2016, Caffarelli, Luis, Dipierro, Serena, Outrata, Jir̆í

We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms. More precisely, for populations that propagate according to a Levy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case. As ambient space, we can consider: bounded domains, periodic environments, transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one. In each of these cases, we analyze the existence/nonexistence of solutions in terms of the spectral properties of the domain. In particular, we give a detailed description of the fact that nonlocal populations may better adapt to sparse resources and small environments.

2013, Bracciali, Cleonice F., Moreno-Balcázar, Juan José

We obtain a Mehler–Heine type formula for a class of generalized hypergeometric polynomials. This type of formula describes the asymptotics of polynomials scale conveniently. As a consequence of this formula, we obtain the asymptotic behavior of the corresponding zeros. We illustrate these results with numerical experiments and some figures.

2009, Bierwirth, E., Wendisch, M., Ehrlich, A., Heese, B., Tesche, M., Althausen, D., Schladitz, A., Müller, D., Otto, S., Trautmann, T., Dinter, T., Von Hoyningen-Huene, W., Kahn, R.

In May-June 2006, airborne and ground-based solar (0.3-2.2 μm) and thermal infrared (4-42 μm) radiation measurements have been performed in Morocco within the Saharan Mineral Dust Experiment (SAMUM). Upwelling and downwelling solar irradiances have been measured using the Spectral Modular Airborne Radiation Measurement System (SMART)-Albedometer. With these data, the areal spectral surface albedo for typical surface types in southeastern Morocco was derived from airborne measurements for the first time. The results are compared to the surface albedo retrieved from collocated satellite measurements, and partly considerable deviations are observed. Using measured surface and atmospheric properties, the spectral and broad-band dust radiative forcing at top-of-atmosphere (TOA) and at the surface has been estimated. The impact of the surface albedo on the solar radiative forcing of Saharan dust is quantified. In the SAMUM case of 19 May 2006, TOA solar radiative forcing varies by 12 W m-2 per 0.1 surface-albedo change. For the thermal infrared component, values of up to +22 W m-2 were derived. The net (solar plus thermal infrared) TOA radiative forcing varies between -19 and +24 W m-2 for a broad-band solar surface albedo of 0.0 and 0.32, respectively. Over the bright surface of southeastern Morocco, the Saharan dust always has a net warming effect. © 2008 The Author Journal compilation © 2008 Blackwell Munksgaard.

2008, Baake, Michael, Birkner, Matthias, Moody, Robert V.

Stochastic point sets are considered that display a diffraction spectrum of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality. Several pairs of autocorrelation and diffraction measures are discussed that show a duality structure that may be viewed as analogues of the Poisson summation formula for lattice Dirac combs.

2018, Schlein, Benjamin, Sohinger, Vedran

In this mini-workshop we brought together leading experts working on the application of Gibbs measures to the study of nonlinear PDEs. This framework is a powerful tool in the probabilistic study of solutions to nonlinear dispersive PDEs, in many ways alternative or complementary to deterministic methods. Among the special topics discussed were the construction of the measures, applications to dynamics, as well as the microscopic derivation of Gibbs measures from many-body quantum mechanics.

2020, Colli, Pierluigi, Gilardi, Gianni, Sprekels, Jürgen

In a recent paper the same authors have proved existence, uniqueness and regularity results for a class of viscous and nonviscous Cahn--Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers in the spectral sense of general linear operators, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space of square-integrable functions on a bounded and smooth three-dimensional domain, and have compact resolvents. Here, for the case of the viscous system, we analyze the asymptotic behavior of the solution as the fractional power coefficient of the second operator tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of the second operator appears.

2011, Götze, Karoline

We study a coupled system of equations describing the movement of a rigid body which is immersed in a viscoelastic fluid. It is shown that under natural assumptions on the data and for general goemetries of the rigid body, excluding contact scenarios, a unique local-in-time strong solution exists.

2015, Dipierro, Serena, Savin, Ovidiu, Valdinoci, Enrico

In this paper we show that a nonlocal minimal surface which is a graph outside a cylinder is in fact a graph in the whole of the space. As a consequence, in dimension 3, we show that the graph is smooth. The proofs rely on convolution techniques and appropriate integral estimates which show the pointwise validity of an Euler-Lagrange equation related to the nonlocal mean curvature.

2010, Bartels, Sören, Müller, Rüdiger

A fully computable upper bound for the finite element approximation error of Allen-Cahn and Cahn-Hilliard equations with logarithmic potentials is derived. Numerical experiments show that for the sharp interface limit this bound is robust past topological changes. Modifications of the abstract results to derive quasi-optimal error estimates in different norms for lowest order finite element methods are discussed and lead to weaker conditions on the residuals under which the conditional error estimates hold.

2011, Schulz, Rüdiger, Buness, Hermann, Beilecke, Thies, von Hartmann, Hartwig, Musmann, Patrick, Bauer, Stefan, Donath, Andreas, Rüter, Horst

[no abstract available]