Asymptotic Analysis - Volume 18, issue 3-4

Authors: Shibata, Tetsutaro

Article Type: Research Article

Abstract: An asymptotic formula of variational eigenvalues of nonlinear multiparameter Sturm–Liouville problems is established. The proof is based on Ljusternik–Schnirelman theory on general level sets due to Zeidler, Pohozaev‐type equality and ODE techniques.

Citation: Asymptotic Analysis, vol. 18, no. 3-4, pp. 173-192, 1998

Authors: Frénod, Emmanuel | Sonnendrücker, Eric

Article Type: Research Article

Abstract: Motivated by the difficulty arising in the numerical simulation of the movement of charged particles in presence of a large external magnetic field, which adds an additional time scale and thus imposes to use a much smaller time step, we perform in this paper a homogenization of the Vlasov equation and the Vlasov–Poisson system which yield approximate equations describing the mean behavior of the particles. The convergence proof is based on the two‐scale convergence tools introduced by N’Guetseng and Allaire. We also consider the case where, in addition to the magnetic field, a large external electric field orthogonal to the …magnetic field and of the same magnitude is applied. Show more

Keywords: Vlasov–Poisson equations, kinetic equations, homogenization, gyrokinetic approximation, multiple time scales, two‐scale convergence

Citation: Asymptotic Analysis, vol. 18, no. 3-4, pp. 193-213, 1998

Authors: Porod, Ursula | Zelditch, Steve

Article Type: Research Article

Abstract: Let (G, \mu) be a symmetric random walk on a compact Lie group G . We will call (G, \mu) a Lagrangean random walk if the step distribution \mu , a probability measure on G , is also a Lagrangean distribution on G with respect to some Lagrangean submanifold \varLambda \subset T^*G . In particular, we are interested in the cases where \mu is a smooth \delta ‐function \delta_C along a ‘positively curved hypersurface’ C of G or where \mu …is a sum of \delta ‐functions \sum_j \delta_{C_j} along a finite union of regular conjugacy classes C_j in G . The Markov (transition) operator T_{\mu} of the Lagrangean random walk is then a Fourier integral operator and our purpose is to apply microlocal techniques to study the convolution powers \mu^{*k} of \mu. In cases where all convolution powers are ‘clean’ (such as for \delta ‐functions on positively curved hypersurfaces), classical FIO methods will be used to determine the Sobolev smoothing order of T_{\mu} on W^s(G) , the minimal power k = k_{\mu} for which \mu^{*k} \in L^2 , the asympotics of the Fourier transform \widehat {\mu}(\rho) of \mu along rays L = \mathbb{N}\rho of representations. In general, convolutions of Lagrangean measures are not ‘clean’ and there can occur a large variety of possible singular behaviour in the convolution powers \mu^{*k} . Classical FIO methods are then no longer sufficient to analyze the asymptotic properties of Lagrangean random walks. However, it is sometimes possible to restore the simple ‘clean convolution’ behaviour by restricting the random walk to a fixed ‘ray of representations’. In such cases, classical Toeplitz methods can be used to determine restricted versions of the above features along the ray. We will illustrate with the case of sums of \delta ‐functions along unions of regular conjugacy classes. Show more

Citation: Asymptotic Analysis, vol. 18, no. 3-4, pp. 215-261, 1998

Authors: Comech, Andrew

Article Type: Research Article

Abstract: We derive damping estimates and asymptotics of L^p operator norms for oscillatory integral operators with finite type singularities. The methods are based on incorporating finite type conditions into L^2 almost orthogonality technique of Cotlar–Stein.

Citation: Asymptotic Analysis, vol. 18, no. 3-4, pp. 263-278, 1998

Authors: Hinder, Rainer | Nazarov, Sergueï A.

