sharp estimates for solutions of multi bubbles in compact riemann surfaces in this paper we consider a sequence of multibubble solutions u k of the equation 0 1 0 u k k he uk f m he u k d 1 0 in m where h is a c 2 positive function in a compact riemann surface m and k is a constant satisfying lim k k 8m for some positive integer m 1 we prove among other things that k 8m 2 m m j 1h 1 p k j 0 log h p k j 8m 2k p k j k j e k j o e k j where p k j are centers of the bubbles of u k and k j are the local maxima of u k after adding a constant this yields a uniform bound of solutions as k converges to 8m from below provided that 0 log h p k j 8m 2k p k j 0 it generalizes a previous result due to ding jost li and wang 18 and nolasco and tarantello 31 which says that any sequence of minimizers u k is uniformly bounded if k 0 for any maximum point p of the sum of 2 log h and the regular part of the green function where k is the gaussian curvature of m the analytic work of this paper is the first step toward computing the topological degree of 0 1 which was initiated by li 24

on liouville systems at critical parameters part 1 one bubble abstract in this paper we consider bubbling solutions to the general liouville system 0 1 g u i k j 1 n a i j j k h j e u j k h j e u j k 1 0 in m i 1 n n 2 where m g is a riemann surface and a a i j n n is a constant non negative matrix and j k j as k among other things we prove the following sharp estimates 1 the location of the blowup point 2 the convergence rate of j k j j 1 n these results are of fundamental importance for constructing bubbling solutions it is interesting to compare the difference between the general liouville system and the su 3 toda system on estimates 1 and 2

a variational analysis of the toda system on compact surfaces in this paper we consider the following toda system of equations on a compact surface we will give existence results by using variational methods in a noncoercive case a key tool in our analysis is a new moser trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of the two components u1 and u2 2012 wiley periodicals inc

statistical mechanics of classical particles with logarithmic interactions the inhomogeneous mean field thermodynamic limit is constructed and evaluated for both the canonical thermodynamic functions and the states of systems of classical point particles with logarithmic interactions in two space dimensions the results apply to various physical models of translation invariant plasmas gravitating systems as well as to planar fluid vortex motion for attractive interactions a critical behavior occurs which can be classified as an extreme case of a second order phase transition to include in particular attractive interactions a new inequality for configurational integrals is derived from the arithmetic geometric mean inequality the method developed in this paper is easily seen to apply as well to systems with fairly general interactions in all space dimensions in addition it also provides us with a new proof of the trudinger moser inequality known from differential geometry in its sharp form

su 3 chern simons vortex theory and toda systems motivated by the study of the asymptotic properties of non topological condensates in the non abelian chern simons vortex theory see 26 we analyze the su 3 toda system p 1 2 zj j e 2i 1kijzi me i 12kijzi 1 on m mzj 0j 1 2 where m r2 z2 k kij 2 1 12 is the su 3 cartan matrix and j are positive parameters we study the variational problem associated with the system p 1 2 in a range of parameters where the trivial solution is a strict local minimum and the corresponding sobolev type inequality fails to apply in this situation a lack of compactness may occur due to concentration phenomena nonetheless we are able to establish the existence of a non trivial solution for p 1 2 which is not a minimizer

uniqueness of topological solutions and the structure of solutions for the chern simons system with two higgs particles the existence of topological solutions for the chern simons equation with two higgs particles has been proved by lin ponce and yang 16 however both the uniqueness problem and the existence of non topological solutions have been left open in this paper we consider the case of one vortex at origin among others we prove the uniqueness of topological solutions and give a complete study of the radial solutions in particular the existence of some non topological solutions

on conformal deformations of metrics onsn abstract on s n there is a naturally metric defined n th order conformal invariant operator p n associated with this operator is a so called q curvature quantity when two metrics are pointwise conformally related their associated operators together with their q curvatures satisfy the natural differential equations this paper is devoted to the question of which function can be a q curvature candidate this is the so called prescribing q curvature problem our main result is that if q is positive nondegenerate and the naturally defined mapping associated with q has nonzero degree then our problem has a solution this is the natural generalization of prescribing gaussian curvature on s 2 into s n

