Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAsymptotic simplification of Aggregation-Diffusion equations towards the heat kernel
66
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We give sharp conditions for the large time asymptotic simplification of aggregation-diffusion equations with linear diffusion. As soon as the interaction potential is bounded and its first and second derivatives decay fast enough at infinity, then the linear diffusion overcomes its effect, either attractive or repulsive, for large times independently of the initial data, and solutions behave like the fundamental solution of the heat equation with some rate. The potential $W(x) sim log |x|$ for $|x| gg 1$ appears as the natural limiting case when the intermediate asymptotics change. In order to obtain such a result, we produce uniform-in-time estimates in a suitable rescaled change of variables for the entropy, the second moment, Sobolev norms and the $C^alpha$ regularity with a novel approach for this family of equations using modulus of continuity techniques.
rate research
Read More
We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not.
In this paper we obtain the precise description of the asymptotic behavior of the solution $u$ of $$ partial_t u+(-Delta)^{frac{theta}{2}}u=0quadmbox{in}quad{bf R}^Ntimes(0,infty), qquad u(x,0)=varphi(x)quadmbox{in}quad{bf R}^N, $$ where $0<theta<2$ and $varphiin L_K:=L^1({bf R}^N,,(1+|x|)^K,dx)$ with $Kge 0$. Furthermore, we develop the arguments in [15] and [18] and establish a method to obtain the asymptotic expansions of the solutions to a nonlinear fractional diffusion equation $$ partial_t u+(-Delta)^{frac{theta}{2}}u=|u|^{p-1}uquadmbox{in}quad{bf R}^Ntimes(0,infty), $$ where $0<theta<2$ and $p>1+theta/N$.
The large time behavior of zero mass solutions to the Cauchy problem for a convection-diffusion equation. We provide conditions on the size and shape of the initial datum such that the large time asymptotics of solutions is given either by the derivative of the Guass-Weierstrass kernel or by a self-similar solution or by a hyperbolic N-wave
We investigate a class of aggregation-diffusion equations with strongly singular kernels and weak (fractional) dissipation in the presence of an incompressible flow. Without the flow the equations are supercritical in the sense that the tendency to concentrate dominates the strength of diffusion and solutions emanating from sufficiently localised initial data may explode in finite time. The main purpose of this paper is to show that under suitable spectral conditions on the flow, which guarantee good mixing properties, for any regular initial datum the solution to the corresponding advection-aggregation-diffusion equation is global if the prescribed flow is sufficiently fast. This paper can be seen as a partial extension of Kiselev and Xu (Arch. Rat. Mech. Anal. 222(2), 2016), and our arguments show in particular that the suppression mechanism for the classical 2D parabolic-elliptic Keller-Segel model devised by Kiselev and Xu also applies to the fractional Keller-Segel model (where $triangle$ is replaced by $-Lambda^gamma$) requiring only that $gamma>1$. In addition, we remove the restriction to dimension $d<4$.
We consider two types of non linear fast diffusion equations in R^N:(1) External drift type equation with general external potential. It is a natural extension of the harmonic potential case, which has been studied in many papers. In this paper we can prove the large time asymptotic behavior to the stationary state by using entropy methods.(2) Mean-field type equation with the convolution term. The stationary solution is the minimizer of the free energy functional, which has direct relation with reverse Hardy-Littlewood-Sobolev inequalities. In this paper, we prove that for some special cases, it also exists large time asymptotic behavior to the stationary state.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