Article Type: Research Article

Abstract: We are interested in finding the velocity distribution at the wings of an aeroplane. Within the scope of a three‐dimensional linear theory we analyse a model which is formulated as a mixed screen boundary value problem for the Helmholtz equation (\Delta + k^{ 2}) \varPhi =0 in \mathbb{R}^{ 3}\backslash \overline{S} , where \varPhi denotes the perturbation velocity potential, induced by the presence of the wings and \overline{S} := \overline{L} \cup \overline{W} with the projection L of the wings onto the (y_{ 1},y_{ 2}) ‐plane and the wake W …. Not all Cauchy data are given explicitly on L , respectively W . These missing Cauchy data depend on the wing circulation \varGamma . \varGamma has to be fixed by the Kutta–Joukovskii condition: \nabla \varPhi should be finite near the trailing edge x_{\rm t} of L . We reduce here this screen problem to an equivalent mixed boundary value problem in \mathbb{R}^{ 3}_{ +} . The main problem is in both cases the calculation of \varGamma . In order to find \varPhi we use the method of matched asymptotics for some small geometrical parameter \varepsilon and the ansatz \varGamma = \varGamma _{ 0} + \varepsilon \varGamma _{ 1} + \cdots which makes it possible to split the problem into a sequence of problems for \varGamma _{ 0},\varGamma _{ 1}, \ldots\, . Concretely, we calculate \varGamma _{ 0} and \varGamma _{ 1} explicitly by the demand of vanishing intensity factors of the solutions of the corresponding mixed problems at the borderline between L and W . Especially, we point out that \varGamma _{ 0} can be obtained by solving a two‐dimensional problem for every cross‐section of L while \varGamma _{ 1} indicates the interaction of these cross‐sections. Show more

Keywords: Lifting surface theory, Kutta–Joukovskii condition, asymptotic analysis, matching procedure

Citation: Asymptotic Analysis, vol. 18, no. 3-4, pp. 279-305, 1998

Authors: Zhang, Ping | Zheng, Yuxi

Article Type: Research Article

Abstract: The wave equation \curpartial _t^2u - c\curpartial _x(c\curpartial _xu)=0 , where c = c(u) is a given function, arises in a number of different physical contexts and is the simplest example of an interesting class of nonlinear hyperbolic partial differential equations. For unidirectional weakly nonlinear waves, an asymptotic equation \curpartial _x(\curpartial _tu + u\curpartial _xu) =(1/2)(\curpartial _xu)^2 has been derived. It has been shown through concrete examples that oscillations in v , v=\curpartial _xu , in the initial data persist into positive time for the asymptotic equation. In the first part of this …paper, we show by applying Young measure theory that no oscillations are generated if there are no oscillations (around a nonnegative state v ) in the initial data, which implies in particular the global existence of weak solutions to the asymptotic equation with nonnegative L^p(\mathbb{R}) initial data v with p>2 . In the second part, we obtain a regularity result for a large class of weak solutions to this equation by using its kinetic formulation. In particular this regularity result applies to both the conservative and dissipative weak solutions. Show more

Keywords: Existence, kinetic formulation, regularity, Young measure

Citation: Asymptotic Analysis, vol. 18, no. 3-4, pp. 307-327, 1998

Authors: Roquejoffre, Jean‐Michel | Vila, Jean‐Paul

Article Type: Research Article

Abstract: This paper is the first of a series aimed at deriving uniform stability results for strong ZND – Zeldovich–Neumann–Döring – detonation waves, in the context of singular perturbations. In this work, the Majda and Majda–Rosales models with small viscosity are examined, and a dispersion relation accounting for the linear stability of the wave is derived, by means of matched asymptotics. A rigorous justification follows, in which the main difficulty is a uniform spectral estimate.

Citation: Asymptotic Analysis, vol. 18, no. 3-4, pp. 329-348, 1998

Authors: Perla Menzala, G. | Zuazua, Enrique

Article Type: Research Article

Abstract: We study the decay of the energy of solutions of the system of magnetoelasticity in a bounded, three‐dimensional conductive medium. We prove that all solutions do decay as t\to\infty in the energy‐space when the domain is simply connected. We also describe the large time behavior of solutions when the domain is not simply connected.

Citation: Asymptotic Analysis, vol. 18, no. 3-4, pp. 349-362, 1998