We consider a coupled Patlak-Keller-Segel-Navier-Stokes system in $\mathbb{R}^2$ that describes the collective motion of cells and fluid flow, where the cells are attracted by a chemical substance and transported by ambient fluid velocity, and the fluid flow is forced by the friction induced by the cells. The main result of the paper is to show the global existence of free-energy solutions to the 2D Patlak-Keller-Segel-Navier-Stokes system with critical and subcritical mass.
We consider the parabolic-parabolic two-dimensional Patlak-Keller-Segel problem. We prove the existence of stable blow-up dynamics in finite time in the radial case. We extend in this article the result of [36] for the parabolic-elliptic case.
We prove a blow-up criterion for the solutions to the $ν$-dimensional Patlak-Keller-Segel equation in the whole space. The condition is new in dimension three and higher. In dimension two it is exactly Dolbeault's and Perthame's blow-up condition, i.e., blow-up occurs if total mass exceeds $8π$ .
We study the global in time existence of solutions to the parabolic-elliptic Patlak-Keller-Segel system of multi-species populations. We prove that if the initial mass satisfies an appropriate notion of sub-criticality, then the system has a solution defined for all time. We explore the gradient flow structure in the Wasserstein space to study the question of existence. Moreover, we show that the obtained solution satisfies energy dissipation inequality.
We consider the parabolic-elliptic Patlak-Keller-Segel (PKS) model of chemotactic aggregation in two space dimensions which describes the aggregation of bacteria under chemo-taxis. When the mass is equal to $8π$ and the second moment is finite (the doubly critical case), we give a precise description of the dynamic as time goes to infinity and extract the limiting profile and speed. The proof shows that this dynamic is stable under perturbations.
In this paper, we proposed a coupled Patlak-Keller-Segel-Navier-Stokes system, which has dissipative free energy. On the plane $\rr^2$, if the total mass of the cells is strictly less than $8π$, classical solutions exist for any finite time, and their $H^s$-Sobolev norms are almost uniformly bounded in time. For the radially symmetric solutions, this $8π$-mass threshold is critical. On the torus $\mathbb{T}^2$, the solutions are uniformly bounded in time under the same mass constraint.
This review is dedicated to recent results on the 2d parabolic-elliptic Patlak-Keller-Segel model, and on its variant in higher dimensions where the diffusion is of critical porous medium type. Both of these models have a critical mass $M_c$ such that the solutions exist globally in time if the mass is less than $M_c$ and above which there are solutions which blowup in finite time. The main tools, in particular the free energy, and the idea of the methods are set out. A number of open questions are also stated.
This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak-Keller-Segel system with $d\ge3$ and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass $M_c$ such that if $M \in (0,M_c]$ solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the ex…
In this paper we consider the parabolic-parabolic Patlak-Keller-Segel models in $\mathbb{T}\times\mathbb{R}$ with advection by a large strictly monotone shear flow. Without the shear flow, the model is $L^1$ critical in two dimensions with critical mass $8π$: solutions with mass less than $8π$ are global in time and there exist solutions with mass larger than $8 π$ which blow up in finite time \cite{Schweyer14}. We show that the additional shear flow, if it is chosen sufficiently large, suppresses one dimension of the dynamics and hence can suppress blow-up. In contrast with the parabolic-elli…
In this paper we investigate qualitative and asymptotic behavior of solutions for a class of diffusion-aggregation equations. Most results except the ones in section 3 and 6 concern radial solutions. The challenge in the analysis consists of the nonlocal aggregation term as well as the degeneracy of the diffusion term which generates compactly supported solutions. The key tools used in the paper are maximum-principle type arguments as well as estimates on mass concentration of solutions.
