strong positivity of solutions to parabolic and elliptic equations on nonsmooth domains nonnegativity of weak solutions of parabolic and elliptic equations on nonsmooth domains is established strong positivity of weak solutions to elliptic equations is proved via a boundary weak harnack inequality

exact solution of a 1d asymmetric exclusion model using a matrix formulation several recent works have shown that the one dimensional fully asymmetric exclusion model which describes a system of particles hopping in a preferred direction with hard core interactions can be solved exactly in the case of open boundaries here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices with this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules they obtain several explicit forms for these non commuting matrices which are in the general case infinite dimensional their approach allows exact expressions to be derived for the current and density profiles finally they discuss briefly two possible generalizations of their results the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles

global dynamics above the ground state energy for the focusing nonlinear klein gordon equation abstract the analysis of global dynamics of nonlinear dispersive equations has a long history starting from small solutions in this paper we study the focusing cubic nonlinear klein gordon equation in r 3 with large radial data in the energy space this equation admits a unique positive stationary solution q called the ground state in 1975 payne and sattinger showed that solutions u t with energy e u u strictly below that of the ground state are divided into two classes depending on a suitable functional k u if k u 0 then one has finite time blow up if k u 0 global existence moreover these sets are invariant under the flow recently ibrahim masmoudi and the first author 22 improved this result by establishing scattering to zero for k u 0 by means of a variant of the kenig merle method kenig and merle 2006 2008 25 26 in this paper we go slightly beyond the ground state energy and we give a complete description of the evolution in that case for example in a small neighborhood of q one encounters the following trichotomy on one side of a center stable manifold one has finite time blow up for t 0 on the other side scattering to zero and on the manifold itself one has scattering to q both as t in total the class of data with energy at most slightly above that of q is divided into nine disjoint non empty sets each displaying different asymptotic behavior as t which includes solutions blowing up in one time direction and scattering to zero on the other the analogue of the solutions found by duyckaerts and merle 2009 2008 13 14 for the energy critical wave and schrodinger equations appear here as the unique one dimensional stable unstable manifolds approaching q exponentially as t or t respectively the main technical ingredient in our proof is a one pass theorem which excludes the existence of almost homoclinic orbits between q as well as q and almost heteroclinic orbits connecting q with q in a companion paper nakanishi and schlag 2010 31 we establish analogous properties for the nls equation

phase transitions in an exactly soluble one dimensional exclusion process we consider an exclusion process with particles injected with rate at the origin and removed with rate at the right boundary of a one dimensional chain of sites the particles are allowed to hop onto unoccupied sites to the right only for the special case of 1 the model was solved previously by derridaet al here we extend the solution to general the phase diagram obtained from our exact solution differs from the one predicted by the mean field approximation

harnack inequality and exponential separation for oblique derivative problems on lipschitz domains abstract we consider the oblique derivative problem for linear nonautonomous second order parabolic equations with bounded measurable coefficients on bounded lipschitz cylinders we derive an optimal elliptic type harnack inequality for positive solutions of this problem and use it to show that each positive solution exponentially dominates any solution which changes sign for all times we show several nontrivial applications of both the exponential estimate and the derived harnack inequality

renormalization and blow up for charge one equivariant critical wave maps we prove the existence of equivariant finite time blow up solutions for the wave map problem from 2 1 s 2 of the form u t r q lambda t r mathcal r t r where u is the polar angle on the sphere q r 2 arctan r is the ground state harmonic map t t 1 and mathcal r t r is a radiative error with local energy going to zero as t 0 the number nu frac 1 2 can be prescribed arbitrarily this is accomplished by first renormalizing the blow up profile followed by a perturbative analysis

