existence uniqueness and lipschitz dependence for patlak keller segel and navier stokes in mathbb r 2 with measure valued initial data 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 mathbb r 2 mu x 8 pi this work improves the small data results of biler stud math 114 2 181 192 1995 and the existence results of senba and suzuki j funct anal 191 17 51 2002 our work is based on that of gallagher and gallay math ann 332 287 327 2005 who prove the uniqueness and log lipschitz continuity of the solution map for the 2d navier stokes equations nse with measure valued initial vorticity we refine their techniques and present an alternative version of their proof which yields existence uniqueness and lipschitz continuity of the solution maps of both pks and nse many steps are more difficult for pks than for nse particularly on the level of the linear estimates related to the self similar spreading solutions

on the existence of local strong solutions to chemotaxis shallow water system with large data and vacuum abstract in this paper motivated by the chemotaxis navier stokes system arising from mathematical biology 43 a modified shallow water type chemotactic model is derived for large initial data allowing vacuum the local existence of strong solutions together with the blow up criterion is established

aggregation blowup and collapse the abc s of taxis in reinforced random walks in many biological systems movement of an organism occurs in response to a diffusible or otherwise transported signal and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by patlak bull of math biophys 15 1953 pp 311 338 however other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages in these systems one example of which is the myxobacteria the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates davis probab theory related fields 84 1990 pp 203 229 has studied this question for a certain class of random walks and here we extend this analysis to the continuum limit of such walks we first derive several general classes of partial differential equations that depend on how the movement rules are

a conservative free energy dissipating and positivity preserving finite difference scheme for multi dimensional nonlocal fokker planck equation abstract in this work we design and analyze a conservative positivity preserving and free energy dissipating finite difference method for the multi dimensional nonlocal fokker planck fp equation based on a non logarithmic landau transformation a central differencing spatial discretization using harmonic mean approximations is developed both forward and backward euler discretizations in time are employed to derive an explicit scheme and a linearized semi implicit scheme respectively three desired properties that are possessed by analytical solutions i mass conservation ii free energy dissipation and iii positivity are proved to be maintained at discrete level remarkably numerical analysis demonstrates that the semi implicit time discretization ensures the property of positivity preserving unconditionally due to the advantages brought by the harmonic mean approximations an estimate on the upper bound of the condition number of the resulting coefficient matrix is further established for the semi implicit scheme extensive numerical tests are performed to validate aforementioned properties numerically

global existence of solutions to a two dimensional attraction repulsion chemotaxis system in the attractive dominant case with critical mass abstract we study the cauchy problem for an attraction repulsion chemotaxis system in two dimensions t u u u 1 v 1 2 v 2 0 v 1 1 v 1 u 0 v 2 2 v 2 u with nonnegative initial data u 0 and positive constants 1 2 1 2 the purpose of this paper is to show that the cauchy problem admits the global nonnegative solutions in time if 1 2 r 2 u 0 d x 8 in the attractive dominant case 1 2

nonlinear aggregation diffusion equations radial symmetry and long time asymptotics we analyze under which conditions equilibration between two competing effects repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction occurs this balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses all stationary states with suitable regularity are shown to be radially symmetric by means of continuous steiner symmetrization techniques calculus of variations tools allow us to show the existence of global minimizers among these equilibria finally in the particular case of newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation diffusion equations towards this unique equilibrium profile up to translations as t rightarrow infty

a class of the first order impulsive dynamic equations on time scales abstract in this paper the first order impulsive dynamic equations on time scales is considered the existence and uniqueness of solution for a class of nonlinear impulsive dynamic equations on time scales is proved

on the fractional fisher information with applications to a hyperbolic parabolic system of chemotaxis we introduce new lower bounds for the fractional fisher information equipped with these bounds we study a hyperbolic parabolic model of chemotaxis and prove the global existence of solutions in certain dissipation regimes

aggregation of a class of interconnected linear dynamical systems abstract in this paper we introduce the notion of a meaningful average of a collection of dynamical systems as distinct from an ensemble average such a notion is useful for the study of a variety of dynamical systems such as traffic flow power systems and econometric systems we also address the associated issue of the existence and computation of such an average for a class of interconnected linear time invariant dynamical systems such an average dynamical system is not only attractive from a computational perspective but also represents the average behavior of the interconnected dynamical systems the problem of analysis and control of heirarchical large scale control systems can be simplified by approximating the lower level dynamics of such systems with such an average dynamical system