the logarithmic hls inequality for systems on compact manifolds abstract we prove an optimal logarithmic hardy littlewood sobolev inequality for systems on compact m dimensional riemannian manifolds for any m 2 we show that a special case of the inequality involving only two functions implies the general case by using an argument from the theory of linear programing

analysis of the poisson nernst planck equation in a ball for modeling the voltage current relation in neurobiological microdomains abstract the electro diffusion of ions is often described by the poisson nernst planck pnp equations which couple nonlinearly the charge concentration and the electric potential this model is used among others to describe the motion of ions in neuronal micro compartments it remains at this time an open question how to determine the relaxation and the steady state distribution of voltage when an initial charge of ions is injected into a domain bounded by an impermeable dielectric membrane the purpose of this paper is to construct an asymptotic approximation to the solution of the stationary pnp equations in a d dimensional ball d 1 2 3 in the limit of large total charge in this geometry the pnp system reduces to the liouville gelfand bratu lgb equation with the difference that the boundary condition is neumann not dirichlet and there is a minus sign in the exponent of the exponential term the entire boundary is impermeable to ions and the electric field satisfies the compatibility condition of poisson s equation these differences replace attraction by repulsion in the lgb equation thus completely changing the solution we find that the voltage is maximal in the center and decreases toward the boundary we also find that the potential drop between the center and the surface increases logarithmically in the total number of charges and not linearly as in classical capacitance theory this logarithmic singularity is obtained for d 3 from an asymptotic argument and cannot be derived from the analysis of the phase portrait these results are used to derive the relation between the outward current and the voltage in a dendritic spine which is idealized as a dielectric sphere connected smoothly to the nerve axon by a narrow neck this is a fundamental microdomain involved in neuronal communication we compute the escape rate of an ion from the steady density in a ball which models a neuronal spine head to a small absorbing window in the sphere we predict that the current is defined by the narrow neck that is connected to the sphere by a small absorbing window as suggested by the narrow escape theory while voltage is controlled by the pnp equations independently of the neck

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

a topological degree counting for some liouville systems of mean field type let a a ij n n be an invertible matrix and a 1 a ij n n be the inverse of a in this paper we consider the generalized liouville system n 0 1 g u i n a ij j h j e u j h j e u j 1 0 in m j 1 where 0 h j c 1 m and j and prove that under the assumptions of h 1 and h 2 see introduction the leray schauder degree of 0 1 is equal to m 1 m n n if 1 n satisfies 8 n n n i 1 i 1 i j n a ij i j 8 n 1 n n i 1 i equation 0 1 is a natural generalization of the classic liouville equation and is the euler lagrangian equation of the nonlinear function u 1 2 m 1 i j n a ij g u i g u j n n i 1 m i u i n n i 1 i log m h i e u i the liouville system 0 1 has arisen in many different research areas in mathematics and physics our counting formulas are the first result in degree theory for liouville systems

moser trudinger inequalities for singular liouville systems in this paper we prove a moser trudinger inequality for the euler lagrange functional of general singular liouville systems on a compact surface we characterize the values of the parameters which yield coercivity for the functional hence the existence of energy minimizing solutions for the system and we give necessary conditions for boundedness from below we also provide a sharp inequality under assuming the coefficients of the system to be non positive outside the diagonal the proofs use a concentration compactness alternative poho aev type identities and blow up analysis

wave equations associated to liouville systems and constant mean curvature equations abstract we study the wave analog of the liouville equation for constant gauss curvature on s 2 we show the global existence in time for the sub critical equation with data in h 1 s 2 l 2 s 2 for the super critical and the critical equation we show the global existence in time assuming the data is symmetric with respect to antipodal points on s 2 use is made of the moser trudinger inequalities for proving this result for general data without symmetry assumptions and for the critical equation we establish that if there is a blow up then it will occur at a single point of s 2 we also consider the wave analog of the constant mean curvature equation in the plateau formalism this is a system we establish finite time blow up if the energy of the initial data is larger than 8 and with an additional subsidiary assumption the threshold energy of 8 arises because it is the energy of the basic bubble discovered by brezis and coron

