Asymptotic Analysis - Volume 107, issue 1-2

Authors: Wilkinson, Mark

Article Type: Research Article

Abstract: We address a question raised in the work of Gallagher, Saint-Raymond and Texier [From Newton to Boltzmann: Hard Spheres and Short-range Potentials , European Mathematical Society (EMS), Zürich, 2013] that concerns the convergence of soft potential dynamics to hard sphere dynamics. In the case of two particles, we establish that hard sphere dynamics is the limit of soft sphere dynamics in the weak-star topology of BV . We view our result as establishing a topological method by which to construct weak solutions to the ODE of hard sphere motion.

Keywords: Hard spheres, kinetic theory, Boltzmann equation, hard particle dynamics, weak solutions

DOI: 10.3233/ASY-171448

Citation: Asymptotic Analysis, vol. 107, no. 1-2, pp. 1-32, 2018

Authors: Shibata, Yoshihiro

Article Type: Research Article

Abstract: This paper deals with the L p -L q decay estimate of the C 0 analytic semigroup { T ( t ) } t ⩾ 0 associated with the perturbed Stokes equations with free boundary conditions in an exterior domain. The problem arises in the study of free boundary problem for the Navier–Stokes equations in an exterior domain. We proved that ‖ ∇ j T ( t ) f ‖ L …p ⩽ C p , q t − j 2 − N 2 ( 1 q − 1 p ) ‖ f ‖ L q ( j = 0 , 1 ) provided that 1 < q ⩽ p ⩽ ∞ and q ≠ ∞ . Compared with the non-slip boundary condition case, the gradient estimate is better, which is important for the application to proving global well-posedness of free boundary problem for the Navier–Stokes equations. In our proof, it is crucial to prove the uniform estimate of the resolvent operator, the resolvent parameter ranging near zero. Show more

Keywords: Exterior domains, Stokes equations, free boundary problem, without surface tension, Lp-Lq decay estimate

DOI: 10.3233/ASY-171449

Citation: Asymptotic Analysis, vol. 107, no. 1-2, pp. 33-72, 2018

Authors: Glizer, Valery Y.

Article Type: Research Article

Abstract: A singularly perturbed linear time-dependent controlled system with multiple point-wise and distributed state delays is considered. The delays in the fast state variable are small of order of the small positive multiplier for a part of the derivatives in the system, which is a parameter of the singular perturbation. The delays in the slow state variable are non-small. Two types of the original singularly perturbed system, standard and nonstandard, are analyzed. For each type, two much simpler parameter-free subsystems (the slow and fast ones) are associated with the original system. It is established in the paper that the approximate state-space …controllability of the slow and fast subsystems yields the approximate state-space controllability of the original system robustly with respect to the parameter of singular perturbation for all its sufficiently small values. Illustrative examples are presented. Show more

Keywords: Singularly perturbed system, state delay, approximate state-space controllability

DOI: 10.3233/ASY-171451

Citation: Asymptotic Analysis, vol. 107, no. 1-2, pp. 73-114, 2018