Lorentz spaces in action on pressureless systems arising from models of collective behavior

Raphaël Danchin,Piotr Bogusław Mucha,Patrick Tolksdorf

doi:10.1007/s00028-021-00668-4

Abstract

We are concerned with global-in-time existence and uniqueness results for models of pressureless gases that come up in the description of phenomena in astrophysics or collective behavior. The initial data are rough: in particular, the density is only bounded. Our results are based on interpolation and parabolic maximal regularity, where Lorentz spaces play a key role. We establish a novel maximal regularity estimate for parabolic systems in L_{q,r}(0,T;L_p(Omega )) spaces.

Highlights

We are concerned with models coming from a special type of hydrodynamical systems, that do not include the effects of the internal pressure
The simplest example is the motion of dust, that is, of free particles evolving in the space like, e.g. in astrophysics [17], or in multi-fluid systems [4,6]
Leaving the world of inanimate matter, one can mention models that describe collective behavior, where particles or rather agents exhibit some intelligence, and for which having a force like internal pressure is not so natural

Summary

Introduction

We are concerned with models coming from a special type of hydrodynamical systems, that do not include the effects of the internal pressure. The first case is a viscous regularization of (1.3) that can be viewed as a simplification of the Euler alignment system It corresponds to the hydrodynamical version of the Cucker–Smale model, namely ρt + div (ρu) = 0, u(t, y) − u(t, x) ρut + ρu · ∇u =. In the theory of regular solutions [9,13,30,33], the effective viscous flux provides the decay properties for the density that are needed for establishing global existence for small data. In the case of pressureless systems, there is no such a possibility, so that we need to resort to more sophisticated techniques to control the density This may partially explain the reason why the mathematical theory of pressureless models is poorer than. Our main results concern global-in-time solvability for the two dimensional case for large velocity, and the three dimensional case in the small data regime

Functional framework

Results

The two dimensional case

The three dimensional case