classification and sharp range of flux pairs for radial solutions to a coupled system abstract in this paper we study a coupled system of two nonlinear partial differential equations in the plane which is related to liouville equations non abelian higgs bps vortex equations or two higgs electroweak model with singularities at the origin in addition to deriving the uniqueness of the so called topological solutions we also clarify the structure of all types of solutions including the blow up ones under various conditions on coefficients and parameters appearing in the system furthermore the sharp range of flux pairs associated with specific types of solutions is considered as well

mass quantization and minimax solutions for neri s mean field equation in 2d turbulence we study the mean field equation derived by neri in the context of the statistical mechanics description of 2d turbulence under a stochastic assumption on the vortex circulations the corresponding mathematical problem is a nonlocal semilinear elliptic equation with exponential type nonlinearity containing a probability measure p m 1 1 which describes the distribution of the vortex circulations unlike the more investigated deterministic version we prove that neri s equation may be viewed as a perturbation of the widely analyzed standard mean field equation obtained by taking p 1 in particular in the physically relevant case where p is non negatively supported and p 1 0 we prove the mass quantization for blow up sequences we apply this result to construct minimax type solutions on bounded domains in r 2 and on compact 2 manifolds without boundary

on the solutions of liouville systems for some prescribed m 0 together with the dirichlet boundary condition u 0 on 0 the liouville equation finds applications in various fields of physics and mathematics to name a few it represents the newtonian potential of a cluster of self gravitating mass distribution where a 0 1 6 36 37 the electric potential induced by charge carriers in the theory of electrolytes where a 0 31 and maximal entropy solutions of the incompressible euler equation 23 9 in the field of differential geometry it stands for the problem of finding a metric corresponding to a prescribed gaussian curvature 3 10 the system analog of l takes the form

qualitative properties of solutions to some nonlinear elliptic equations in r 2 o introduction elliptic equations in this paper we investigate properties of the solutions to the au r x e u x x r 2 for functions r x which are positive near infinity equations of this kind arise from a variety of situations such as from prescribing gaussian curvature in geometry i 2 and from combustion theory in physics 3 recently a series ofworks have been sought to understand the existence and the qualitative properties of the solutions of ni i 5 and ni cheng 4 1 considered the case where r x is nonpositive mcowen 6 1 and aviles 7 investigated the situation where r x 0 in some order as ixl in our previous paper 1 we consider a special case where r is a constant we proved that the solutions are radially symmetric and hence classified all the solutions in this paper we consider more general functions r x first we obtain the asymptotic behavior of the solution near infinity consequently we prove that all the solutions satisfy an identity which is somewhat of a generalization of the well known kazdan warner condition finally using the asymptotic behavior together with the further development of the method employed in our previous paper i 1 we show that all the solutions are radially symmetric provided r is radially symmetric and nonincreasing this part can be viewed as the completion of 1 throughout this paper we assume that the function r x is positive near infinity in 1 we study the asymptotic behavior of the solution u x of let fl 1 2n r2 r x e dx under some appropriate conditions we show that the solutions approach

existence and uniqueness for mean field equations on multiply connected domains at the critical parameter we consider the mean field equation begin aligned left begin array l quad l vardelta u rho frac e u int varomega e u 0 hbox in varomega u 0 hbox on partial varomega end array right end aligned 1 where varomega subset mathbb r 2 is an open and bounded domain of class c 1 in his 1992 paper suzuki proved that if varomega is a simply connected domain then eq 1 admits a unique solution for rho in 0 8 pi this result for varomega a simply connected domain has been extended to the case rho 8 pi by chang chen and the second author however the uniqueness result for varomega a multiply connected domain has remained a long standing open problem which we solve positively here for rho in 0 8 pi to obtain this result we need a new version of the classical bol s inequality suitable to be applied on multiply connected domains our second main concern is the existence of solutions for 1 when rho 8 pi we obtain a necessary and sufficient condition for the solvability of the mean field equation at rho 8 pi which is expressed in terms of the robin s function gamma for varomega for example if eq 1 has no solution at rho 8 pi then gamma has a unique nondegenerate maximum point as a by product of our results we solve the long standing open problem of the equivalence of canonical and microcanonical ensembles in the onsager s statistical description of two dimensional turbulence on multiply connected domains