We consider a special case of the Patlak-Keller-Segel system in a disc, which arises in the modelling of chemotaxis phenomena. For a critical value of the total mass, the solutions are known to be global in time but with density becoming unbounded, leading to a phenomenon of mass-concentration in infinite time. We establish the precise grow-up rate and obtain refined asymptotic estimates of the solutions. Unlike in most of the similar, recently studied, grow-up problems, the rate is neither polynomial nor exponential. In fact, the maximum of the density behaves like $e^{\sqrt{2t}}$ for large t…
We study the regularity and large-time behavior of a crowd of species driven by chemo-tactic interactions. What distinguishes the different species is the way they interact with the rest of the crowd: the collective motion is driven by different chemical reactions which end up in a coupled system of parabolic Patlak-Keller-Segel equations. We show that the densities of the different species diffuse to zero provided the chemical interactions between the different species satisfy certain sub-critical condition; the latter is intimately related to a log-Hardy-Littlewood-Sobolev inequality for sys…
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…
The L^1-critical parabolic-elliptic Patlak-Keller-Segel system is a classical model of chemotactic aggregation in micro-organisms well-known to have critical mass phenomena. In this paper we study this critical mass phenomenon in the context of Patlak-Keller-Segel models with spatially varying diffusivity and decay rate of the chemo-attractant. The primary tool for the proof of global existence below the critical mass is the use of pseudo-differential operators to precisely evaluate the leading order quadratic portion of the potential energy (interaction energy). Under the assumption of radial…
Aggregation equations and Patlak-Keller-Segel (PKS) models for chemotaxis with nonlinear diffusion are popular models for nonlocal aggregation phenomenon and are a source of a number of interesting mathematical problems in nonlinear PDE. The purpose of this work is twofold. First, we continue our previous work, which focused on nonlocal aggregation, modeled with a convolution. The goal was to unify the local and global theory of these convolution-type models, including the identification of a sharp critical mass; however, some cases involving unbounded domains were left open. In particular, th…
We study the global in time existence and long time asymptotics of solutions to the parabolic-elliptic Patlak-Keller-Segel system for the multi-species populations in the whole Euclidean space $\mathbb{R}^2.$ We prove that at the borderline case of critical mass there exists a global {\it free energy solution} subject to initial data with finite entropy and second moment. Moreover, we show that as time $t$ approaches to infinity, all the components of the solutions concentrate in the form of a Dirac measure at a single point. Our approach utilizes the gradient flow structure in Wasserstein spa…
We establish a new local well-posedness result in the space of finite Borel measures for mild solutions of the parabolic-elliptic Patlak-Keller-Segel (PKS) model of chemotactic aggregation in two dimensions. Our result only requires that the initial measure satisfy the necessary assumption $\max_{x \in \Real^2}μ(\set{x}) < 8π$. This work improves the small-data results of Biler and the existence results of Senba and Suzuki. Our work is based on that of Gallagher and Gallay, who prove the uniqueness and log-Lipschitz continuity of the solution map for the 2D Navier-Stokes equations (NSE) with m…
Recently, there has been a wide interest in the study of aggregation equations and Patlak-Keller-Segel (PKS) models for chemotaxis with degenerate diffusion. The focus of this paper is the unification and generalization of the well-posedness theory of these models. We prove local well-posedness on bounded domains for dimensions $d\geq 2$ and in all of space for $d\geq 3$, the uniqueness being a result previously not known for PKS with degenerate diffusion. We generalize the notion of criticality for PKS and show that subcritical problems are globally well-posed. For a fairly general class of p…
Based on some elementary estimates for the space-time derivatives of the heat kernel, we use a bootstrapping approach to establish the optimal decay rates for the $L^q(\mathbb{R}^d)$ ($1\leq q\leq\infty$, $d\in\mathbb{N}$) norm of the space-time derivatives of solutions to the (modified) Patlak-Keller-Segel equations with initial data in $L^1(\mathbb{R}^d)$, which implies the joint space-time analyticity of solutions. When the $L^1(\mathbb{R}^d)$ norm of the initial datum is small, the upper bound for the decay estimates is global in time, which yields a lower bound on the growth rate of the r…
This paper is concerned with a chemotaxis aggregation model for cells, more precisely with a parabolic-elliptic semilinear Patlak-Keller-Segel system in a ball of $\mathbb{R}^N$ for $N\geq 2$. For $N=2$, this system is well known for its critical mass $8π$. It has been proved in \cite{Montaru2} that it also exhibits a critical mass phenomenon for $N\geq 3$. The main result of this paper is the exponential speed of uniform convergence of radial solutions toward the unique steady state in the subcritical case for $N\geq 2$. We stress that this covers in particular the classical Keller-Segel syst…
