An aggregation equation with a nonlocal flux

In this paper we study an aggregation equation with a general nonlocal flux. We study the local well-posedness and some conditions ensuring global existence. We are also interested in the differences arising when the nonlinearity in the flux changes. Thus, we perform some numerics corresponding to different convexities for the nonlinearity in the equation

The confined Muskat problem: differences with the deep water regime

We study the evolution of the interface given by two incompressible fluids with different densities in the porous strip \RR\times[-l,l]. This problem is known as the Muskat problem and is analogous to the two phase Hele-Shaw cell. The main goal of this paper is to compare the qualitative properties between the model when the fluids move without boundaries and the model when the fluids are confined. We find that, in a precise sense, the boundaries decrease the diffusion rate and the system becomes more singular.Comment: Revised version. 32 pages, 4 figure

Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality

9 pages, no figures.-- MSC2000 code: 33C45.MR#: MR2431543 (2009g:41009)Zbl#: Zbl 1155.33006We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same $n$th root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with respect to a Sobolev inner product with exponential weights of the form e−φ(x), giving a unified treatment for the so-called Freud (i.e., when φ has polynomial growth at infinity) and Erdös (when φ grows faster than any polynomial at infinity) cases. In addition, we provide a new proof for the bound of the distance of the zeros to the convex hull of the support for these Sobolev orthogonal polynomials.Research by first two authors (C.D.M. and R.O.) was partially supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología of Spain, under grants MTM2005-08571 and MTM2007-68114. Research by third author (H.P.) was partially supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología of Spain, under grant MTM2006-13000-C03-02, by Comunidad de Madrid-Universidad Carlos III de Madrid, under grants CCG06-UC3M/EST-0690 and CCG07-UC3M/ESP-3339, by Centro de Investigación Matemática de Canarias (CIMAC) and by Vicerrectorado de Investigación de La Universidad de La Laguna: Convocatoria 2005 de Ayudas a Profesores Invitados.Publicad

Zeros and logarithmic asymptotics of Sobolev orthogonal polynomials for exponential weights

8 pages, no figures.-- MSC2000 code: 42C05.Zbl#: Zbl 1177.42020We obtain the (contracted) weak zero asymptotics for orthogonal polynomials with respect to Sobolev inner products with exponential weights in the real semiaxis, of the form x\sp \gamma e\sp {\-varphi(x)} with γ>0, which include as particular cases the counterparts of the so-called Freud (i.e., when φ has a polynomial growth at infinity) and Erdös (when φ grows faster than any polynomial at infinity) weights. In addition, the boundness of the distance of the zeros of these Sobolev orthogonal polynomials to the convex hull of the support and, as a consequence, a result on logarithmic asymptotics are derived.The research of C.D.M. and R.O. was partially supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología of Spain, under grants MTM2005-08571 and MTM2007-68114. The research of H.P. was partially supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología of Spain, under grant MTM2006-13000-C03-02, by Comunidad de Madrid- Universidad Carlos III de Madrid, under grant CCG06-UC3M/EST-0690 and by Centro de Investigación Matemática de Canarias (CIMAC).Publicad

The best Sobolev trace constant in periodic media for critical and subcritical exponents

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 aɛ(x, ∇u) with non-linear Neumann boundary conditions. In particular, we can deal with the embedding W1,p(Ω) ↪ Lq(∂Ω).Fil: Fernandez Bonder, Julian. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Orive, Rafael. Universidad Autónoma de Madrid; EspañaFil: Rossi, Julio Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Asymptotic structure of the spectrum in a Dirichlet-strip with double periodic perforations

We address a spectral problem for the Dirichlet-Laplace operator in a waveguideis obtained from repsilon an unbounded two-dimensional strip ?? which is periodically perforated by a family of holes, which are also periodically distributed along a line, the so-called "perforation string". We assume that the two periods are different, namely, O(1)O(1) and O(?)O(?) respectively, where 0<??10<??1. We look at the band-gap structure of the spectrum . We derive asymptotic formulas for the endpoints of the spectral bands and show that has a large number of short bands of length ) which alternate with wide gaps of width O(1)O(1)