marginalization and linear opinion pools abstract suppose a decision maker has asked a group of experts to assess subjective probability distributions over some space and he wishes to find a consensus probability distribution as a function of these the assumption that finding the consensus distribution commutes with marginalization of the distributions implies that the consensus must be found using a linear opinion pool weighted average provided the space being considered contains three or more points some of the consequences of this result are discussed

homogeneous quantization and multiplicities of group representations let b be a compact manifold a cone over b is a principal r bundle x with base b let a x a x be the mapping associated with the action of a r on x x is called a symplectic cone if it possesses a symplectic form such that a a a compact lie group g is said to act in a homogeneous fashion on x if it acts on x in such a way that both and the principal bundle structure are preserved it is known that to such an action one can associate in a fairly canonical way a representation of g on a hilbert space h see 3 in this paper we propose a symplectic recipe for the multiplicities with which h decomposes into g irreducibles and show that this recipe is correct generically

convergence rate location and z2 condition for fully bubbling solutions to su n 1 toda systems abstract it is well known that the study of su n 1 toda systems is important not only to chern simons models in physics but also to the understanding of holomorphic curves harmonic sequences or harmonic maps from riemann surfaces to c p n one major goal in the study of su n 1 toda system on riemann surfaces is to completely understand the asymptotic behavior of fully bubbling solutions in this article we use a unified approach to study fully bubbling solutions to general su n 1 toda systems and we prove three major sharp estimates important for constructing bubbling solutions the closeness of blowup solutions to entire solutions the location of blowup points and a z 2 condition

a compactness result for periodic multivortices in the electroweak theory we derive a priori uniform bounds for solutions of an elliptic system of liouville type equations first analyzed by j spruck and y yang comm math phys 144 1992 1 yielding periodic multivortices in the classical electroweak theory of glashow salam weinberg our proof is based on a concentration quantization result in the same spirit of brezis merle comm partial differential equations 16 8 9 1991 1223 and li shafrir indiana univ math j 43 4 1994 1255 for mean field equations on riemannian compact 2 manifolds

a special class of stationary flows for two dimensional euler equations a statistical mechanics description we consider the canonical gibbs measure associated to an vortex system in a bounded domain at inverse temperature widetilde beta and prove that in the limitn widetilde beta n n 1 where 8 here denotes the vorticity intensity of each vortex the one particle distribution function n nx x converges to a superposition of solutions of the following mean field equation left begin array 20 c varrho beta x frac e beta psi mathop smallint limits lambda e beta psi delta psi varrho beta in lambda psi partial lambda 0 end array right

sharp conditions for blowup of solutions of a chemotactical model for two species in r2 abstract we consider a model system of keller segel type for the evolution of two species in the whole space r 2 which are driven by chemotaxis and diffusion it is well known that this problem admits global and blowup solutions we show that there exists a sharp condition which allows to distinguish global and blowup solutions in the radially symmetric case more precisely let m and n be the total masses of the species then there exists a critical curve in the m n plane such that the solution blows up if and only if m n is above this gives an answer to a question raised by conca et al 2011 in 8 we also study the asymptotic behaviour of global solutions in the subcritical case showing that they are asymptotically self similar

random rotations characters and random walks on so n we analyze a random walk on the orthogonal group so n given by repeatedly rotating by a fixed angle through randomly chosen planes of r we derive estimates of the rate at which this random walk will converge to haar measure on so n using character theory and the upper bound lemma of diaconis and shashahani in some cases we are able to establish the existence of a cut off phenomenon for the random walk this is the first interesting such result on a non finite group

a necessary and sufficient condition for reaching a consensus using degroot s method abstract degroot 1974 proposed a model in which a group of k individuals might reach a consensus on a common subjective probability distribution for an unknown parameter this paper presents a necessary and sufficient condition under which a consensus will be reached by using degroot s method this work corrects an incorrect statement in the original paper about the conditions needed for a consensus to be reached the condition for a consensus to be reached is straightforward to check and yields the value of the consensus if one is reached

