Convergence of the Mass-Transport Steepest Descent Scheme for the Subcritical Patlak–Keller–Segel Model

Abstract

Abstract. Variational steepest descent approximation schemes for the modified Patlak-Keller-Segel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean Wasserstein distance, associated to this equation for sub-critical masses. As a consequence, we recover the recent result about the global in time existence of weak-solutions to the modified Patlak-Keller-Segel equation for the logarithmic interaction kernel in any dimension in the sub-critical case. Moreover, we show how this method performs numerically in one dimension. In this particular case, this numerical scheme corresponds to a standard implicit Euler method for the pseudo-inverse of the cumulative distribution function. We demonstrate its capabilities to reproduce easily without the need of mesh-refinement the blow-up of solutions for super-critical masses.

Уравнение Агрегации С Анизотропной Диффузией

“…Thus, this equation bears the structure of being a gradient flow of the free energy functional in the sense of probability measures, see [2,8,10,26] and the references therein.…”

“…It was shown in [48,19] that degenerate diffusion with m > 1 is able to regularize the 2D classical Keller-Segel problem, where solutions exist globally in time regardless of its mass, and each solution remain uniformly bounded in time. For the Newtonian attraction interaction in dimension d ≥ 3, the authors in [8] show that the value of the degeneracy of the diffusion that allows the mass to be the critical quantity for dichotomy between global existence and finite time blow-up is given by m = 2 − 2/d. In fact, based on scaling arguments it is easy to argue that for m > 2 − 2/d, the diffusion term dominates when density becomes large, leading to global existence of solutions for all masses.…”

Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction

“…6) which satisfies the dual equation Figure 3 illustrates the relationship between ρ ε and u ε . As it turns out, u ε is much better behaved than ρ ε in the limit ε → 0: if the initial datum for u ε is bounded above and below, then the same holds for u ε by the comparison principle, since constants are solutions of (1.7).…”

“…[3,6,8,9,15,19,26,37], just to name a few), and therefore any method that uses only the properties of this structure has the potential of application to a wide range of problems. Consequently, our approach here is to limit our use of information to those properties that follow directly from the gradient-flow structure.…”

Scalar conservation laws seen as gradient flows: known results and new perspectives

“…The gradient flow approach of [45] was also used in chemotaxis modelling, see [6], and in nonlocal interaction equations with non smooth kernels, see [8,20]. Already in their review paper [28], Carrillo and Toscani started investigating the role of the Lagrangian formulation for a continuity equation of the form (1), with the aim of proving simple 'quasi-contraction inequalities' of the form d 2 (ρ 1 (t), ρ 2 (t)) ≤ d 2 (ρ 1 (0), ρ 2 (0))e Ct , C ∈ R.…”