Asymptotic Analysis - Volume 108, issue 1-2

Authors: Biswas, Animikh | Hudson, Joshua | Larios, Adam | Pei, Yuan

Article Type: Research Article

Abstract: We propose several continuous data assimilation (downscaling) algorithms based on feedback control for the 2D magnetohydrodynamic (MHD) equations. We show that for sufficiently large choices of the control parameter and resolution and assuming that the observed data is error-free, the solution of the controlled system converges exponentially (in L 2 and H 1 norms) to the reference solution independently of the initial data chosen for the controlled system. Furthermore, we show that a similar result holds when controls are placed only on the horizontal (or vertical) variables, or …on a single Elsässer variable, under more restrictive conditions on the control parameter and resolution. Finally, using the data assimilation system, we show the existence of abridged determining modes, nodes and volume elements. Show more

Keywords: Data assimilation, feedback control, magnetohydrodynamics

DOI: 10.3233/ASY-171454

Citation: Asymptotic Analysis, vol. 108, no. 1-2, pp. 1-43, 2018

Authors: Lacave, C. | Lopes Filho, M.C. | Nussenzveig Lopes, H.J.

Article Type: Research Article

Abstract: In this article, we study the limit of a family of solutions to the incompressible 2D Euler equations in the exterior of a family of n k disjoint disks with centers { z i k } and radii ε k . We assume that the initial velocities u 0 k are smooth, divergence-free, tangent to the boundary and that they vanish at infinity. We allow, but we do not require, n k …→ ∞ , and we assume ε k → 0 as k → ∞ . Let γ i k be the circulation of u 0 k around the circle { | x − z i k | = ε k } . We prove that the limit as k → ∞ retains information on the circulations as a time-independent coefficient. More precisely, we assume that: (1) ω 0 k = curl u 0 k has a uniform compact support and converges weakly in L p 0 , for some p 0 > 2 , to ω 0 ∈ L c p 0 ( R 2 ) , (2) ∑ i = 1 n k γ i k δ z i k ⇀ μ weak-∗ in BM ( R 2 ) for some bounded Radon measure μ , and (3) the radii ε k are sufficiently small. Then the corresponding solutions u k converge strongly to a weak solution u of a modified Euler system in the full plane. This modified Euler system is given, in vorticity formulation, by an active scalar transport equation for the quantity ω = curl u , with initial data ω 0 , where the transporting velocity field is generated from ω , so that its curl is ω + μ . As a byproduct, we obtain a new existence result for this modified Euler system. Show more

Keywords: Euler equations, homogenization in perforated domain, shrinking obstacles and porous medium

DOI: 10.3233/ASY-171456

Citation: Asymptotic Analysis, vol. 108, no. 1-2, pp. 45-83, 2018

Authors: Montalto, Riccardo

Article Type: Research Article

Abstract: In this paper we consider time dependent Schrödinger equations on the one-dimensional torus T : = R / ( 2 π Z ) of the form ∂ t u = i V ( t ) [ u ] where V ( t ) is a time dependent, self-adjoint pseudo-differential operator of the form V ( t ) = V ( t , x ) | D | M + W ( t ) , M > 1 , | D | …: = − ∂ x x , V is a smooth function uniformly bounded from below and W is a time-dependent pseudo-differential operator of order strictly smaller than M . We prove that the solutions of the Schrödinger equation ∂ t u = i V ( t ) [ u ] grow at most as t ε , t → + ∞ for any ε > 0 . The proof is based on a reduction to constant coefficients up to smoothing remainders of the vector field i V ( t ) which uses Egorov type theorems and pseudo-differential calculus. Show more

Keywords: Growth of Sobolev norms, linear Schrödinger equations, pseudo-differential operators

DOI: 10.3233/ASY-181470

Citation: Asymptotic Analysis, vol. 108, no. 1-2, pp. 85-114, 2018