Optimal waiting time bounds for some flux-saturated diffusion equations

We consider the Cauchy problem for two prototypes of flux-saturated diffusion equations. In arbitrary space dimension, we give an optimal condition on the growth of the initial datum which discriminates between occurrence or nonoccurrence of a waiting time phenomenon. We also prove optimal upper bounds on the waiting time. Our argument is based on the introduction of suitable families of subsolutions and on a comparison result for a general class of flux-saturated diffusion equations.Comment: Comm. Partial Differential Equations, to appea

Nonlinear diffusion in transparent media: the resolvent equation

We consider the partial differential equation $$u-f={\rm div}\left(u^m\frac{\nabla u}{|\nabla u|}\right)$$ with $f$ nonnegative and bounded and $m\in\mathbb{R}$. We prove existence and uniqueness of solutions for both the Dirichlet problem (with bounded and nonnegative {boundary datum}) and the homogeneous Neumann problem. Solutions, which a priori belong to a space of truncated bounded variation functions, are shown to have zero jump part with respect to the ${\mathcal H}^{N-1}$ Haussdorff measure. Results and proofs extend to more general nonlinearities

Anisotropic tempered diffusion equations

We introduce a functional framework which is specially suited to formulate several classes of anisotropic evolution equations of tempered diffusion type. Under an amenable set of hypothesis involving a very natural potential function, these models can be shown to belong to the entropy solution framework devised by [F. Andreu, V. Caselles, J. M. Mazo ́n, Nonlinear Anal. 61 (2005), J. Eur. Math. Soc. 7 (2005)], therefore ensuring well-posedness. We connect the properties of this potential with those of the associated cost function, thus providing a link with optimal transport theory and a supply of new examples of relativistic cost functions. Moreover, we characterize the anisotropic spreading properties of these models and we determine the Rankine–Hugoniot conditions that rule the temporal evolution of jump hypersurfaces under the given anisotropic flows.“Plan Propio de Investigación, programa 9” (funded by Universidad de Granada and european FEDER (ERDF) funds)Project RTI2018-098850-B-I00 (funded by MICINN and european FEDER funds)Project A-FQM-311-UGR18 (funded by Junta de Andalucía and european FEDER funds)Project P18-RT-2422 (funded by Junta de Andalucía and european FEDER funds

Anisotropic tempered diffusion equations

We introduce a functional framework which is specially suited to formulate several classes of anisotropic evolution equations of tempered diffusion type. Under an amenable set of hypothesis involving a very natural potential function, these models can be shown to belong to the entropy solution framework devised by 4, 5, therefore ensuring well-posedness. We connect the properties of this potential with those of the associated cost function, thus providing a link with optimal transport theory and a supply of new examples of relativistic cost functions. Moreover, we characterize the anisotropic spreading properties of these models and we determine the Rankine-Hugoniot conditions that rule the temporal evolution of jump hypersurfaces under the given anisotropic flows.Comment: 43 page

Variational Methods for Evolution (hybrid meeting)

Variational principles for evolutionary systems take advantage of the rich toolbox provided by the theory of the calculus of variations. Such principles are available for Hamiltonian systems in classical mechanics, gradient flows for dissipative systems, but also time-incremental minimization techniques for more general evolutionary problems. The new challenges arise via the interplay of two or more functionals (e.g. a free energy and a dissipation potential), new structures (systems with nonlocal transport, gradient flows on graphs, kinetic equations, systems of equations) thus encompassing a large variety of applications in the modeling of materials and fluids, in biology, in multi-agent systems, and in data science. This workshop brought together a broad spectrum of researchers from calculus of variations, partial differential equations, metric geometry, and stochastics, as well as applied and computational scientists to discuss and exchange ideas. It focused on variational tools such as minimizing movement schemes, optimal transport, gradient flows, and large-deviation principles for time-continuous Markov processes, $\Gamma$-convergence and homogenization

Singular patterns in Keller–Segel-type models

The aim of this paper is to elucidate the existence of patterns for Keller-Segel-type models that are solutions of the traveling pulse form. The idea is to search for transport mechanisms that describe this type of waves with compact support, which we find in the so-called nonlinear diffusion through saturated flux mechanisms for the movement cell. At the same time, we analyze various transport operators for the chemoattractant. The techniques used combine the analysis of the phase diagram in dynamic systems together with its counterpart in the system of partial differential equations through the concept of entropic solution and the admissible jump conditions of the Rankine-Hugoniot type. We found traveling pulse waves of two types that correspond to those found experimentally.MICINN- Feder RTI2018-098850-B-I00Junta de Andalucía B-FQM-580-UGR20 & PY18-RT-242