37 research outputs found
On the ferromagnetism equations with large variations solutions
We exhibit some large variations solutions of the Landau-Lifschitz equations
as the exchange coefficient ε^2 tends to zero. These solutions are
described by some asymptotic expansions which involve some internals layers by
means of some large amplitude fluctuations in a neighborhood of width of order
ε of an hypersurface contained in the domain. Despite the nonlinear
behaviour of these layers we manage to justify locally in time these asymptotic
expansions
An optimal penalty method for a hyperbolic system modeling the edge plasma transport in a tokamak
The penalization method is used to take account of obstacles, such as the
limiter, in a tokamak. Because of the magnetic confinement of the plasma in a
tokamak, the transport occurs essentially in the direction parallel to the
magnetic field lines. We study a 1D nonlinear hyperbolic system as a simplified
model of the plasma transport in the area close to the wall. A penalization
which cuts the flux 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 distribution sense.
Hence, a new penalty method is proposed for this hyperbolic system. For this
penalty method, an asymptotic expansion and numerical tests give an optimal
rate of convergence without spurious boundary layer. Another two-fields
penalization has also been implemented and the numerical convergence analysis
when the penalization parameter tends to reveals the presence of a boundary
layer
Asymptotic-preserving methods for an anisotropic model of electrical potential in a tokamak
A 2D nonlinear model for the electrical potential in the edge plasma in a
tokamak generates a stiff problem due to the low resistivity in the direction
parallel to the magnetic field lines. An asymptotic-preserving method based on
a micro-macro decomposition is studied in order to have a well-posed problem,
even when the parallel resistivity goes to . Numerical tests with a finite
difference scheme show a bounded condition number for the linearised discrete
problem solved at each time step, which confirms the theoretical analysis on
the continuous problem.Comment: 8 page
Counter-Examples to the Concentration-Cancellation Property
International audienceWe study the existence and the asymptotic behavior of large amplitude high-frequency oscillating waves subjected to the 2D Burger equation. This program is achieved by developing tools related to supercritical WKB analysis. By selecting solutions which are divergence free, we show that incompressible or compressible 2D Euler equations are not locally closed for the weak topology
Resonant leading order geometric optics expansions for quasilinear hyperbolic fixed and free boundary problems
International audienceWe provide a justification with rigorous error estimates showing that the leading term in weakly nonlinear geometric optics expansions of highly oscillatory reflecting wavetrains is close to the uniquely determined exact solution for small wavelengths. Waves reflecting off of fixed noncharacteristic boundaries and off of multidimensional shocks are considered under the assumption that the underlying fixed (respectively, free) boundary problem is uniformly spectrally stable in the sense of Kreiss (respectively, Majda). Our results apply to a general class of problems that includes the compressible Euler equations; as a corollary we rigorously justify the leading term in the geometric optics expansion of highly oscillatory multidimensional shock solutions of the Euler equations. An earlier stability result of this type was obtained by a method that required the construction of high-order approximate solutions. That construction in turn was possible only under a generically valid (absence of) small divisors assumption. Here we are able to remove that assumption and avoid the need for high-order expansions by studying associated singular fixed and free boundary problems. The analysis applies equally to systems that cannot be written in conservative form
Singular pseudodifferential calculus for wavetrains and pulses
International audienceWe develop a singular pseudodifferential calculus. The symbols that we consider do not satisfy the standard decay with respect to the frequency variables. We thus adopt a strategy based on the Calderon-Vaillancourt Theorem. The remainders in the symbolic calculus are bounded operators on , whose norm is measured with respect to some small parameter. Our main improvement with respect to an earlier work by Williams consists in showing a regularization effect for the remainders. Due to a nonstandard decay in the frequency variables, the regularization takes place in a scale of anisotropic, and singular, Sobolev spaces. Our analysis allows to extend previous results on the existence of highly oscillatory solutions to nonlinear hyperbolic problems. The results are also used in a companion work to justify nonlinear geometric optics with boundary amplification, which corresponds to a more singular regime than any other one considered before. The analysis is carried out with either an additional real or periodic variable in order to cover problems for pulses or wavetrains in geometric optics
Semilinear geometric optics with boundary amplification
International audienceWe study weakly stable semilinear hyperbolic boundary value problems with highly oscillatory data. Here weak stability means that exponentially growing modes are absent, but the so-called uniform Lopatinskii condition fails at some boundary frequency in the hyperbolic region. As a consequence of this degeneracy there is an amplification phenomenon: outgoing waves of amplitude and wavelength give rise to reflected waves of amplitude , so the overall solution has amplitude . Moreover, the reflecting waves emanate from a radiating wave that propagates in the boundary along a characteristic of the Lopatinskii determinant. An approximate solution that displays the qualitative behavior just described is constructed by solving suitable profile equations that exhibit a loss of derivatives, so we solve the profile equations by a Nash-Moser iteration. The exact solution is constructed by solving an associated singular problem involving singular derivatives of the form , being the tangential variables with respect to the boundary. Tame estimates for the linearization of that problem are proved using a first-order calculus of singular pseudodifferential operators constructed in the companion article \cite{CGW2}. These estimates exhibit a loss of one singular derivative and force us to construct the exact solution by a separate Nash-Moser iteration. The same estimates are used in the error analysis, which shows that the exact and approximate solutions are close in on a fixed time interval independent of the (small) wavelength . The approach using singular systems allows us to avoid constructing high order expansions and making small divisor assumptions