the cut off phenomenon for random reflections for many random walks on sufficiently large finite groups the so called cut off phenomenon occurs roughly stated there exists a number k 0 depending on the size of the group such that k 0 steps are necessary and sufficient for the random walk to closely approximate uniformity as a first example on a continuous group rosenthal recently proved the occurrence of this cut off phenomenon for a specific random walk on so n here we present and for the case of o n prove results for random walks on o n u n and sp n where the one step distribution is a suitable probability measure concentrated on reflections in all three cases the cut off phenomenon occurs at k 0 1 2n log n

semiclassical spectra of gauge fields abstract we study the asymptotic behavior of the eigenvalues of the schrodinger operator with a vector potential on a compact manifold as planck s constant tends to zero we obtain estimates in terms of periodic trajectories of wong s flow which are uniform in the charge parameter

classification and nondegeneracy of su n 1 toda system with singular sources we consider the following toda system open image in new window where i 1 0 is dirac measure at 0 and the coefficients a ij form the standard tri diagonal cartan matrix in this paper i we completely classify the solutions and obtain the quantization result sum j 1 n a ij int mathbb r 2 e u j dx 4 pi 2 gamma i gamma n 1 i quad forall 1 leq i leq n this generalizes the classification result by jost and wang for i 0 forall 1 leq i leq n ii we prove that if i i 1 j for all 1 i j n then any solution u i is radially symmetric w r t 0 iii we prove that the linearized equation at any solution is non degenerate these are fundamental results in order to understand the bubbling behavior of the toda system

gauss polynomials and the rotation algebra newton s binomial theorem is extended to an interesting noncommutative setting as follows if in a ring ba ab with commuting witha andb then the generalized binomial coefficient left begin array 20 c n k end array right r arising in the expansion left a b right n sum limits k 0 n left begin array 20 c n k end array right gamma a n k b k resulting from these relations is equal to the value at of the gaussian polynomial left begin array 20 c n k end array right frac left n right left k right left n k right where m 1 x m 1 x m 1 1 x this is of course known in the case 1 from this it is deduced that in the universal c algebraa gq generated by unitariesu andv such thatvu e 2 i uv the spectrum of the self adjoint element u v u v has all the gaps that have been predicted to exist provided that either is rational or is a liouville number in the latter case the gaps are labelled in the natural way viak theory by the set of all non zero integers and the spectrum is a cantor set

supercritical mean field equations on convex domains and the onsager s statistical description of two dimensional turbulence we are motivated by the study of the microcanonical variational principle within onsager s description of two dimensional turbulence in the range of energies where the equivalence of statistical ensembles fails we obtain sufficient conditions for the existence and multiplicity of solutions for the corresponding mean field equation on convex and thin enough domains in the supercritical with respect to the moser trudinger inequality regime this is a brand new achievement since existence results in the supercritical region were previously known only on multiply connected domains we then study the structure of these solutions by the analysis of their linearized problems and we also obtain a new uniqueness result for solutions of the mean field equation on thin domains whose energy is uniformly bounded from above finally we evaluate the asymptotic expansion of those solutions with respect to the thinning parameter and combining it with all the results obtained so far we solve the microcanonical variational principle in a small range of supercritical energies where the entropy is shown to be concave

a sup inf inequality for some nonlinear elliptic equations involving exponential nonlinearities let u be a solution of the equation u v x eu in a domain r2 where 0 a v b and v is lipschitz continuous we prove that sup u can be controlled in terms of inf u more precisely supku inf u c a b k v l for any compact subset k this extends an earlier result of shafrir who obtained a similar conclusion when v 1

blow up analysis for a cosmic strings equation abstract in this paper we develop a blow up analysis for solutions of an elliptic pde of liouville type over the plane such solutions describe the behavior of cosmic strings parallel in a given direction for a w boson model coupled with einstein s equation we show how the blow up behavior of the solutions is characterized according to the physical parameters involved by new and surprising phenomena for example in some cases after a suitable scaling the blow up profile of the solution is described in terms of an equations that bares a geometrical meaning in the context of the uniformization of the riemann sphere with conical singularities

on steady distributions of self attracting clusters under friction and fluctuations we consider the vlasov fokker planek equation with a newtonian attracting potential and study its stationary solutions given by the generalized lane emden equation in a two dimensional domain we obtain the existence of a critical mass beyond which the system may admit a gravitational collapse for a one dimensional model we prove some results on existence uniqueness stability and symmetry breaking of stationary solutions

