Ground States for Diffusion Dominated Free Energies with Logarithmic Interaction

Abstract

Abstract.Replacing linear diffusion by a degenerate diffusion of porous medium type is known to regularize the classical two-dimensional parabolic-elliptic Keller-Segel model [V. Calvez and J. A. Carrillo, J. Math. Pures Appl. (9), 86 (2006), pp. 155-175]. The implications of nonlinear diffusion are that solutions exist globally and are uniformly bounded in time. We analyze the stationary case showing the existence of a unique, up to translation, global minimizer of the associated free energy. Furthermore, we prove that this global minimizer is a radially decreasing compactly supported continuous density function which is smooth inside its support, and it is characterized as the unique compactly supported stationary state of the evolution model. This unique profile is the clear candidate to describe the long time asymptotics of the diffusion dominated classical Keller-Segel model for general initial data. 1. Introduction. Ground states of free energies play a crucial role in the long time asymptotics of nonlinear aggregation diffusion models. These nonlocal partial differential equations (PDEs) are ubiquitous in the mathematical modeling of phenomena which involve a large number of particles. For instance, nonlocal drift-diffusion equations show up naturally in semiconductor modeling, bacterial chemotaxis, granular media, and many other areas; see [18,9,13] and the references therein. These equations are just based on two competing mechanisms, namely the attraction, modeled by a nonlocal force, and the repulsion, modeled by a nonlinear diffusion.One of the archetypal models of nonlinear aggregation diffusion is the so-called classical parabolic-elliptic Keller-Segel model. This model was classically introduced as the simplest description for chemotactic bacteria movement in which the tendency of linear diffusion to spread repels the attraction due to the logarithmic kernel interaction in two dimensions. Although there is a large amount of literature in this field, many advances have been made in the last 10 years thanks to the combination of different ideas ranging from functional inequalities to entropy-entropy dissipation techniques passing through optimal transport. We refer the reader to [9, 8, 6, 11] and

Уравнение Агрегации С Анизотропной Диффузией

“…The proof is based on moving plane techniques, where the compact support of the stationary solution seems crucial, and it also relies on the fact that the Newtonian potential in 3D converges to zero at infinity. Similar results are obtained in [22] for 2D Newtonian potential with m > 1 using an adapted moving plane technique. Again, the uniqueness result is based on showing radial symmetry of compactly supported stationary solutions.…”

“…In section 3, we show that these global minimizers are in fact compactly supported radially decreasing continuous functions. These results fully generalize the results in [62,22]. Putting together Sections 2 and 3, the uniqueness and full characterization of the stationary states is reduced to uniqueness among the class of radial solutions.…”

“…Parts of these arguments are reminiscent from those done in [62,22] in the case of attractive Newtonian potentials.…”

“…Namely, under (K1)-(K4) and one of the extra assumptions (K5) or (K6) below, we will show that for any given mass, there indeed exists a stationary solution satisfying the above conditions. We will generalize the arguments of [22] to show that there exists a radially decreasing global minimizer ρ of the functional (2.5) given by…”

“…The first step will be to show radial symmetry of stationary solutions to (1.1) under quite general assumptions on W and the class of stationary solutions. Let us point out that standard rearrangement techniques fail in trying to show radial symmetry of general stationary states to (1.1) and they are only useful for showing radial symmetry of global minimizers, see [22]. Comparison arguments for radial solutions allow to prove uniqueness of radial stationary solutions in particular cases [50,46].…”