consider the problem of collisions. In particular we establish rigorously the existence of a pair of almost parallel ...

35Q35 ; 76B47

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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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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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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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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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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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In this talk, I will present the global solvability of the primitive equations for the atmosphere coupled to moisture dynamics with phase changes for warm clouds, where water is present in the form of water vapor and in the liquid state as cloud water and rain water. This moisture model, which has been used by Klein-Majda in [1] and corresponds to a basic form of the bulk microphysics closure in the spirit of Kessler [2] and Grabowski-Smolarkiewicz [3], contains closures for the phase changes condensation and evaporation, as well as the processes of autoconversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. The moisture balances are strongly coupled to the thermodynamic equation via the latent heat associated to the phase changes. The global well-posedness was proved by combining the technique used in Hittmeir-Klein-Li-Titi [4], where global well-posedness was established for the pure moisture system for given velocity, and the ideas of Cao-Titi [5], who succeeded in proving the global solvability of the primitive equations without coupling to the moisture.
In this talk, I will present the global solvability of the primitive equations for the atmosphere coupled to moisture dynamics with phase changes for warm clouds, where water is present in the form of water vapor and in the liquid state as cloud water and rain water. This moisture model, which has been used by Klein-Majda in [1] and corresponds to a basic form of the bulk microphysics closure in the spirit of Kessler [2] and Grabowski-S...

35A01 ; 35B45 ; 35D35 ; 35M86 ; 35Q30 ; 35Q35 ; 35Q86 ; 76D03 ; 76D09

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