Descripción: We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations ut = Δu, vt = Δv in Ω × (0, T); fully coupled by the boundary conditions ∂u/∂η = up11vp12, ∂v/∂η = up21vp22 on ∂Ω × (0, T), where Ω is a bounded smooth domain in ℝd. We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation (U, V). We prove that if U blows up in finite time then V can fail to blow up if and only if p11 > 1 and p21 < 2(p11 - 1), which is the same condition as the one for non-simultaneous blow-up in the continuous problem. Moreover, we find that if the continuous problem has non-simultaneous blow-up then the same is true for the discrete one. We also prove some results about the convergence of the scheme and the convergence of the blow-up times.

Descripción: In this paper we study homogenisation problems for Sobolev trace embedding H1(Ω) Lq(∂Ω) in a bounded smooth domain. When q = 2 this leads to a Steklov-like eigenvalue problem. We deal with the best constant of the Sobolev trace embedding in rapidly oscillating periodic media, and we consider H1 and Lq spaces with weights that are periodic in space. We find that extremals for these embeddings converge to a solution of a homogenised limit problem, and the best trace constant converges to a homogenised best trace constant. Our results are in fact more general; we can also consider general operators of the form aε(x, ∇u) with non-linear Neumann boundary conditions. In particular, we can deal with the embedding W1,p(Ω) Lq(∂Ω). © 2009 Glasgow Mathematical Journal Trust.

Descripción: Context: Observations of magnetic clouds (MCs) are consistent with the presence of flux ropes detected in the solar wind (SW) a few days after their expulsion from the Sun as coronal mass ejections (CMEs). Aims: Both the in situ observations of plasma velocity profiles and the increase of their size with solar distance show that MCs are typically expanding structures. The aim of this work is to derive the expansion properties of MCs in the inner heliosphere from 0.3 to 1 AU. Methods: We analyze MCs observed by the two Helios spacecraft using in situ magnetic field and velocity measurements. We split the sample in two subsets: those MCs with a velocity profile that is significantly perturbed from the expected linear profile and those that are not. From the slope of the in situ measured bulk velocity along the Sun-Earth direction, we compute an expansion speed with respect to the cloud center for each of the analyzed MCs. Results: We analyze how the expansion speed depends on the MC size, the translation velocity, and the heliocentric distance, finding that allMCs in the subset of non-perturbed MCs expand with almost the same non-dimensional expansion rate (ζ).We find departures from this general rule for ζ only for perturbed MCs, and we interpret the departures as the consequence of a local and strong SW perturbation by SW fast streams, affecting the MC even inside its interior, in addition to the direct interaction region between the SW and the MC. We also compute the dependence of the mean total SW pressure on the solar distance and we confirm that the decrease of the total SW pressure with distance is the main origin of the observed MC expansion rate. We found that ζ was 0.91 ± 0.23 for non-perturbed MCs while ζ was 0.48 ± 0.79 for perturbed MCs, the larger spread in the last ones being due to the influence of the solar wind local environment conditions on the expansion. © ESO 2010.