In this paper we consider the parabolic-elliptic Patlak-Keller-Segel models in $\mathbb T^d$ with $d=2,3$ with the additional effect of advection by a large shear flow. Without the shear flow, the model is $L^1$ critical in two dimensions with critical mass $8π$; solutions with mass less than $8π$ are global and solutions with mass larger than $8 π$ with finite second moment, all blow up in finite time. In three dimensions, the model is $L^{3/2}$ critical and $L^1$ supercritical; there exists solutions with arbitrarily small mass which blow up in finite time arbitrarily fast. We show that the …
We complete previous results about the incompressible limit of both the $n$-dimensional $(n\geq3)$ compressible Patlak-Keller-Segel (PKS) model and its stationary state. As in previous works, in this limit, we derive the weak form of a geometric free boundary problem of Hele-Shaw type, also called congested flow. In particular, we are able to take into account the unsaturated zone, and establish the complementarity relation which describes the limit pressure by a degenerate elliptic equation. Not only our analysis uses a completely different framework than previous approaches, but we also esta…
We investigate the long time behavior of the critical mass Patlak-Keller-Segel equation. This equation has a one parameter family of steady-state solutions $ρ_λ$, $λ>0$, with thick tails whose second moment is not bounded. We show that these steady state solutions are stable, and find basins of attraction for them using an entropy functional ${\mathcal H}_λ$ coming from the critical fast diffusion equation in $\R^2$. We construct solutions of Patlak-Keller-Segel equation satisfying an entropy-entropy dissipation inequality for ${\mathcal H}_λ$. While the entropy dissipation for ${\mathcal H}_λ…
We present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased random walk. In order to take into account possible alterations in cellular motility observed at high cell densities (i.e. volume-filling), we let the probabilities of cell movement be modulated by a decaying function of the cell density. We formally show that a general form of the celebrated Patlak-Keller-Segel (PKS) model of chemotaxis can be formally derived as the appropriate continuum limit of this …
We study the solutions of the two-dimensional Keller-Segel system describing chemotaxis. The Keller-Segel system as well as the properties of the blow-up set has been extensively studied. In this paper we obtain generalized solutions for the Keller-Segel system in the sense of measures as the limit of two-different regularizations of it. We will show that the resulting limit measures, that are in some suitable sense global weak solutions of the Keller-Segel system, depend in the regularization.
We consider the stationary Keller-Segel system from chemotaxis in a ball and we show the existence of a solution concentrating at the boundary of the ball.
The existence and nonexistence of global in time solutions is studied for a class of equations generalizing the chemotaxis model of Keller and Segel. These equations involve Lévy diffusion operators and general potential type nonlinear terms.
Hele-Shaw flow with surface tension and kinetic undercooling as a sharp interface limit of a fully parabolic Patlak-Keller-Segel system with nonlinear diffusion
The focus of this paper is on solutions to a two-dimensional Keller-Segel system containing sub-logistic sources. We show that the presence of sub-logistic terms is adequate to prevent blow-up phenomena even in strongly degenerate Keller-Segel systems. Our proof relies on several techniques, including parabolic regularity theory in Orlicz spaces, variational arguments, interpolation inequalities, and the Moser iteration method.
The long range movement of certain organisms in the presence of a chemoattractant can be governed by long distance runs, according to an approximate Levy distribution. This article clarifies the form of biologically relevant model equations: We derive Patlak-Keller-Segel-like equations involving nonlocal, fractional Laplacians from a microscopic model for cell movement. Starting from a power-law distribution of run times, we derive a kinetic equation in which the collision term takes into account the long range behaviour of the individuals. A fractional chemotactic equation is obtained in a bi…
We investigate nonlinear dynamics near an unstable constant equilibrium in the classical Keller-Segel model. Given any general perturbation of magnitude $δ$, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes, over a time period of $ln(1/δ)$. Our result can be interpreted as a rigourous mathematical characterization for early pattern formation in the Keller-Segel model.
We examine the long-term asymptotic behavior of dissipating solutions to aggregation equations and Patlak-Keller-Segel models with degenerate power-law and linear diffusion. The purpose of this work is to identify when solutions decay to the self-similar spreading solutions of the homogeneous diffusion equations. Combined with strong decay estimates, entropy-entropy dissipation methods provide a natural solution to this question and make it possible to derive quantitative convergence rates in $L^1$. The estimated rate depends only on the nonlinearity of the diffusion and the strength of the in…
We construct radial self-similar solutions of the, so called, minimal parabolic-elliptic Keller--Segel model in several space dimensions with radial, nonnegative initial conditions with are below the Chandrasekhar solution -- the singular stationary solution of this system.
