In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure. These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential. The proposed scheme is able to cope with non-smooth stationary states, different time scales including metastability, as well as concentrations and self-similar behavior induced by singular nonlocal kernels. We use the scheme to explore properties of these equations beyond their present theoretical knowledge.

A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure

This survey summarizes and illustrates the main qualitative properties of hydrodynamics models for collective behavior. These models include a velocity consensus term together with attractive–repulsive potentials leading to non-trivial flock profiles. The connection between the underlying particle systems and the swarming hydrodynamic equations is performed through kinetic theory modeling arguments. We focus on Lagrangian schemes for the hydrodynamic systems showing the different qualitative behaviors of the systems and its capability of keeping properties of the original particle models. We illustrate the known results concerning large-time profiles and blowup in finite time of the hydrodynamic systems to validate the numerical scheme. We finally explore the unknown situations making use of the numerical scheme showcasing a number of conjectures based on the numerical results.

A Review on Attractive–Repulsive Hydrodynamics for Consensus in Collective Behavior

The existence of compactly supported global minimisers for continuum models of particles interacting through a potential is shown under almost optimal hypotheses The main assumption on the potential is that it is catastrophic, or not H-stable, which is the complementary assumption to that in classical results on thermodynamic limits in statistical mechanics The proof is based on a uniform control on the local mass around each point of the support of a global minimiser, together with an estimate on the size of the "gaps" it may have The class of potentials for which we prove the existence of global minimisers includes power-law potentials and, for some range of parameters, Morse potentials, widely used in applications We also show that the support of local minimisers is compact under suitable assumptions

https://link.springer.com/content/pdf/10.1007%2Fs00205-015-0852-3.pdf

Existence of Compactly Supported Global Minimisers for the Interaction Energy

A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity

This work considers the Keller–Segel system of parabolic–parabolic type in for n ≥ 2. We prove existence results in a new framework and with initial data in . This initial data class is larger than the previous ones, e.g., Kozono–Sugiyama (2008 Indiana Univ. Math. J. 57 1467–500) and Biler (1998 Adv. Math. Sci. Appl. 8 715–43), and covers physical cases of initial aggregation at points (Diracs) and on filaments. Self-similar solutions are obtained for initial data with the correct homogeneity and a certain value of parameter γ. We also show an asymptotic behaviour result, which provides a basin of attraction around each self-similar solution.

Existence and asymptotic behaviour for the parabolic–parabolic Keller–Segel system with singular data

In this paper, we show a local-in-time existence result for the 3D micropolar fluid system in the framework of Besov–Morrey spaces. The initial data class is larger than the previous ones and contains strongly singular functions and measures.

Existence of solutions for the 3D-micropolar fluid system with initial data in Besov-Morrey spaces

Existence and symmetries of solutions in Besov–Morrey spaces for a semilinear heat-wave type equation

This paper considers a semilinear integro-differential equation of Volterra type which interpolates semilinear heat and wave equations. Global existence of solutions is showed in spaces of Besov type based in Morrey spaces, namely Besov-Morrey spaces. Our initial data is larger than the previous works and our results provide a maximal existence class for semilinear interpolated heat-wave equation. Some symmetries, self-similarity and asymptotic behavior of solutions are also investigated in the framework of Besov-Morrey spaces.

/pdf/existence-and-symmetries-of-solutions-in-besov-morrey-spaces-52xc3o68n3.pdf

Juliana Conceição Precioso

Papers

A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity

Existence and asymptotic behaviour for the parabolic–parabolic Keller–Segel system with singular data

Existence of solutions for the 3D-micropolar fluid system with initial data in Besov-Morrey spaces

Existence and symmetries of solutions in Besov–Morrey spaces for a semilinear heat-wave type equation

Existence and symmetries of solutions in Besov-Morrey spaces for a semilinear heat-wave type equation