198 research outputs found
Symétrie et compacité dans les espaces de Sobolev
RésuméDans cet article, on met en évidence divers résultats de compacité pour les sousespaces d'espaces de Sobolev très généraux constitués par les fonctions ayant un certain nombre de symétries: symétrie sphérique, symétrie cylindrique, etc.AbstractIn this paper, we show several compactness results concerning the subspaces of general Sobolev spaces formed by the functions possessing some symmetry: spherical symmetry, cylindrical symmetry, etc
On the uniqueness of the solution of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity
We propose two different proofs of the fact that Oseen's vortex is the unique
solution of the two-dimensional Navier-Stokes equation with a Dirac mass as
initial vorticity. The first argument, due to C.E. Wayne and the second author,
is based on an entropy estimate for the vorticity equation in self-similar
variables. The second proof is new and relies on symmetrization techniques for
parabolic equations.Comment: 9 pages, no figur
Stochastic averaging lemmas for kinetic equations
We develop a class of averaging lemmas for stochastic kinetic equations. The
velocity is multiplied by a white noise which produces a remarkable change in
time scale. Compared to the deterministic case and as far as we work in ,
the nature of regularity on averages is not changed in this stochastic kinetic
equation and stays in the range of fractional Sobolev spaces at the price of an
additional expectation. However all the exponents are changed; either time
decay rates are slower (when the right hand side belongs to ), or
regularity is better when the right hand side contains derivatives. These
changes originate from a different space/time scaling in the deterministic and
stochastic cases. Our motivation comes from scalar conservation laws with
stochastic fluxes where the structure under consideration arises naturally
through the kinetic formulation of scalar conservation laws
Scalar conservation laws with rough (stochastic) fluxes
We develop a pathwise theory for scalar conservation laws with quasilinear
multiplicative rough path dependence, a special case being stochastic
conservation laws with quasilinear stochastic dependence. We introduce the
notion of pathwise stochastic entropy solutions, which is closed with the local
uniform limits of paths, and prove that it is well posed, i.e., we establish
existence, uniqueness and continuous dependence, in the form of pathwise
-contraction, as well as some explicit estimates. Our approach is
motivated by the theory of stochastic viscosity solutions, which was introduced
and developed by two of the authors, to study fully nonlinear first- and
second-order stochastic pde with multiplicative noise. This theory relies on
special test functions constructed by inverting locally the flow of the
stochastic characteristics. For conservation laws this is best implemented at
the level of the kinetic formulation which we follow here
Scalar conservation laws with rough (stochastic) fluxes; the spatially dependent case
We continue the development of the theory of pathwise stochastic entropy
solutions for scalar conservation laws in with quasilinear
multiplicative ''rough path'' dependence by considering inhomogeneous fluxes
and a single rough path like, for example, a Brownian motion. Following our
previous note where we considered spatially independent fluxes, we introduce
the notion of pathwise stochastic entropy solutions and prove that it is well
posed, that is we establish existence, uniqueness and continuous dependence in
the form of a (pathwise) -contraction. Our approach is motivated by the
theory of stochastic viscosity solutions, which was introduced and developed by
two of the authors, to study fully nonlinear first- and second-order stochastic
pde with multiplicative noise. This theory relies on special test functions
constructed by inverting locally the flow of the stochastic characteristics.
For conservation laws this is best implemented at the level of the kinetic
formulation which we follow here
user's guide to viscosity solutions of second order partial differential equations
The notion of viscosity solutions of scalar fully nonlinear partial
differential equations of second order provides a framework in which startling
comparison and uniqueness theorems, existence theorems, and theorems about
continuous dependence may now be proved by very efficient and striking
arguments. The range of important applications of these results is enormous.
This article is a self-contained exposition of the basic theory of viscosity
solutions.Comment: 67 page
Mean Field Games and Applications.
This text is inspired from a “Cours Bachelier” held in January 2009 and taught by Jean-Michel Lasry. This course was based upon the articles of the three authors and upon unpublished materials they developed. Proofs were not presented during the conferences and are now available. So are some issues that were only rapidly tackled during class.Mean Field Games;
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