On the convergence of quadrature formulas connected with multipoint Padé-type approximants

29 pages, no figures.-- MSC2000 codes: 41A55, 41A21.MR#: MR1408352 (97e:41066)Zbl#: Zbl 0856.41027^aLet $I(F)= \int^1_{- 1} F(x)\omega(x) dx$, where $\omega$ is a complex valued integrable function. We consider quadrature formulas for $I(F)$ which are exact with respect to rational functions with prescribed poles contained in \overline{\bbfC}\backslash [- 1, 1]. Their rate of convergence is studied.The research by the first three authors (P.G.-V., M.J.P., R.O.) was partially supported by the HCM project ROLLS, under Contract CHRX-CT93-0416. Research by the fourth author (G.L.L.) was carried out while on a visit at Universidad de La Laguna. This visit was made possible by a travel grant from CDE-IMU.Publicad

Spectral gaps in a double-periodic perforated Neumann waveguide

We examine the band-gap structure of the spectrum of the Neumann problem for the Laplace operator in a strip with periodic dense transversal perforation by identical holes of a small diameter ε>0. The periodicity cell itself contains a string of holes at a distance O(ε) between them. Under assumptions on the symmetry of the holes, we derive and justify asymptotic formulas for the endpoints of the spectral bands in the low-frequency range of the spectrum as ε→0. We demonstrate that, for ε small enough, some spectral gaps are open. The position and size of the opened gaps depend on the strip width, the perforation period, and certain integral characteristics of the holes. The asymptotic behavior of the dispersion curves near the band edges is described by means of a 'fast Floquet variable' and involves boundary layers in the vicinity of the perforation string of holes. The dependence on the Floquet parameter of the model problem in the periodicity cell requires a serious modification of the standard justification scheme in homogenization of spectral problems. Some open questions and possible generalizations are listed.The work has been partially supported by MICINN through PGC2018-098178-B-I00, PID2020-114703GB-I00 and Severo Ochoa Programme for Centres of Excellence in R&D (CEX2019-000904-S)

Bloch Approximation in Homogenization and Applications

The classical problem of homogenization of elliptic operators with periodically oscillating coefficients is revisited in this paper. As is well known, the homogenization process in a classical framework is concerned with the study of asymptotic behavior of solutions $u^\varepsilon$ of boundary value problems associated with such operators when the period $\varepsilon>0$ of the coefficients is small. In a previous work by C. Conca and M. Vanninathan [SIAM J. Appl. Math., 57 (1997), pp. 1639--1659], a new proof of weak convergence as $\varepsilon\to 0$ towards the homogenized solution was furnished using Bloch wave decomposition. Following the same approach here, we go further and introduce what we call Bloch approximation, which will provide energy norm approximation for the solution $u^\varepsilon$. We develop several of its main features. As a simple application of this new object, we show that it contains both the first and second order correctors. Necessarily, the Bloch approximation will have to capture the oscillations of the solution in a sharper way. The present approach sheds new light and offers an alternative for viewing classical results

Incompressible flow in porous media with fractional diffusion

In this paper we study the heat transfer with a general fractional diffusion term of an incompressible fluid in a porous medium governed by Darcy's law. We show formation of singularities with infinite energy and for finite energy we obtain existence and uniqueness results of strong solutions for the sub-critical and critical cases. We prove global existence of weak solutions for different cases. Moreover, we obtain the decay of the solution in $L^p$, for any $p\geq2$, and the asymptotic behavior is shown. Finally, we prove the existence of an attractor in a weak sense and, for the sub-critical dissipative case with $\alpha\in (1,2]$, we obtain the existence of the global attractor for the solutions in the space $H^s$ for any $s > (N/2)+1-\alpha$