analytic aspects of the toda system i a moser trudinger inequality in this paper we analyze solutions of the open toda system and establish an optimal moser trudinger type inequality for this system let be a closed surface with area 1 and k aij n n the cartan matrix for su n 1 i e we show that has a lower bound in h1 n if and only if this inequality is optimal as a direct consequence if mj for 4 for j 1 2 n m has a minimizer u that satisfies 2001 john wiley sons inc

semi classical limit for random walks let g u be a discrete symmetric random walk on a compact lie group g with step distribution and let t be the associated transition operator on l2 g the irreducibles vp of the left regular representation of g on l2 g are finite dimensional invariant subspaces for t and the spectrum of t is the union of the sub spectra a t 1vp on the irreducibles which consist of real eigenvalues apl ap dim vp our main result is an asymptotic expansion for the spectral measures dim vp m dim vp j along rays of representations in a positive weyl chamber t i e for sequences of representations kp k c n with k oo as a corollary we obtain some estimates on the spectral radius of the random walk we also analyse the fine structure of the spectrum for certain random walks on u n for which tm is essentially a direct sum of harper operators

symmetry properties in systems of semilinear elliptic equations abstract we investigate symmetry properties of solutions of systems of semilinear elliptic equations the two main tools we use consist of the maximum principle and the device of moving parallel planes to a critical position and then showing that the solution is symmetric about the limiting plane an application to the belousov zhabotinskii chemical reaction is given

symmetry results for semilinear elliptic systems in the whole space in this paper we obtain symmetry results of gidas ni nirenberg type for cooperative nonlinear elliptic systems in the whole space

nonradial entire solutions for liouville systems abstract we consider the following system of liouville equations u 1 2 e u 1 e u 2 in r 2 u 2 e u 1 2 e u 2 in r 2 r 2 e u 1 r 2 e u 2 we show the existence of at least n n 3 global branches of nonradial solutions bifurcating from u 1 x u 2 x u x log 64 2 8 x 2 2 at the values 2 n 2 n 2 n 2 n 2 for any n n

the conformal plate buckling equation the linear equation 2 u 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over r 2 has the particularly interesting nonlinear generalization g 2 u 1 where ag e 2u is the laplace beltrami operator for the metric g e 2u g 0 with g 0 the standard euclidean metric on r 2 this conformal elliptic pde of fourth order is equivalent to the nonlinear system of elliptic pdes of second order u x k g x exp 2u x 0 and k g x exp 2u x 0 with x r 2 describing a conformally flat surface with a gauss curvature function k g that is generated self consistently through the metric s conformal factor we study this conformal plate buckling equation under the hypotheses of finite integral curvature f kg exp 2u dx k finite area exp 2u dx a and the mild compactness condition k l 1 b 1 y uniformly with respect to y r 2 we show that asymptotically for x all solutions behave like u x 2 ln x c o 1 and k x 2 ln x c o 1 with 2 4 and 2 4 we also show that for each k 2 4 there exists a k and a radially symmetric solution pair u k satisfying k u k and max k k which is unique modulo translation of the origin and scaling of x coupled with a translation of u

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

chemotactic systems in the presence of conflicts a new functional inequality abstract the evolution of a chemotactic system involving a population of cells attracted to self produced chemicals is described by the keller segel system in dimension 2 this system demonstrates a balance between the spreading effect of diffusion and the concentration due to self attraction as a result there exists a critical mass i e total cell s population above which the solution of this system collapses in a finite time while below this critical mass there is global existence in time in particular subcritical mass leads under certain additional conditions to the existence of steady states corresponding to the solution of an elliptic liouville equation the existence of this critical mass is related to a functional inequality known as the moser trudinger inequality an extension of the keller segel model to several cells populations was considered before in the literature here we review some of these results and in particular consider the case of conflict between two populations that is when population one attracts population two while at the same time population two repels population one this assumption leads to a new functional inequality which generalizes the moser trudinger inequality as an application of this inequality we derive sufficient conditions for the existence of steady states corresponding to solutions of an elliptic liouville system