Piotr Biler, Alexandre Boritchev, Grzegorz Karch, Philippe Laurençot
We consider the drift-diffusion equation $$ u_t-\varepsilon Δu+\nabla\cdot(u\nabla K\star u)=0 $$ in the whole space with global-in-time bounded solutions. Mass concentration phenomena for radially symmetric solutions of this equation with small diffusivity are studied.
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The nonexistence of global in time solutions is studied for a class of aggregation equations involving Lévy diffusion operators and general interaction kernels.
We consider the drift-diffusion equation $u_t-εΔu + \nabla \cdot(u\nabla K^*u)=0$ in the whole space with global-in-time solutions bounded in all Sobolev spaces; for simplicity, we restrict ourselves to the model case $K(x)=-|x|$. We quantify the mass concentration phenomenon, a genuinely nonlinear effect, for radially symmetric solutions of this equation for small diffusivity $ε$ studied in our previous paper [3], obtaining optimal sharp upper and lower bounds for Sobolev norms.
We study the long-time effect of noise on pattern formation for the aggregation model. We consider aggregation kernels that generate patterns consisting of two delta-concentrations. Without noise, there is a one-parameter family of admissible equilibria that consist of two concentrations whose mass is not necessary equal. We show that when a small amount of noise is added, the heavier concentration "leaks" its mass towards the lighter concentration over a very long time scale, eventually resulting in the equilibration of the two masses. We use exponentially small asymptotics to derive the long…
Concentration of mass of solutions to aggregation-diffusion equations
研究全空间中带轻度奇异相互作用核的聚集-扩散方程，以非负、有界且可积初值为补充有全局解，证明非负径向对称解在共同有限时间区间内随ε趋于0在原点附近聚集。
We investigate the diffusion-aggregation equations with degenerate diffusion $Δu^m$ and singular interaction kernel $\mathcal{K}_s = (-Δ)^{-s}$ with $s\in(0,\frac{d}{2})$. We analyze the regime %($m>2-2s/d$, $d$ is the dimension) where the diffusive forces are stronger than the aggregation forces. In such regime, we show existence, uniform boundedness and Hölder regularity of solutions in the case that either $s>\frac{1}{2}$ or $m<2$. Uniqueness is proved for kernels with $s>1$.
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-a…
We study the multidimensional aggregation equation $u_t+\Div(uv)=0$, $v=-\nabla K*u$ with initial data in $\cP_2(\bR^d)\cap L_{p}(\bR^d)$. We prove that with biological relevant potential $K(x)=|x|$, the equation is ill-posed in the critical Lebesgue space $L_{d/(d-1)}(\bR^d)$ in the sense that there exists initial data in $\cP_2(\bR^d)\cap L_{d/(d-1)}(\bR^d)$ such that the unique measure-valued solution leaves $L_{d/(d-1)}(\bR^d)$ immediately. We also extend this result to more general power-law kernels $K(x)=|x|^α$, $0<α<2$ for $p=p_s:=d/(d+α-2)$, and prove a conjecture in Bertozzi, Laurent …
Typically, aggregation-diffusion is modeled by parabolic equations that combine linear or nonlinear diffusion with a Fokker-Planck convection term. Under very general suitable assumptions, we prove that radial solutions of the evolution process converge asymptotically in time towards a stationary state representing the balance between the two effects. Our parabolic system is the gradient flow of an energy functional, and in fact we show that the stationary states are minimizers of a relaxed energy. Here, we study radial solutions of an aggregation-diffusion model that combines nonlinear fast d…
Partial mass concentration for fast-diffusions with non-local aggregation terms
研究快扩散范围下非局部聚集-扩散方程的适定性与长时间行为，利用粘性解证明径向初值的部分质量集中现象，并给出实例证实无穷时间内发生部分质量集中。
We prove the existence of non-trivial global minimizers of a class of free energies related to aggregation equations with degenerate diffusion on $\Real^d$. Such equations arise in mathematical biology as models for organism group dynamics which account for competition between the tendency to aggregate into groups and nonlinear diffusion to avoid over-crowding. The existence of non-zero optimal free energy stationary solutions representing coherent groups in $\Real^d$ is therefore of interest. The primary contribution is the investigation of a notion of criticality associated with the global m…
Ground states and phase transitions for an aggregation model with fast diffusion on sphere
本文研究了球面上具有快速扩散的聚集体模型的地面状态与相变，探讨了非局部相互作用强度对相变的影响。
We are concerned with the dynamics of one fold symmetric patches for the two-dimensional aggregation equation associated to the Newtonian potential. We reformulate a suitable graph model and prove a local well-posedness result in sub-critical and critical spaces. The global existence is obtained only for small initial data using a weak damping property hidden in the velocity terms. This allows to analyze the concentration phenomenon of the aggregation patches near the blow up time. In particular, we prove that the patch collapses to a collection of disjoint segments and we provide a descriptio…
We provide a quantitative asymptotic analysis for the nonlinear Vlasov--Poisson--Fokker--Planck system with a large linear friction force and high force-fields. The limiting system is a diffusive model with nonlocal velocity fields often referred to as aggregation-diffusion equations. We show that a weak solution to the Vlasov--Poisson--Fokker--Planck system strongly converges to a strong solution to the diffusive model. Our proof relies on the modulated macroscopic kinetic energy estimate based on the weak-strong uniqueness principle together with a careful analysis of the Poisson equation.
