57 research outputs found
Unique Implementation in Auctions and in Public Goods Problems
Des conditions nouvelles sont prĂ©sentĂ©es, assurant lâexistence de mĂ©canismes dâenchĂšres ou de production de biens publics menant Ă un Ă©quilibre unique ou essentiellement unique, lorsque les prĂ©fĂ©rences des agents sont quasi linĂ©aires. Ces conditions portent exclusivement sur la croyance des agents.We present new conditions that guarantee the existence of mechanism with a unique or essentially unique equilibrium in auction and public goods problems with quasi-linear utility functions. These conditions bear only on the information structures of the agents
A Kato type Theorem for the inviscid limit of the Navier-Stokes equations with a moving rigid body
The issue of the inviscid limit for the incompressible Navier-Stokes
equations when a no-slip condition is prescribed on the boundary is a famous
open problem. A result by Tosio Kato says that convergence to the Euler
equations holds true in the energy space if and only if the energy dissipation
rate of the viscous flow in a boundary layer of width proportional to the
viscosity vanishes. Of course, if one considers the motion of a solid body in
an incompressible fluid, with a no-slip condition at the interface, the issue
of the inviscid limit is as least as difficult. However it is not clear if the
additional difficulties linked to the body's dynamic make this issue more
difficult or not. In this paper we consider the motion of a rigid body in an
incompressible fluid occupying the complementary set in the space and we prove
that a Kato type condition implies the convergence of the fluid velocity and of
the body velocity as well, what seems to indicate that an answer in the case of
a fixed boundary could also bring an answer to the case where there is a moving
body in the fluid
On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations
We consider an active scalar equation that is motivated by a model for
magneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusive
equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast,
the critically diffusive equation is well-posed. In this case we give an
example of a steady state that is nonlinearly unstable, and hence produces a
dynamo effect in the sense of an exponentially growing magnetic field.Comment: We have modified the definition of Lipschitz well-posedness, in order
to allow for a possible loss in regularity of the solution ma
Time scales separation for dynamo action
The study of dynamo action in astrophysical objects classically involves two
timescales: the slow diffusive one and the fast advective one. We investigate
the possibility of field amplification on an intermediate timescale associated
with time dependent modulations of the flow. We consider a simple steady
configuration for which dynamo action is not realised. We study the effect of
time dependent perturbations of the flow. We show that some vanishing low
frequency perturbations can yield exponential growth of the magnetic field on
the typical time scale of oscillation. The dynamo mechanism relies here on a
parametric instability associated with transient amplification by shear flows.
Consequences on natural dynamos are discussed
Existence of global strong solutions to a beam-fluid interaction system
We study an unsteady non linear fluid-structure interaction problem which is
a simplified model to describe blood flow through viscoleastic arteries. We
consider a Newtonian incompressible two-dimensional flow described by the
Navier-Stokes equations set in an unknown domain depending on the displacement
of a structure, which itself satisfies a linear viscoelastic beam equation. The
fluid and the structure are fully coupled via interface conditions prescribing
the continuity of the velocities at the fluid-structure interface and the
action-reaction principle. We prove that strong solutions to this problem are
global-in-time. We obtain in particular that contact between the viscoleastic
wall and the bottom of the fluid cavity does not occur in finite time. To our
knowledge, this is the first occurrence of a no-contact result, but also of
existence of strong solutions globally in time, in the frame of interactions
between a viscous fluid and a deformable structure
Well-posedness of boundary layer equations for time-dependent flow of non-Newtonian fluids
We consider the flow of an upper convected Maxwell fluid in the limit of high
Weissenberg and Reynolds number. In this limit, the no-slip condition cannot be
imposed on the solutions. We derive equations for the resulting boundary layer
and prove the well-posedness of these equations. A transformation to Lagrangian
coordinates is crucial in the argument
The Navier wall law at a boundary with random roughness
We consider the Navier-Stokes equation in a domain with irregular boundaries.
The irregularity is modeled by a spatially homogeneous random process, with
typical size \eps \ll 1. In a parent paper, we derived a homogenized boundary
condition of Navier type as \eps \to 0. We show here that for a large class
of boundaries, this Navier condition provides a O(\eps^{3/2} |\ln
\eps|^{1/2}) approximation in , instead of O(\eps^{3/2}) for periodic
irregularities. Our result relies on the study of an auxiliary boundary layer
system. Decay properties of this boundary layer are deduced from a central
limit theorem for dependent variables
Uniform regularity for the Navier-Stokes equation with Navier boundary condition
We prove that there exists an interval of time which is uniform in the
vanishing viscosity limit and for which the Navier-Stokes equation with Navier
boundary condition has a strong solution. This solution is uniformly bounded in
a conormal Sobolev space and has only one normal derivative bounded in
. This allows to get the vanishing viscosity limit to the
incompressible Euler system from a strong compactness argument
The Inviscid Limit and Boundary Layers for Navier-Stokes Flows
The validity of the vanishing viscosity limit, that is, whether solutions of
the Navier-Stokes equations modeling viscous incompressible flows converge to
solutions of the Euler equations modeling inviscid incompressible flows as
viscosity approaches zero, is one of the most fundamental issues in
mathematical fluid mechanics. The problem is classified into two categories:
the case when the physical boundary is absent, and the case when the physical
boundary is present and the effect of the boundary layer becomes significant.
The aim of this article is to review recent progress on the mathematical
analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of
Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final
publication is available at http://www.springerlink.co
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