lp theory for the multidimensional aggregation equation we consider well posedness of the aggregation equation tu div uv 0 v k u with initial data in input amssym cal p 2 rm bbb r d rm cap l p bbb r d in dimensions 2 and higher we consider radially symmetric kernels where the singularity at the origin is of order x 2 d and prove local well posedness in input amssym cal p 2 bbb r d rm cap l p bbb r d for sufficiently large p ps in the special case of k x x the exponent ps d d 1 is sharp for local well posedness in that solutions can instantaneously concentrate mass for initial data in input amssym cal p 2 bbb r d rm cap l p bbb r d with p ps we also give an osgood condition on the potential k x that guarantees global existence and uniqueness in input amssym cal p 2 bbb r d rm cap l p bbb r d 2010 wiley periodicals inc

numerical simulation of nonlinear continuity equations by evolving diffeomorphisms abstract in this paper we present a numerical scheme for nonlinear continuity equations which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance it can be applied to a large class of nonlinear continuity equations whose dynamics are driven by internal energies given external potentials and or interaction energies the solver is based on its variational formulation as a gradient flow with respect to the wasserstein distance positivity of solutions as well as energy decrease of the semi discrete scheme are guaranteed by its construction we illustrate this property with various examples in spatial dimension one and two

on the convergence rate of diffusion in the bidirectional erdos renyi networks an h2 norm perspective abstract we investigate the convergence rates of diffusion in bidirectional erdos renyi networks we analytically and numerically demonstrate that the asymptotic distribution is far from a bell shape further we numerically verify that the dynamical networks generated by the barabasi albert model which is a standard scale free network model exhibiting faster convergence of average consensus

uniqueness and symmetry of equilibria in a chemotaxis model we consider in a disc of r a class of parameter dependent nonlocal elliptic boundary value problems that describe the steady states of some chemotaxis systems if the appearing parameter is less than an explicit critical value we establish several uniqueness results for solutions that are invariant under a group of rotations furthermore we discuss the associated consequences for the time asymptotic behavior of the solutions to the corresponding time dependent chemotaxis systems our results also provide optimal constants in some moser trudinger type inequalities mathematics subject classification 2000 35j60 35j50 92c17

uniqueness of weak solutions to a high dimensional keller segel equation with degenerate diffusion and nonlocal aggregation abstract this paper considers weak solutions to the degenerate keller segel equation with nonlocal aggregation u t u m u b u in r d r where b u 2 u d 3 2 d 1 m 2 d in a previous paper of the authors hong et al 2015 a criterion was established for global existence versus finite time blow up of weak solutions to the problem a natural question is whether the uniqueness is true for the weak solutions obtained a positive answer is given in this paper that the global weak solutions must be unique provided the second moment of initial data is finite which means that the weak solutions are weak entropy solutions in fact the framework of the proof is based on the optimal transportation method

the aggregation equation with newtonian potential the vanishing viscosity limit abstract the viscous and inviscid aggregation equation with newtonian potential models a number of different physical systems and has close analogs in 2d incompressible fluid mechanics we consider a slight generalization of these equations in the whole space establishing well posedness of the viscous and inviscid equations spatial decay of the viscous solutions and the convergence of viscous solutions to the inviscid solution as the viscosity goes to zero

equilibria of homogeneous functionals in the fair competition regime we consider macroscopic descriptions of particles where repulsion is modelled by non linear power law diffusion and attraction by a homogeneous singular non singular kernel leading to variants of the keller segel model of chemotaxis we analyse the regime in which both homogeneities scale the same with respect to dilations that we coin as fair competition in the singular kernel case we show that existence of global equilibria can only happen at a certain critical value and they are characterised as optimisers of a variant of hls inequalities we also study the existence of self similar solutions for the sub critical case or equivalently of optimisers of rescaled free energies these optimisers are shown to be compactly supported radially symmetric and non increasing stationary solutions of the non linear keller segel equation on the other hand we show that no radially symmetric non increasing stationary solutions exist in the non singular kernel case implying that there is no criticality however we show the existence of positive self similar solutions for all values of the parameter under the condition that diffusion is not too fast we finally illustrate some of the open problems in the non singular kernel case by numerical experiments

stability of moving invariant sets and uncertain dynamic systems on time scales abstract in this paper we study the stability of moving invariant sets and uncertain dynamic systems on time scale a control application is considered

