Asymptotic Analysis - Volume 14, issue 4

Authors: Temam, Roger | Wang, Xiaoming

Article Type: Research Article

Abstract: We continue our study of the asymptotic behavior of the Navier–Stokes equations linearized around the rest state as viscostiy ε approaches zero. We study the convergence as ε→0 to the inviscid type equations. Suitable correctors are obtained which resolve the boundary layer and we obtain convergence results valid up to the boundary. Explicit asymptotic expansion formulas are given which display the boundary layer phenomena. We improve our previous by treating here the general smooth bounded domain in R2 instead of two-dimensional channels. Curvilinear coordinates are used to resolve the complex geometry.

Keywords: Asymptotic expansions, correctors, boundary layer, Navier–Stokes equations, curvilinear coordinates

DOI: 10.3233/ASY-1997-14401

Citation: Asymptotic Analysis, vol. 14, no. 4, pp. 293-321, 1997

Authors: Berrone, Lucio R. | Mannucci, Paola

Article Type: Research Article

Abstract: We study the asymptotic behaviour of the diffraction problem in the one-dimensional case when the conductivity of one of the two media goes to infinity. By means of heat potentials and Laplace transforms we deduce that the temperature of the good conductor (well-stirred fluid) is spatially constant and we find the related contact conditions between the two media involving time derivatives.

DOI: 10.3233/ASY-1997-14402

Citation: Asymptotic Analysis, vol. 14, no. 4, pp. 323-337, 1997

Authors: Komornik, Vilmos | Rao, Bopeng

Article Type: Research Article

Abstract: Using a new method, we prove the uniform exponential stability of a linear system of compactly coupled wave equations. We also estimate the energy decay rate in case of nonlinear boundary feedbacks.

DOI: 10.3233/ASY-1997-14403

Citation: Asymptotic Analysis, vol. 14, no. 4, pp. 339-359, 1997

Authors: Fournie, Eric | Lebuchoux, Jérôme | Touzi, Nizar

Article Type: Research Article

Abstract: Consider the second-order differential operators L0 =−∂t −a1 (y)∂x −a2 (y)∂xx and $\tilde{L}$ =−b1 (y)∂y −b2 (y)∂yy −c(y)∂xy and let uε (t,x,y) be the solution of the parabolic problem L0 +ε$\tilde{L}$ u=0 on [0,T)×R2 with terminal condition uε (T,x,y)=ϕ(x), for given ε∈R. We provide an explicit asymptotic expansion of the solution uε around the value ε=0. The expansion coefficients of any order are determined by an explicit induction scheme involving the derivatives of u0 with respect to x. The results are applied for the computation of European contingent claim prices by Monte Carlo …simulations in stochastic volatility models, which are popular in the financial literature. The asymptotic expansion is used as accelerator in an importance sampling variance reduction procedure. Show more

DOI: 10.3233/ASY-1997-14404

Citation: Asymptotic Analysis, vol. 14, no. 4, pp. 361-376, 1997

Authors: Combescure, M. | Robert, D.

Article Type: Research Article

Abstract: Precise semiclassical estimates for the spreading of quantum wave packets are derived, when the initial wave packet is in a coherent state. We find asymptotics for the quantum evolution of coherent states, at any order in the Planck constant ħ, with a control in large time of the remainder term depending explicitely on ħ and on the stability matrix. Our results extend Hagedorn's work on the propagation of Gaussian coherent states. We present here a proof simplifying Hagedorn's arguments, and extending it to general, possibly time-dependent Hamiltonians, not necessarily in the form of kinetic energy plus potential energy (p2 …+ V(x)). Our proof also works for more general coherent states and extends recent results by Paul and Uribe. As a first application of our semiclassical estimates we show that, if the initial quantum state is a coherent state located around an unstable fixed point α of the classical flow, the spreading of the quantum wave packet at time t grows like e2λt for times not larger than (γ/λ)log (1/ħ), where λ is the classical instability exponent associated to the fixed point α and γ is a numerical constant. Show more

DOI: 10.3233/ASY-1997-14405

Citation: Asymptotic Analysis, vol. 14, no. 4, pp. 377-404, 1997

Article Type: Other

Citation: Asymptotic Analysis, vol. 14, no. 4, pp. 405-406, 1997