In this paper we derive matched asymptotic expansions for a solution of the Keller-Segel system in two space dimensions for which the amount of mass aggregation is $8πN$, where $N=1,2,3,...$ Previously available asymptotics had been computed only for the case in which N=1.
We consider a class of $L^1$ critical nonlocal aggregation equations with linear or nonlinear porous media-type diffusion which are characterized by a long-range interaction potential that decays faster than the Newtonian potential at infinity. The fast decay breaks the $L^1$ scaling symmetry and we prove that `sufficiently spread out' initial data, regardless of the mass, result in global spreading solutions. This is in contrast to the classical parabolic-elliptic PKS for which essentially all solutions with more than critical mass are known to blow up in finite time. In all cases, the long-t…
In this paper, we present local Hölder estimates for the degenerate Keller-Segel system \eqref{eq-cases-aligned-main-problem-of-Keller-Segel-System} below in the range of $m>1$ and $q>1$ before a blow-up of solutions. To deal with difficulties caused by the degeneracy of the operator, we find uniform estimates depending sup-norm of the density function and modified the energy estimates and intrinsic scales considered in Porous Medium Equation. As its application, the uniqueness of weak solution to \eqref{eq-cases-aligned-main-problem-of-Keller-Segel-System} is also showed in the class of Hölde…
This paper generalizes and extends to the case of nonlinear effects and logistic perturbations some results recently developed in the literature where, for the linear counterpart and in absence of logistics, criteria toward boundedness for an attraction-repulsion Keller-Segel system with double saturation are derived.
We study a new nonlocal approach to the mathematical modelling of the Chemotaxis problem, which describes the random motion of a certain population due a substance concentration. Considering the initial-boundary value problem for the fractional hyperbolic Keller-Segel model, we prove the solvability of the problem. The solvability result relies mostly on fractional calculus and kinetic formulation of scalar conservation laws.
We study a Keller-Segel type chemotaxis model with a modified sensitivity function in a bounded domain $Ω\subset \mathbb{R}^N$, $N\geq2$. The global existence of classical solutions to the fully parabolic system is established provided that the ratio of the chemotactic coefficient to the motility of cells is not too large.
In this paper we study the initial boundary value problem for the system $u_t-Δu^m=-\mbox{div}(u^{q}\nabla v),\ v_t-Δv+v=u$. This problem is the so-called Keller-Segel model with nonlinear diffusion. Our investigation reveals that nonlinear diffusion can prevent overcrowding. To be precise, we show that solutions are bounded as long as $m>q>0$, thereby substantially generalizing the known results in this area. Furthermore, our result seems to imply that the Keller-Segel model can have bounded solutions and blow-up ones simultaneously.
In two space dimensions, the parabolic-parabolic Keller--Segel system shares many properties with the parabolic-elliptic Keller--Segel system. In particular, solutions globally exist in both cases as long as their mass is less than 8?. However, this threshold is not as clear in the parabolic-parabolic case as it is in the parabolic-elliptic case, in which solutions with mass above 8? always blow up. Here we study forward self-similar solutions of the parabolic-parabolic Keller--Segel system and prove that, in some cases, such solutions globally exist even if their total mass is above 8?, which…
We construct weak global in time solutions to the classical Keller-Segel system cell movement by chemotaxis in two dimensions when the total mass is below the well-known critical value. Our construction takes advantage of the fact that the Keller-Segel system can be realized as a gradient flow in a suitable functional product space. This allows us to employ a hybrid variational principle which is a generalisation of the minimising implicit scheme for Wasserstein distances introduced by Jordan, Kinderlehrer and Otto (1998).
We introduce the planar helical flows on three dimensional torus and study the dissipation enhancement of such flows. We then use such flows as transport flows to solve the three dimensional advective Kuramoto-Sivashinsky and Keller-Segel equations. The global well-posedness of the Kuramoto-Sivashinsky equation is achieved when the linearized operator does not have growing mode in the direction orthogonal to the flow. The global classical solution of the three dimensional Keller-Segel is ensured for any size of the torus with arbitrarily large initial data.
We consider the simplest parabolic-elliptic model of chemotaxis in the whole space in several dimensions. Criteria for the existence of radial global-in-time solutions in terms of suitable Morrey norms are derived.