existence of global bounded classical solution to a quasilinear attraction repulsion chemotaxis system with logistic source abstract this paper concerns the global existence of bounded classical solution to a quasilinear attraction repulsion chemotaxis system with logistic source u t d u u u v u w u u 2 in v t v u v in w t w u w in under homogeneous neumann boundary condition with positive parameters and d s c 0 s m 1 for s 0 d 0 0 we prove that in dimension three there exists a unique global bounded classical solution provided that m 6 5 and 0 with an additional assumption d s c 0 s m 1 1 for s 0 we prove that for any m 1 6 5 there exists a constant 0 0 m such that for all 0 the above problem admits a unique global bounded classical solution

hamilton jacobi theory over time scales and applications to linear quadratic problems abstract in this paper we first derive the verification theorem for nonlinear optimal control problems over time scales that is we show that the value function is the only solution of the hamilton jacobi equation in which the minimum is attained at an optimal feedback controller applications to the linear quadratic regulator problem lqr problem gives a feedback optimal controller form in terms of the solution of a generalized time scale riccati equation and that every optimal solution of the lqr problem must take that form a connection of the newly obtained riccati equation with the traditional one is established problems with shift in the state variable are also considered as an important tool for the latter theory we obtain a new formula for the chain rule on time scales finally the corresponding lqr problem with shift in the state variable is analyzed and the results are related to previous ones

blow up dynamics for the aggregation equation with degenerate diffusion abstract we study radially symmetric finite time blow up dynamics for the aggregation equation with degenerate diffusion u t u m u k u in r d where the kernel k x is of power law form x depending on m d and the initial data the solution exhibits three kinds of blow up behavior self similar with no mass concentrated at the core imploding shock solution and near self similar blow up with a fixed amount of mass concentrated at the core computations are performed for different values of m d and using an arbitrary lagrangian eulerian method with adaptive mesh refinement

global existence for a kinetic model of pattern formation with density suppressed motilities abstract in this paper we consider global existence of classical solutions to the following kinetic model of pattern formation 0 1 u t v u u 1 u v v u in a smooth bounded domain r n n 1 with no flux boundary conditions here 0 is any given constant the function represents a signal dependent diffusion motility and is decreasing in v which models a density suppressed motility in the process of stripe pattern formation through the self trapping mechanism 9 20 the major difficulty in analysis lies in the possible degeneracy of diffusion as v in the present contribution based on a subtle observation of the nonlinear structure we develop a new method to rule out finite time degeneracy in any spatial dimension for all smooth motility functions satisfying v 0 and v 0 for v 0 then we prove global existence of classical solution for 0 1 in the two dimensional setting with any 0 moreover the global solution is proven to be uniform in time bounded if either 1 satisfies certain polynomial growth condition or 0 besides we pay particular attention to the specific case v e v with 0 under the circumstances system 0 1 becomes of great interest because it shares the same set of equilibria as well as the lyapunov functional with the classical keller segel model a novel critical phenomenon in the two dimensional setting is observed that with any initial datum of sub critical mass the global solution is proved to be uniform in time bounded while with certain initial datum of super critical mass the global solution will become unbounded as time goes to infinity namely blowup takes place in infinite time rather than finite time in our model which is distinct from the well known fact that certain initial data of super critical mass will enforce a finite time blowup for the classical keller segel system

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

asymptotic behaviour for small mass in the two dimensional parabolic elliptic keller segel model abstract the keller segel system describes the collective motion of cells that are attracted by a chemical substance and are able to emit it in its simplest form it is a conservative drift diffusion equation for the cell density coupled to an elliptic equation for the chemo attractant concentration this paper deals with the rate of convergence towards a unique stationary state in self similar variables which describes the intermediate asymptotics of the solutions in the original variables although it is known that solutions globally exist for any mass less 8 a smaller mass condition is needed in our approach for proving an exponential rate of convergence in self similar variables

global existence of solutions to the cauchy problem for an attraction repulsion chemotaxis system in r2 in the attractive dominant case abstract we consider the cauchy problem for an attraction repulsion chemotaxis system in r 2 with the chemotactic coefficient of the attractant 1 and that of the repellent 2 it is known that in the repulsive dominant case 1 2 or the balance case 1 2 the nonnegative solutions to the cauchy problem exist globally in time whereas in the attractive dominant case 1 2 there are blowing up solutions in finite time under the assumption 1 2 r 2 u 0 d x 8 on the nonnegative initial data u 0 in this paper we show the global existence of nonnegative solutions to the cauchy problem under the assumption 1 2 r 2 u 0 d x 8 in the attractive dominant case

