26 research outputs found
Penalty Methods for the Hyperbolic System Modelling the Wall-Plasma Interaction in a Tokamak
The penalization method is used to take account of obstacles in a tokamak,
such as the limiter. We study a non linear hyperbolic system modelling the
plasma transport in the area close to the wall. A penalization which cuts the
transport term of the momentum is studied. We show numerically that this
penalization creates a Dirac measure at the plasma-limiter interface which
prevents us from defining the transport term in the usual sense. Hence, a new
penalty method is proposed for this hyperbolic system and numerical tests
reveal an optimal convergence rate without any spurious boundary layer.Comment: 8 pages; International Symposium FVCA6, Prague : Czech Republic
(2011
Multiphase weakly nonlinear geometric optics for Schrodinger equations
We describe and rigorously justify the nonlinear interaction of highly
oscillatory waves in nonlinear Schrodinger equations, posed on Euclidean space
or on the torus. Our scaling corresponds to a weakly nonlinear regime where the
nonlinearity affects the leading order amplitude of the solution, but does not
alter the rapid oscillations. We consider initial states which are
superpositions of slowly modulated plane waves, and use the framework of Wiener
algebras. A detailed analysis of the corresponding nonlinear wave mixing
phenomena is given, including a geometric interpretation on the resonance
structure for cubic nonlinearities. As an application, we recover and extend
some instability results for the nonlinear Schrodinger equation on the torus in
negative order Sobolev spaces.Comment: 29 page
Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions
Under natural spectral stability assumptions motivated by previous
investigations of the associated spectral stability problem, we determine sharp
estimates on the linearized solution operator about a multidimensional
planar periodic wave of a system of conservation laws with viscosity, yielding
linearized stability for all and dimensions and nonlinear stability and
-asymptotic behavior for and . The behavior can in
general be rather complicated, involving both convective (i.e., wave-like) and
diffusive effects
Spectral stability of noncharacteristic isentropic Navier-Stokes boundary layers
Building on work of Barker, Humpherys, Lafitte, Rudd, and Zumbrun in the
shock wave case, we study stability of compressive, or "shock-like", boundary
layers of the isentropic compressible Navier-Stokes equations with gamma-law
pressure by a combination of asymptotic ODE estimates and numerical Evans
function computations. Our results indicate stability for gamma in the interval
[1, 3] for all compressive boundary-layers, independent of amplitude, save for
inflow layers in the characteristic limit (not treated). Expansive inflow
boundary-layers have been shown to be stable for all amplitudes by Matsumura
and Nishihara using energy estimates. Besides the parameter of amplitude
appearing in the shock case, the boundary-layer case features an additional
parameter measuring displacement of the background profile, which greatly
complicates the resulting case structure. Moreover, inflow boundary layers turn
out to have quite delicate stability in both large-displacement and
large-amplitude limits, necessitating the additional use of a mod-two stability
index studied earlier by Serre and Zumbrun in order to decide stability
ON SOME GEOMETRY OF PROPAGATION IN DIFFRACTIVE TIME SCALES
International audienceIn this article, we develop a non linear geometric optics which presents the two main following features. It is valid in diffractive times and it extends the classical approaches to the case of fast variable coefficients. In this context, we can show that the energy is transported along the rays associated with some non usual long-time hamiltonian. Our analysis needs structural assumptions and initial data suitably polrarized to be implemented. All the required conditions are met concerning a current model arising in fluid mechanics and which was the original motivation of our work. As a by product, we get results complementary to the litterature concerning the propagation of the Rossby waves which play a part in the description of large oceanic currents, like Gulf stream or Kuroshio
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
Multidimensional Conservation Laws: Overview, Problems, and Perspective
Some of recent important developments are overviewed, several longstanding
open problems are discussed, and a perspective is presented for the
mathematical theory of multidimensional conservation laws. Some basic features
and phenomena of multidimensional hyperbolic conservation laws are revealed,
and some samples of multidimensional systems/models and related important
problems are presented and analyzed with emphasis on the prototypes that have
been solved or may be expected to be solved rigorously at least for some cases.
In particular, multidimensional steady supersonic problems and transonic
problems, shock reflection-diffraction problems, and related effective
nonlinear approaches are analyzed. A theory of divergence-measure vector fields
and related analytical frameworks for the analysis of entropy solutions are
discussed.Comment: 43 pages, 3 figure
Local existence of stratified solutions to systems of balance laws
We prove the local existence of stratified solutions to systems of balance laws. We also provide specific physical examples