Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2405.12134

Two-dimensional signal-dependent parabolic-elliptic Keller-Segel system and its mean-field derivation

Lukas Bol·Li Chen·Yue Li
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, the well-posedness of two-dimensional signal-dependent Keller-Segel system and its mean-field derivation from a interacting particle system on the whole space are investigated. The signal dependence effect is reflected by the fact that the diffusion coefficient in the particle system depends non-linearly on the interactions between the individuals. Therefore, the mathematical challenge in studying the well-posedness of this system lies in the possible degeneracy and the aggregation effect when the concentration of signal becomes unbounded. The well-established method on bounded domains, to obtain the appropriate estimates for the signal concentration, is invalid for the whole space case. Motivated by the entropy minimization method and Onofri's inequality, which has been successfully applied for the parabolic-parabolic Keller-Segel system, we establish a complete entropy estimate benefited from the linear diffusion term, which plays an important role in obtaining the L^p estimates for the solution. Furthermore, the upper bound for the concentration of signal is obtained. Based on the estimates we obtained for the density of bacteria, the rigorous mean-field derivation is proved by introducing an intermediate particle system with a mollified interaction potential with logarithmic scaling. By using this mollification, we obtain the convergence of the particle trajectories in expectation, which implies the weak propagation of chaos. Additionally, under a regularity assumption of the initial data, we infer higher regularity for the solutions, which allows us to use the relative entropy method to derive the strong L^1 convergence for the propagation of chaos.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Lukas Bol, Li Chen, Yue Li. 2025-06-04. Two-dimensional signal-dependent parabolic-elliptic Keller-Segel system and its mean-field derivation. https://arxiv.org/abs/2405.12134

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Multilayered fluid-structure interactions: existence of weak solutions for time-periodic and initial-value problems

We establish the existence of weak solutions for a class of fully coupled multilayered fluid-structure interaction systems in a three-dimensional spatial setting. The model consists of an incompressible viscous fluid interacting with a thin elastic shell, which is in turn coupled to a three-dimensional elastic solid, yielding a nonstandard $3D/2D/3D$ coupling configuration. The system is driven by time-periodic boundary forcing through Bernoulli-type pressure conditions. For sufficiently small forcing in $L^2$, we prove the existence of at least one time-periodic weak solution. A central analytical difficulty stems from the strong nonlinear coupling across interfaces of different dimensionality and the absence of classical compactness mechanisms. This challenge is overcome through a carefully designed energy framework combined with and new $L^{2}$ compactness arguments adapted to the multilayered geometry. A key structural assumption is the viscoelasticity of the three-dimensional solid, which yields additional diffusion estimates and ensures energy stability. In the purely elastic case, we establish the global-in-time existence of weak solutions to the corresponding initial-value problem, provided that no degeneration (self-contact) of the fluid domain occurs. These results extend existing theories for two-dimensional and reduced-dimensional configurations to a genuinely three-dimensional multilayered setting, providing new analytical insight into complex coupled PDE systems arising in fluid-structure interaction.

math.AP↗
arXiv2026

A linear test approach to global controllability of third- and fifth-order nonlinear dispersive equations

We investigate third- and fifth-order nonlinear dispersive equations of KdV type on the torus and establishes approximate controllability by a fixed four-dimensional control; rather than relying solely on the saturation machinery, the analysis exploits the finite-dimensional controllability of the inviscid Burgers equation linearized around a carefully constructed return trajectory, with the trajectory itself obtained from an observable family. This ``linear test" strategy, yields more information about the structure of the control than the standard approach. In particular, the constructed control is shown to depend continuously on the initial and target states, a property that is by no means automatic in nonlinear control problems, and to decompose as a bounded linear operator applied to the data plus a fixed remainder, with the operator part interestingly independent of the order of dispersion.

math.AP↗
arXiv2026

Global in-time rough large data solution to complex-valued semilinear damped evolution equations

We study the semilinear Cauchy problem for complex-valued damped evolution equations \begin{align*} \partial_t^2u+(-Δ)^σu+(-Δ)^δ\partial_tu=u^p,\ \ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x), \end{align*} with $δ\in[0,σ]$, $σ\in\mathbb{R}_+$ and $p\in\mathbb{N}_+\backslash\{1\}$, where the initial data belong to the rough space $E^α_s$ endowed with the norm \begin{align*} \|f\|_{E^α_s}=\big\|\langleξ\rangle^s\,2^{α|ξ|}\widehat{f}(ξ)\big\|_{L^2}\ \ \mbox{with}\ \ α<0, \ s\in\mathbb{R}. \end{align*} Concerning $(u_0,u_1)\in E^α_{s+\barκ}\times E^α_s$ when $s\geqslant\frac{n}{2}-\frac{2κ+\barκ-2δ}{p-1}-\barκ$ with $κ=\min\{2δ,σ\}$ and $\barκ=\max\{2δ,σ\}$ whose Fourier transforms are supported in a suitable subset of first octant, we prove a global in-time existence result without requiring the smallness of rough initial data.

math.AP↗