global existence versus blow up in a high dimensional keller segel equation with degenerate diffusion and nonlocal aggregation abstract in this paper we study the degenerate keller segel equation with nonlocal aggregation u t u m u b u in r d r where m 1 d 3 and b u 2 u with 2 d by analyzing the interaction between the degenerate diffusion and the nonlocal aggregation we determine the conditions for initial data under which weak solutions globally exist or blow up in finite time with m 1 d d d furthermore a sharper criterion is given for global existence and finite time blow up of weak solutions with m in the subrange 2 d 2 d d d 1 d d

functional inequalities thick tails and asymptotics for the critical mass patlak keller segel model 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 rhohls lambda 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 fcfd 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 fcfd while the entropy dissipation for fcfd is strictly positive it turns out to be a difference of two terms neither of which need to be small when the dissipation is small we introduce a strategy of em controlled concentration to deal with this issue and then use the regularity obtained from the entropy entropy dissipation inequality to prove the existence of basins of attraction for each stationary state composed by certain initial data converging towards rhohls in the present paper we do not provide any estimate of the rate of convergence but we discuss how this would result from a stability result for a certain sharp gagliardo nirenberg sobolev inequality

decay estimates of the coupled chemotaxis fluid equations in r 3 abstract in this paper we are concerned with a chemotaxis navier stokes model arising from biology which is a coupled system of the chemotaxis equations and the viscous incompressible fluid equations with transport and external force the optimal convergence rates of classical solutions to the chemotaxis navier stokes system for small initial perturbation around constant states are obtained by pure energy method under the assumption the initial data belong to h s h n n 3 0 s 3 2 the h s 0 s 3 2 negative sobolev norms are shown to be preserved along time evolution compared to the result in 5 we obtain the optimal decay rates of the higher order spatial derivatives of the solutions

algebraic connectivity and degree sequences of trees we investigate the structure of trees that have minimal algebraic connectivity among all trees with a given degree sequence we show that such trees are caterpillars and that the vertex degrees are non decreasing on every path on non pendant vertices starting at the characteristic set of the fiedler vector author s abstract

asymptotic behavior of higher order nonlinear equations on time scales abstract utilizing the theory of dynamic systems on time scales which unifies the theory of continuous and discrete dynamic systems a necessary and sufficient condition is given for the asymptotic behavior of solutions of higher order nonlinear equations

global well posedness and blow up for the 2 d patlak keller segel equation abstract in this paper we prove that for every nonnegative initial data in l 1 r 2 the patlak keller segel equation is globally well posed if and only if the total mass m 8 our proof is based on some monotonicity formulas of nonnegative mild solutions

on explosions of solutions to a system of partial differential equations modelling chemotaxis a system of partial differential equations modelling chemotactic aggregation is analysed keller segel model conditions on the system of paramaters are given implying global existence of smooth solutions in two space dimensions and radially symmetric situations explosion of the bacteria concentration in finite time is shown for a class of initial values

existence uniqueness and asymptotic behavior of the solutions to the fully parabolic keller segel system in the plane abstract in the present article we consider several issues concerning the doubly parabolic keller segel system 1 1 1 2 in the plane when the initial data belong to critical scaling invariant lebesgue spaces more specifically we analyze the global existence of integral solutions their optimal time decay uniqueness and positivity together with the uniqueness of self similar solutions in particular we prove that there exist integral solutions of any mass provided that e 0 is sufficiently large with those results at hand we are then able to study the large time behavior of global solutions and prove that in the absence of the degradation term 0 the solutions behave like self similar solutions while in the presence of the degradation term 0 the global solutions behave like the heat kernel

point dynamics in a singular limit of the keller segel model 2 formation of the concentration regions this paper continues the analysis started in the first part of this article cf j j l velazquez siam j appl math 64 2004 pp 1198 1223 it was seen there using the method of matched asymptotics that a regularized version of the keller segel system admits for a suitable asymptotic limit solutions with some regions of high concentrations for the cell density this paper considers the relation between the phenomenon of blow up for the limit problem and the dynamics of the concentration regions described in j j l velazquez siam j appl math 64 2004 pp 1198 1223 in particular this paper analyzes the precise way in which the regularization introduced in the keller segel system stops the aggregation process and yields the formation of concentration regions

weak solutions to a parabolic elliptic system of chemotaxis abstract we study a parabolic elliptic system of partial differential equations which describes the chemotactic feature of slime molds it is known that the blowup solution forms singularities such as delta functions referred to as the collapses here we study the case that the domain is a flat torus and show that the post blowup continuation of the solution is possible only when those collapses are quantized with the mass 8