We investigate the effect of linear diffusion and interactions with the domain boundary on swarm equilibria by analyzing critical points of the associated energy functional. Through this process we uncover two properties of energy minimization that depend explicitly on the spatial domain: (i) unboundedness from below of the energy due to an imbalance between diffusive and aggregative forces depends explicitly on a certain volume filling property of the domain, and (ii) metastable mass translation occurs in domains without sufficient symmetry. From the first property, we present a sharp conditi…
In this article, we will derive the differential equations considering the concentration polarization (\textsc{cp}) in membrane separations and purifications, which involve convective-diffusive equations of the concentration polarization layer and the Spiegler-Kedem-Katchalsky (\textsc{skk}) model of the membrane layer. Analytical solutions of these differential equations are straightforward and given. The film theory coupled \textsc{skk} model can be used to evaluate the degree of concentration polarization, and predict flux and rejection rates.
We show that partial mass concentration can happen for stationary solutions of aggregation-diffusion equations with homogeneous attractive kernels in the fast diffusion range. More precisely, we prove that the free energy admits a radial global minimizer in the set of probability measures which may have part of its mass concentrated in a Dirac delta at a given point. In the case of the quartic interaction potential, we find the exact range of the diffusion exponent where concentration occurs in space dimensions $N\geq6$. We then provide numerical computations which suggest the occurrence of ma…
In this paper we study the long time asymptotic behavior for a class of diffusion-aggregation equations. Most results except the ones in Section 3.3 concern radial solutions. The main tools used in the paper are maximum-principle type arguments on mass concentration of solutions, as well as energy method. For the Patlak-Keller-Segel problem with critical power $m=2-2/d$, we prove that all radial solutions with critical mass would converge to a family of stationary solutions, while all radial solutions with subcritical mass converge to a self-similar dissipating solution algebraically fast. For…
We study a nonlocal aggregation equation with degenerate diffusion, set in a periodic domain. This equation represents the generalization to $m > 1$ of the McKean-Vlasov equation where here the "diffusive" portion of the dynamics are governed by Porous medium self-interactions. We focus primarily on $m\in(1,2]$ with particular emphasis on $m = 2$. In general, we establish regularity properties and, for small interaction, exponential decay to the uniform stationary solution. For $m=2$, we obtain essentially sharp results on the rate of decay for the entire regime up to the (sharp) transitional …
We propose a mathematical model, namely a reaction-diffusion system, to describe social behaviour of cockroaches. An essential new aspect in our model is that the dispersion behaviour due to overcrowding effect is taken into account {as a counterpart to commonly studied aggregation}. This consideration leads to an intriguing new phenomenon which has not been observed in the literature. Namely, due to the competition between aggregation towards areas of higher concentration of pheromone and dispersion avoiding overcrowded areas, the cockroaches aggregate more at the transition area of pheromone…
Formation of multiple flocks in a simple nonlocal aggregation model
该研究针对具有“势阱-势垒”核的非局部聚集模型，利用集中紧性原理的二分情形求解能量极小值，解释了大种群中多鸟群形成的机制。
Aggregation of chemotactic bacteria under a unimodal distribution of chemical cues was investigated by Monte Carlo (MC) simulation based on a kinetic transport equation, which considers an internal adaptation dynamics as well as a finite tumbling duration. It was found that there exist two different regimes of the adaptation time, between which the effect of the adaptation time on the aggregation behavior is reversed; that is, when the adaptation time is as small as the running duration, the aggregation becomes increasingly steeper as the adaptation time increases, while, when the adaptation t…
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