Nonlinear Absorptions

AbstractLet (Ω,Bμ) be a σ-finite measure space, A be an m-completely accretive operator in L1(Ω) (A is m-accretive with resolvant Jλ= (I+λA)−1 satisfying u≤û+k ⇒Jλ≤Jλû + k for k≥0), and j: Ω × R → [0,∞] be measurable in x ∈ Ω. convex, and l.s.c. in r ∈ R with j(x, 0) = 0. We consider the operator A + B where B is the operator in L1(Ω) defined by w ∈ Bu iff j(r) ≥j(u(x)) + (r − u(x)) w(x) for any r ∈ R, a.e. x ∈ Ω.We define natural m-completely accretive extensions of A + B in L1(Ω) and study their dependence with respect to j and different cases where A + B is itself m-completely accretive; we consider also the evolution problem du/dt + Au + Bu ∋ ƒ, u(0)= u0

Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case

We consider a possibly degenerate porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. The main idea consists in approximating the equation by equations with monotone non-degenerate coefficients and deriving some new analytical properties of the solution

A new infrared band in the Interstellar and Circumstellar Clouds: C_4 or C_4H?

We report on the detection with the Infrared Space Observatory (ISO) of a molecular band at 57.5 microns (174 cm^{-1}) in carbon-rich evolved stars and in Sgr B2. Taking into account the chemistry of these objects the most likelihood carrier is a carbon chain. We tentatively assign the band to the nu_5 bending mode of C_4 for which a wavenumber of 170-172.4 cm^{-1} has been derived in matrix experiments (Withey et al. 1991). An alternate carrier might be C_4H, although the frequency of its lowest energy vibrational bending mode, nu_7, is poorly known (130-226 cm^{-1}). If the carrier is C_4, the derived maximum abundance is nearly similar to that found for C_3 in the interstellar and circumstellar media by Cernicharo, Goicoechea & Caux (2000). Hence, tetra-atomic carbon could be one of the most abundant carbon chain molecules in these media.Comment: 11 pages, 1 figure, accepted in ApJ Letter

Mathematical analysis of a model of river channel formation.

The study of overland flow of water over an erodible sediment leads to a coupled model describing the evolution of the topographic elevation and the depth of the overland water film. The spatially uniform solution of this model is unstable, and this instability corresponds to the formation of rills, which in reality then grow and coalesce to form large-scale river channels. In this paper we consider the deduction and mathematical analysis of a deterministic model describing river channel formation and the evolution of its depth. The model involves a degenerate nonlinear parabolic equation (satisfied on the interior of the support of the solution) with a super-linear source term and a prescribed constant mass. We propose here a global formulation of the problem (formulated in the whole space, beyond the support of the solution) which allows us to show the existence of a solution and leads to a suitable numerical scheme for its approximation. A particular novelty of the model is that the evolving channel self-determines its own width, without the need to pose any extra conditions at the channel margin

A review on sparse solutions in optimal control of partial differential equations

In this paper a review of the results on sparse controls for partial differential equations is presented. There are two different approaches to the sparsity study of control problems. One approach consists of taking functions to control the system, putting in the cost functional a convenient term that promotes the sparsity of the optimal control. A second approach deals with controls that are Borel measures and the norm of the measure is involved in the cost functional. The use of measures as controls allows to obtain optimal controls supported on a zero Lebesgue measure set, which is very interesting for practical implementation. If the state equation is linear, then we can carry out a complete analysis of the control problem with measures. However, if the equation is nonlinear the use of measures to control the system is still an open problem, in general, and the use of functions to control the system seems to be more appropriate.This work was partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2014-57531-P

The Need for Laboratory Measurements and Ab Initio Studies to Aid Understanding of Exoplanetary Atmospheres

Laboratory astrophysics and astrochemistr