on asymptotic behaviors of solutions to parabolic systems modelling chemotaxis abstract this paper deals with large time behaviors of solutions to a keller segel system which possesses self similar solutions by taking into account the invariant properties of the equation with respect to a scaling and translations we show that suitably shifted self similar solutions give more precise asymptotic profiles of general solutions at large time the convergence rate is also computed in details

suppression of blow up by a logistic source in 2d keller segel system with fractional dissipation we consider a two dimensional parabolic elliptic keller segel equation with a logistic forcing and a fractional diffusion of order we obtain existence of global in time regular solution for arbitrary initial data with no size restrictions and c 2 where c 0 2 depends on the equation s parameters for an even wider range of s we prove existence of global in time weak solution for general initial data

system theoretic aspects of influenced consensus single input case this technical note examines the dynamics of networked multi agent systems operating with a consensus type algorithm under the influence of an attached node or external agent depending on the specific scenario the attached node can be viewed as a network intruder or an administrator we introduce an influence scheme naive of the network topology involving predictable excitation of the network with the objective of manipulating disrupting or steering its evolution the spectrum of the corresponding dirichlet matrix provides bounds on the system theoretic properties of the resulting influenced network quantifying its security or viewed differently its manageability finally the controllability gramian for influenced consensus is examined providing insights into its h2 norm and controllability properties

stability of some mechanisms of chemotactic aggregation in this paper the stability of a blow up mechanism for a particular keller segel model that yields chemotactic aggregation is considered it is shown using matched asymptotic methods that the considered blow up mechanism is stable more precisely small perturbations of the initial data produce a small shifting on the blow up time and the blow up point but not modification in the blow up mechanism

positive solutions for nonlinear first order periodic boundary value problems of impulsive dynamic equations on time scales in this paper existence criteria of positive solutions to a class of nonlinear first order periodic boundary value problems of impulsive dynamic equations on time scales are obtained the main tool used in this paper is the well known guo krasnoselskii fixed point theorem

the optimal temporal decay estimates for the fractional power dissipative equation in negative besov spaces in this paper we first generalize a new energy approach developed by guo and wang commun partial differ equations 37 2165 2208 2012 in the framework of homogeneous besov spaces for proving the optimal temporal decay rates of solutions to the fractional power dissipative equation then we apply this approach to the critical and supercritical surface quasi geostrophic equation and the critical keller segel system we show that certain weighted negative besov norm of solutions is preserved along time evolution and obtain the optimal temporal decay rates of the spatial derivatives of solutions by the fourier splitting approach and the interpolation techniques

the derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many particle systems the chemotaxis equations are a well known system of partial differential equations describing aggregation phenomena in biology in this paper they are rigorously derived from an interacting stochastic many particle system where the interaction between the particles is rescaled in a moderate way as population size tends to infinity the novelty of this result is that in all previous applications of this kind of limiting procedure the principal part of the system is assumed to fulfill an ellipticity condition which is not satisfied in our case new techniques which deal with this difficulty are presented

propagation of chaos for the keller segel equation over bounded domains abstract in this paper we rigorously justify the propagation of chaos for the parabolic elliptic keller segel equation over bounded convex domains the boundary condition under consideration is the no flux condition as intermediate steps we establish the well posedness of the associated stochastic equation as well as the well posedness of the keller segel equation for bounded weak solutions

l2 spaces and boundary value problems on time scales abstract in this paper we consider a second order sturm liouville type boundary value operator of the form l u p u q u on an arbitrary bounded time scale t for suitable functions p q together with suitable boundary conditions operators of this type on time scales have normally been considered in a setting involving banach spaces of continuous functions on t in this paper we introduce a space l 2 t of square integrable functions on t and sobolev type spaces h n t n 1 consisting of l 2 t functions with nth order generalised l 2 t type derivatives we prove some basic functional analytic results for these spaces and then formulate the operator l in this setting in particular we allow p h 1 t while q l 2 t this generalises the usual conditions that p c rd 1 t q c rd 0 t 2 we give some immediate applications of the functional analytic results to l such as positivity injectivity invertibility and compactness of the inverse we also construct a green s function for l the analogues of these results on real intervals are well known and are fundamental to the usual sturm liouville theory on such intervals

global existence and convergence rates to a chemotaxis fluids system with mixed boundary conditions abstract in this paper we investigate the large time behavior of strong solutions to a chemotaxis fluids system in an unbounded domain with mixed boundary conditions based on the anisotropic l p technique the elliptic estimates and stokes estimates we first establish the global existence of strong solution around the equilibrium state 0 c satn 0 with the help of the continuity arguments where c satn is the saturation value of oxygen inside the fluid then we use de giorgi s technique and energy method to show that such a solution will converge to 0 c satn 0 with an explicit convergence rate in the chemotaxis free case our assumptions and results are consistent with the experimental descriptions and the numerical analysis the novelty here consists of deriving some new elliptic estimates and stokes estimates and choosing a suitable weight in de giorgi s technique to deal with the mixed boundary conditions

