48 research outputs found
Long time behaviour of viscous scalar conservation laws
This paper is concerned with the stability of stationary solutions of the
conservation law , where
the flux is periodic with respect to its first variable. Essentially two
kinds of asymptotic behaviours are studied here: the case when the equation is
set on , and the case when it is endowed with periodic boundary conditions.
In the whole space case, we first prove the existence of viscous stationary
shocks - also called standing shocks - which connect two different periodic
stationary solutions to one another. We prove that standing shocks are stable
in , provided the initial disturbance satisfies some appropriate
boundedness conditions. We also extend this result to arbitrary initial data,
but with some restrictions on the flux . In the periodic case, we prove that
periodic stationary solutions are always stable. The proof of this result
relies on the derivation of uniform bounds on the solution of the
conservation law, and on sub- and super-solution techniques.Comment: 36 page
Mathematical study of degenerate boundary layers: A Large Scale Ocean Circulation Problem
This paper is concerned with a complete asymptoticanalysis as of the stationary Munk equation in a domain , supplemented with
boundaryconditions for and . This equation is a
simplemodel for the circulation of currents in closed basins, the variables
and being respectively the longitude and the latitude. A crudeanalysis
shows that as , the weak limit of satisfiesthe
so-called Sverdrup transport equation inside the domain, namely, while boundary layers appear in the vicinity ofthe boundary.These
boundary layers, which are the main center of interest of thepresent paper,
exhibit several types of peculiar behaviour. First, thesize of the boundary
layer on the western and eastern boundary, whichhad already been computed by
several authors, becomes formally verylarge as one approaches northern and
southern portions of the boudary,i.e. pieces of the boundary on which the
normal is vertical. Thisphenomenon is known as geostrophic degeneracy. In order
to avoid suchsingular behaviour, previous studies imposed restrictive
assumptionson the domain and on the forcing term . Here, we
provethat a superposition of two boundary layers occurs in the vicinity ofsuch
points: the classical western or eastern boundary layers, andsome northern or
southern boundary layers, whose mathematicalderivation is completely new. The
size of northern/southern boundarylayers is much larger than the one of western
boundary layers( vs. ). We explain in
detail how the superpositiontakes place, depending on the geometry of the
boundary.Moreover, when the domain is not connex in the
direction, is not continuous in , and singular layers appear
inorder to correct its discontinuities. These singular layers areconcentrated
in the vicinity of horizontal lines, and thereforepenetrate the interior of the
domain . Hence we exhibit some kindof boundary layer separation.
However, we emphasize that we remainable to prove a convergence theorem, so
that the singular layerssomehow remain stable, in spite of the
separation.Eventually, the effect of boundary layers is non-local in
severalaspects. On the first hand, for algebraic reasons, the boundary
layerequation is radically different on the west and east parts of theboundary.
As a consequence, the Sverdrup equation is endowed with aDirichlet condition on
the East boundary, and no condition on the Westboundary. Therefore western and
eastern boundary layers have in factan influence on the whole domain ,
and not only near theboundary. On the second hand, the northern and southern
boundary layerprofiles obey a propagation equation, where the space variable
plays the role of time, and are therefore not local.Comment: http://www.ams.org/books/memo/1206/memo1206.pd
Well-posedness of the Stokes-Coriolis system in the half-space over a rough surface
This paper is devoted to the well-posedness of the stationary d
Stokes-Coriolis system set in a half-space with rough bottom and Dirichlet data
which does not decrease at space infinity. Our system is a linearized version
of the Ekman boundary layer system. We look for a solution of infinite energy
in a space of Sobolev regularity. Following an idea of G\'erard-Varet and
Masmoudi, the general strategy is to reduce the problem to a bumpy channel
bounded in the vertical direction thanks a transparent boundary condition
involving a Dirichlet to Neumann operator. Our analysis emphasizes some strong
singularities of the Stokes-Coriolis operator at low tangential frequencies.
One of the main features of our work lies in the definition of a Dirichlet to
Neumann operator for the Stokes-Coriolis system with data in the Kato space
.Comment: 64 page
Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux
This article investigates the long-time behaviour of parabolic scalar
conservation laws of the type , where and the flux is periodic in . More
specifically, we consider the case when the initial data is an
disturbance of a stationary periodic solution. We show, under polynomial growth
assumptions on the flux, that the difference between and the stationary
solution vanishes for large times in norm. The proof uses a self-similar
change of variables which is well-suited for the analysis of the long time
behaviour of parabolic equations. Then, convergence in self-similar variables
follows from arguments from dynamical systems theory. One crucial point is to
obtain compactness in on the family of rescaled solutions; this is
achieved by deriving uniform bounds in weighted spaces.Comment: 37 page