Asymptotic Analysis - Volume 95, issue 3-4

Authors: Hillairet, M. | Kelaï, T.

Article Type: Research Article

Abstract: We consider the stationary Stokes problem in a three-dimensional fluid domain F with non-homogeneous Dirichlet boundary conditions. We assume that this fluid domain is the complement of a bounded obstacle B in a bounded or an exterior smooth container Ω . We compute sharp asymptotics of the solution to the Stokes problem when the distance between the obstacle and the container boundary is small.

Keywords: fluid/solid interactions, Stokes problem, lubrication approximation

DOI: 10.3233/ASY-151319

Citation: Asymptotic Analysis, vol. 95, no. 3-4, pp. 187-241, 2015

Authors: Barles, Guy | Briani, Ariela | Chasseigne, Emmanuel | Tchou, Nicoletta

Article Type: Research Article

Abstract: We consider homogenization problems in the framework of deterministic optimal control when the dynamics and running costs are completely different in two (or more) complementary domains of the space R N . For such optimal control problems, the three first authors have shown that several value functions can be defined, depending, in particular, if the choice is to use only “regular strategies” or to use also “singular strategies”. We study the homogenization problem in these two different cases. It is worth pointing out that, if the second one can be handled by usual partial differential …equations method “à la Lions–Papanicolaou–Varadhan” with suitable adaptations, the first case has to be treated by control methods (dynamic programming). Show more

Keywords: homogenization, deterministic optimal control, discontinuous dynamic, cell problem, Bellman equation, viscosity solutions

DOI: 10.3233/ASY-151322

Citation: Asymptotic Analysis, vol. 95, no. 3-4, pp. 243-278, 2015

Authors: Sellama, Hocine

Article Type: Research Article

Abstract: We consider the discretization q ( t + ε ) + q ( t − ε ) − 2 q ( t ) = ε 2 sin ( q ( t ) ) , ε > 0 a small parameter, of the pendulum equation q ″ = sin ( q ) ; in system form, we have the discretization q ( t + ε ) − q ( t ) = ε p ( t + ε ) , p …( t + ε ) − p ( t ) = ε sin ( q ( t ) ) of the system q ′ = p , p ′ = sin ( q ) . The latter system of ordinary differential equations has two saddle points at A = ( 0 , 0 ) , B = ( 2 π , 0 ) and near both, there exist stable and unstable manifolds. It also admits a heteroclinic orbit connecting the stationary points B and A parametrized by q 0 ( t ) = 4 arctan ( e − t ) and which contains the stable manifold of this system at A as well as its unstable manifold at B . We prove that the stable manifold of the point A and the unstable manifold of the point B do not coincide for the discretization. More precisely, we show that the vertical distance between these two manifolds is exponentially small but not zero and in particular we give an asymptotic estimate of this distance. For this purpose we use a method adapted from the article of Schäfke and Volkmer [J. Reine Angew. Math. 425 (1992), 9–60] using formal series and accurate estimates of the coefficients. Our result is a variant of the results of Gelfreich [Comm. Math. Phys. 201 (1999), 155–216], Lazutkin et al. [Physica D 40 (1989), 235–248] for the pendulum problem and our method of proof, however, is quite different. This method will be useful for other problems of this type. Show more

Keywords: difference equation, manifolds, linear operator, formal solution, Gevrey asymptotic, quasi-solution

DOI: 10.3233/ASY-151326

Citation: Asymptotic Analysis, vol. 95, no. 3-4, pp. 279-324, 2015

Authors: Ichim, Andrei

Article Type: Research Article

Abstract: The aim of this paper is to study the asymptotic behaviour of a wide class of incompressible quasi-Newtonian fluids flowing through a thin 3D pipe, with prescribed pressure variance at the ends. The small parameter is given by the ratio between the diameter of the cross section and the length. We prove that, if the domain satisfies a particular geometric condition at the ends, a complete asymptotic expansion can be obtained.

Keywords: quasi-Newtonian fluids, thin pipes, asymptotic expansion

DOI: 10.3233/ASY-151327

Citation: Asymptotic Analysis, vol. 95, no. 3-4, pp. 325-344, 2015

Authors: Lototsky, S.V. | Moers, M.

Article Type: Research Article

Abstract: Using asymptotic analysis of the Laplace transform, we establish almost sure divergence of certain integrals and derive logarithmic asymptotic of small ball probabilities for quadratic forms of Gaussian diffusion processes. The large time behavior of the quadratic forms exhibits little dependence on the drift and diffusion matrices or the initial conditions, and, if the noise driving the equation is not degenerate, then similar universality also holds for small ball probabilities. On the other hand, degenerate noise leads to a variety of different asymptotics of small ball probabilities, including unexpected influence of the initial conditions.

Keywords: algebraic Riccati equation, logarithmic asymptotic, Ornstein–Uhlenbeck process, small ball constant, small ball rate

DOI: 10.3233/ASY-151328

Citation: Asymptotic Analysis, vol. 95, no. 3-4, pp. 345-374, 2015

Article Type: Other

Citation: Asymptotic Analysis, vol. 95, no. 3-4, pp. 375-376, 2015