the basin of attraction of the steady states for a chemotaxis model in mathbb r 2 with critical mass we consider the cauchy problem for a parabolic elliptic system in mathbb r 2 which is amathematical model of chemotaxis and also amodel of self attracting particles in the critical mass case we determine the basin of attraction of the steady states for the cauchy problem through a lyapunov functional

the thomas fermi von weizsacker theory of atoms and molecules we place the thomas fermi von weizsacker model of atoms on a firm mathematical footing we prove existence and uniqueness of solutions of the thomas fermi von weizsacker equation as well as the fact that they minimize the thomas fermi von weizsacker energy functional moreover we prove the existence of binding for two very dissimilar atoms in the frame of this model

on a parabolic elliptic system with gradient dependent chemotactic coefficient abstract we consider a second order pdes system of parabolic elliptic type with chemotactic terms the system describes the evolution of a biological species u moving towards a higher concentration of a chemical stimuli v in a bounded and open domain of r n in the system considered the chemotaxis sensitivity depends on the gradient of v i e the chemotaxis term has the following expression d i v u v p 2 v where is a positive constant and p satisfies p 1 if n 1 and p 1 n n 1 if n 2 we obtain uniform bounds in l and the existence of global in time solutions for the one dimensional case we prove the existence of infinitely many non constant steady states for p 1 2 for any positive and a given positive mass

stochastic particle approximation for measure valued solutions of the 2d keller segel system we construct an approximation to the measure valued global in time solutions to the patlak keller segel model in 2d based on systems of stochastic interacting particles the advantage of our approach is that it reproduces the well known dichotomy in the qualitative behavior of the system and moreover captures the solution even after the possible blow up events we present a numerical method based on this approach and show some numerical results moreover we make a first step toward the convergence analysis of our scheme by proving the convergence of the stochastic particle approximation for the keller segel model with a regularized interaction potential the proof is based on a bbgky like approach for the corresponding particle distribution function

solvability for a drift diffusion system with robin boundary conditions abstract this paper focuses on a drift diffusion system subjected to boundedly non dissipative robin boundary conditions a general existence result with large initial conditions is established by using suitable l 1 l 2 and trace estimates finally two examples coming from the corrosion and the self gravitation model are analyzed

uniqueness and long time asymptotic for the keller segel equation the parabolic elliptic case the present paper deals with the parabolic elliptic keller segel equation in the plane in the general framework of weak or free energy solutions associated to initial datum with finite mass m finite second moment and finite entropy the aim of the paper is threefold 1 we prove the uniqueness of the free energy solution on the maximal interval of existence 0 t with t in the case when m 8 and t 8 the proof uses a diperna lions renormalizing argument which makes it possible to get the optimal regularity as well as an estimate of the difference of two possible solutions in the critical l4 3 lebesgue norm similarly to the 2d vorticity navier stokes equation 2 we prove the immediate smoothing effect and in the case m 8 we prove the sobolev norm bound uniformly in time for the rescaled solution corresponding to the self similar variables 3 in the case m 8 we also prove the weighted l4 3 linearized stability of the self similar profile and then the universal optimal rate of convergence of the solution to the self similar profile the proof is mainly based on an argument of enlargement of the functional space for semigroup spectral gap

the continuous coagulation fragmentation equations with diffusion existence of global weak solutions to the continuous coagulation fragmentation equations with diffusion is investigated when the kinetic coefficients satisfy a detailed balance condition or the coagulation coefficient enjoys a monotonicity condition our approach relies on weak and strong compactness methods in l 1 in the spirit of the diperna lions theory for the boltzmann equation under the detailed balance condition the large time behaviour is also studied

infinite time aggregation for the critical patlak keller segel model in ℝ2 we analyze the two dimensional parabolic elliptic patlak keller segel model in the whole euclidean space r2 under the hypotheses of integrable initial data with finite second moment and entropy we first show local in time existence for any mass of free energy solutions namely weak solutions with some free energy estimates we also prove that the solution exists as long as the entropy is controlled from above the main result of the paper is to show the global existence of free energy solutions with initial data as before for the critical mass 8 actually we prove that solutions blow up as a delta dirac at the center of mass when t keeping constant their second moment at any time furthermore all moments larger than 2 blow up as t if initially bounded

