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

Abstract

We consider a one dimensional transport model with nonlocal velocity given by the Hilbert transform and develop a global well-posedness theory of probability measure solutions. Both the viscous and non-viscous cases are analyzed. Both in original and in self-similar variables, we express the corresponding equations as gradient flows with respect to a free energy functional including a singular logarithmic interaction potential. Existence, uniqueness, self-similar asymptotic behavior and inviscid limit of solutions are obtained in the space P2(R) of probability measures with finite second moments, without any smallness condition. Our results are based on the abstract gradient flow theory developed in [2]. An important byproduct of our results is that there is a unique, up to invariance and translations, global in time self-similar solution with initial data in P2(R), which was already obtained in [17,6] by different methods. Moreover, this self-similar solution attracts all the dynamics in self-similar variables. The crucial monotonicity property of the transport between measures in one dimension allows to show that the singular logarithmic potential energy is displacement convex. We also extend the results to gradient flow equations with negative power-law locally integrable interaction potentials.

Gradient flows of time-dependent functionals in metric spaces and applications to PDEs

“…[2]). We also quote the paper [4] where a 1D non-local fluid mechanics model with velocity coupled via Hilbert transform was analyzed by using gradient flow theory in P 2 . In [13], the authors dealt with nonlinear diffusion equations in the form…”

Solutions of a non‐local aggregation equation: Universal bounds, concavity changes, and efficient numerical solutions

“…Remark 1 (Existence of solutions). Note that the existence theory for the one-dimensional aggregation Equation (2) is well established for initial data in the space of probability measures with bounded second moment and subject to a large class of interaction potentials (including 5); see, eg, Burger and Di Francesco, 29 Carrillo et al, 30,31 and the references therein.…”

“…For the results of the present paper, we only require solutions (x, t) such that (t) ∈ C[0, ∞) ∩ C 1 (0, ∞). Without relying on the previous existence theory results, [29][30][31] the required continuity and smoothness of (t) follows from the results of Sections 3 and 4 for non-negative probability densities (x, 0) ∈ L 1 + (1 + x 2 , dx) (that is, non-negative L 1 functions with finite second-order moment ∫ ∞ −∞ x 2 (x)dx) via standard fixed-point-type arguments. More precisely, the method of characteristics (12 and 13) is globally well posed thanks to (21) for continuous (t), which follows in return from formula (27) and the global bound (34).…”

“…We begin by differentiating the expression (27) for (t) that we obtained in Lemma 2 and using Equation (31), that is, 3 :…”

Convergence of a linearly transformed particle method for aggregation equations

“…Equation (1.1) has attracted lots of attention in the recent years for three reasons: its gradient flow structure [2, 32, 33, 61, 73], the blow-up dynamics for fully attractive potentials [12, 14, 26, 31], and the rich variety of steady states and their bifurcations both at the discrete (1.2) and the continuous (1.1) level of descriptions [3–5, 11, 14, 22, 25, 27, 28, 49, 50, 67, 74, 75]. Furthermore, these systems are ubiquitous in mathematical modelling appearing in granular media models [10, 61], swarming models for animal collective behavior [30, 46, 59], equilibrium states for self-assembly and molecules [47, 54, 70, 76], and mean-field games in socioeconomics [17, 43] among others.…”