integration on time scales in this paper we study the process of riemann and lebesgue integration on time scales the relationship of the riemann and lebesgue integrals is considered and a criterion for riemann integrability is established

convergence of the mass transport steepest descent scheme for the subcritical patlak keller segel model variational steepest descent approximation schemes for the modified patlak keller segel equation with a logarithmic interaction kernel in any dimension are considered we prove the convergence of the suitably interpolated in time implicit euler scheme defined in terms of the euclidean wasserstein distance associated with this equation for subcritical masses as a consequence we recover the recent result about the global in time existence of weak solutions to the modified patlak keller segel equation for the logarithmic interaction kernel in any dimension in the subcritical case moreover we show how this method performs numerically in dimension one in this particular case this numerical scheme corresponds to a standard implicit euler method for the pseudoinverse of the cumulative distribution function we demonstrate its capabilities to reproduce the blow up of solutions for supercritical masses easily without the need of mesh refinement

well posedness for chemotaxis dynamics with nonlinear cell diffusion abstract this paper investigates the well posedness of a reaction diffusion system of chemotaxis type with a nonlinear diffusion coefficient and a dynamics growth death of the cell population b and a stationary equation for the chemoattractant c with respect to other works in which a nonlinear diffusion for cells has been considered we treat here two distinct cases for this diffusion coefficient the first in which it is a positive bounded function on r and the other in which it may display a singularity at a finite value of the cell density essentially the latter model is new and describes the saturation of the cell population in the neighborhood of a critical value for its diffusion coefficient the chemotactic sensitivity function is supposed to depend both on the cell and chemoattractant densities for homogeneous neumann boundary conditions for the cell population and chemoattractant we prove the existence of a local in time solution when the l 2 norm of the initial datum b 0 is sufficiently small and compute the maximum time interval for which the solution is bounded and smooth under a stronger assumption related to the chemotactic sensitivity we show that there exists a global in time solution for arbitrarily large initial data in a case when the diffusion coefficient is singular we focus on a model expressed by a variational inequality describing the saturation of the cell population at the blowing up diffusion value here the proof requires the study of an intermediate problem with robin boundary conditions which may be interesting by itself in all situations uniqueness follows on a time interval included not necessarily strictly in that of the solution existence under sufficient conditions

on patlak keller segel system for several populations a gradient flow approach abstract 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

on spectra of linearized operators for keller segel models of chemotaxis abstract we consider the phenomenon of collapse in the critical keller segel equation ks which models chemotactic aggregation of micro organisms underlying many social activities e g fruiting body development and biofilm formation also ks describes the collapse of a gas of self gravitating brownian particles we find the fluctuation spectrum around the collapsing family of steady states for these equations which is instrumental in the derivation of the critical collapse law to this end we develop a rigorous version of the method of matched asymptotics for the spectral analysis of a class of second order differential operators containing the linearized keller segel operators and as we argue linearized operators appearing in nonlinear evolution problems we explain how the results we obtain are used to derive the critical collapse law as well as for proving its stability

on the stability of critical chemotactic aggregation we consider the two dimensional parabolic elliptic keller segel model of chemotactic aggregation for radially symmetric initial data we show the existence of a stable mechanism of singularity formation and obtain a complete description of the associated aggregation process

global in time weak measure solutions and finite time aggregation for nonlocal interaction equations in this paper we provide a well posedness theory for weak measure solutions of the cauchy problem for a family of nonlocal interaction equations these equations are continuum models for interacting particle systems with attractive repulsive pairwise interaction potentials the main phenomenon of interest is that even with smooth initial data the solutions can concentrate mass in finite time we develop an existence theory that enables one to go beyond the blow up time in classical norms and allows for solutions to form atomic parts of the measure in finite time the weak measure solutions are shown to be unique and exist globally in time moreover in the case of sufficiently attractive potentials we show the finite time total collapse of the solution onto a single point for compactly supported initial measures our approach is based on the theory of gradient flows in the space of probability measures endowed with the wasserstein metric in addition to classical tools we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations

non trivial ω limit sets and oscillating solutions in a chemotaxis model in r2 with critical mass abstract this paper studies the cauchy problem for a parabolic elliptic system in r 2 modeling chemotaxis as well as self attracting particles in the critical mass case the fine dynamics of the model is ascertained in terms of the structure of the underlying limit sets according to the results of this paper any nonnegative radially symmetric bounded solution either stabilizes to a steady state as t or oscillates between two steady states moreover a rather general class of nonnegative initial data not necessarily radially symmetric for which the associated solutions exhibit a complex oscillatory behavior is constructed their limit sets consist of a nontrivial topological continuum of steady states besides the technical difficulties inherent to the lack of compactness of the resolvent operators one has to add the challenge that the problem is utterly non local consequently thought the basic ideas on the foundations of this paper might be considered classical most of the proofs throughout are extremely sophisticated and absolutely new in their full generality

singularity patterns in a chemotaxis model the authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time such a finite time blow up of solutions implies chemotactic collapse namely concentration of species to form sporae the model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity which has the form ut u u v 0 v u 1 on r2 where is an open disc with no flux homogeneous neumann boundary conditions the initial conditions are continuous functions u x 0 u0 x 0 v x 0 v0 x 0 for x under these conditions the authors prove there exists a radially symmetric solution u r t which blows up at r 0 t t a specific description of such a solution is presented the authors also discuss the strong similarity between the chemotactic model they study and the classical stefan problem

exponential speed of uniform convergence of the cell density toward equilibrium for subcritical mass in a patlak keller segel model 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 rn exhibiting a critical mass phenomenon for n 2 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 we stress that this covers in particular the classical keller segel system with n 2 well known for its critical mass 8 and that the result improves on the known results even for this most studied problem a key tool is an associated one dimensional degenerate parabolic problem the proof exploits its formal gradient flow structure ut f u t on an infinite dimensional riemannian manifold in particular we show a new hardy type inequality equivalent to the strict convexity of f at any steady state of subcritical mass which heuristically explains the exponential speed of convergence

metrics for probability distributions and the trend to equilibrium for solutions of the boltzmann equation this paper deals with the trend to equilibrium of solutions to the spacehomogeneous boltzmann equation for maxwellian molecules with angular cutoff as well as with infinite range forces the solutions are considered as densities of probability distributions the tanaka functional is a metric for the space of probability distributions which has previously been used in connection with the boltzmann equation our main result is that if the initial distribution possesses moments of order 2 e then the convergence to equilibrium in his metric is exponential in time in the proof we study the relation between several metrics for spaces of probability distributions and relate this to the boltzmann equation by proving that the fourier transformed solutions are at least as regular as the fourier transform of the initial data this is also used to prove that even if the initial data only possess a second moment then v r f v t v 2 dv 0 asr and this convergence is uniform in time

an aggregation equation with a nonlocal flux abstract in this paper we study an aggregation equation with a general nonlocal flux we study the local well posedness and some conditions ensuring global existence we are also interested in the differences arising when the nonlinearity in the flux changes thus we perform some numerics corresponding to different convexities for the nonlinearity in the equation

non existence of finite time self similar solutions of the keller segel system in the scaling invariant class let us consider the problem whether there does exist a finite time self similar solution of the backward type to the semilinear keller segel system in the case of parabolic elliptic type for n 3 we show that there is no such a solution with a finite mass in the scaling invariant class on the other hand in the case of parabolic parabolic type for n 2 non existence of finite time self similar solutions is proved in a larger class of a finite mass with some local bounds

nonlocal interactions by repulsive attractive potentials radial ins stability abstract we investigate nonlocal interaction equations with repulsive attractive radial potentials such equations describe the evolution of a continuum density of particles in which they repulse resp attract each other in the short resp long range we prove that under some conditions on the potential radially symmetric solutions converge exponentially fast in some transport distance toward a spherical shell stationary state otherwise we prove that it is not possible for a radially symmetric solution to converge weakly toward the spherical shell stationary state we also investigate under which condition it is possible for a non radially symmetric solution to converge toward a singular stationary state supported on a general hypersurface finally we provide a detailed analysis of the specific case of the repulsive attractive power law potential as well as numerical results

necessary conditions of optimality for a class of optimal control problems on time scales a class of optimal control problems of a system governed by linear dynamic equations on time scales with quadratic cost functional is considered by the lebesgue d integral theory and the sobolev type space h 1 on time scales the weak solution of linear dynamic equations on time scales for both initial value problem and backward problem are introduced therefore the necessary conditions of optimality are presented some typical examples are given for demonstration

preventing blow up in a chemotaxis model abstract this paper deals with a chemotactic model of keller segel type the main feature of the keller segel model is the possibility of blow up of solutions in a finite time to eliminate the possibility of blow up a modified version of keller segel model is introduced the blow up control relies on the presence of a pressure function which increases faster than a logarithm for high enough cells densities for such a pressure function the solutions cannot blow up in a finite time some other conditions are introduced to ensure the global boundedness