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

on the formation of singularities in the critical o 3 sigma model we study the phenomena of energy concentration for the critical o 3 sigma model also known as the wave map flow from 2 1 minkowski space into the sphere s 2 we establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities the construction and analysis are done in the context of the k equivariant symmetry reduction and we restrict to maps with homotopy class k 4 the concentration mechanism we uncover is essentially due to a resonant self focusing shrinking of a corresponding harmonic map we show that the phenomenon is generic e g in certain sobolev spaces in that it persists under small perturbations of initial data while the resulting blowup is bounded by a log modified self similar asymptotic

the application of the yang lee theory to study a phase transition in a non equilibrium system we study a phase transition in a non equilibrium model first introduced in ref 5 using the yang lee description of equilibrium phase transitions in terms of both canonical and grand canonical partition function zeros the model consists of two different classes of particles hopping in opposite directions on a ring on the complex plane of the diffusion rate we find two regions of analyticity for the canonical partition function of this model which can be identified by two different phases the exact expressions for both distribution of the canonical partition function zeros and their density are obtained in the thermodynamic limit the fact that the model undergoes a second order phase transition at the critical point is confirmed we have also obtained the grand canonical partition function zeros of our model numerically the similarities between the phase transition in this model and the bose einstein condensation has also been studied

pointwise estimates for oblique derivative problems in nonsmooth domains abstract there is a long history of studying boundary value problems for elliptic and parabolic differential equations when the equation is posed in a domain with sufficiently smooth boundary in this paper we prove some pointwise estimates and regularity results for solutions when a directional derivative is prescribed on the boundary which satisfies an interior cone condition we consider very weak hypotheses on the data of the problem in particular we consider an optimal relation between the interior cone and the direction of the prescribed directional derivative and we assume very little smoothness of the coefficients in the equation and the boundary condition

blowup dynamics for smooth data equivariant solutions to the critical schrodinger map problem we consider the energy critical schrodinger map problem with the 2 sphere target for equivariant initial data of homotopy index k 1 we show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm which generates finite time blowup solutions we give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy

two way traffic flow exactly solvable model of traffic jam we study completely asymmetric two channel exclusion processes in one dimension it describes a two way traffic flow with cars moving in opposite directions the interchannel interaction makes cars slow down in the vicinity of approaching cars in the other lane particularly we consider in detail the system with a finite density of cars on one lane and a single car on the other when the interchannel interaction reaches a critical value a traffic jam occurs which turns out to be of first order phase transition we derive exact expressions for the average velocities the current the density profile and the k point density correlation functions we also obtain the exact probability of two cars being in one lane of distance r apart provided there is a finite density of cars on the other lane and show that the two cars form a weakly bound state in the jammed phase

stability in h 1 of the sum of k solitary waves for some nonlinear schrodinger equations in this article we consider nonlinear schrodinger nls equations in rd for d 1 2 and 3 we consider nonlinearities satisfying a flatness condition at zero and such that solitary waves are stable let rk t x be k solitary wave solutions of the equation with different speeds v1 v2 vk provided that the relative speeds of the solitary waves vk vk 1 are large enough and that no interaction of two solitary waves takes place for positive time we prove that the sum of the rk t is stable for t 0 in some suitable sense in h1 to prove this result we use an energy method and a new monotonicity property on quantities related to momentum for solutions of the nonlinear schrodinger equation this property is similar to the l2 monotonicity property that has been proved by martel and merle for the generalized korteweg de vries gkdv equations see 12 lem 16 proof of prop 6 and that was used to prove the stability of the sum of k solitons of the gkdv equations by the authors of the present article see 15 th 1 i

transverse size critical exponent of directed percolation from yang lee zeros of survival probability by using transfer matrix method we compute survival probabilities for the directed percolation problem on strips of a square lattice and get very precise estimates of their yang lee zeros lying closest to the real axis in the complex plane of occupation probability this allows us to get accurate values for transverse size critical exponent and percolation threshold

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

the instability of bourgain wang solutions for the l2 critical nls we consider the two dimensional l 2 critical nonlinear schr odinger equation i partial tu delta u uu 2 0 in their pioneering 1997 work bourgain and wang have constructed smooth solutions which blow up in finite time t nabla u t l 2 sim 1 over t t and which display some decoupling between the regular and the singular part of the solution at blow up time we prove that this dynamic is unstable more precisely we show that any such solution with small super critical l 2 mass lies on the boundary of both h 1 open sets of global solutions that scatter forward and backwards in time and solutions that blow up in finite time on the right in the log log regime exhibited in work by f merle and p rapha e l we moreover exhibit some continuation properties of the scattering solution after blow up time and recover the chaotic phase behavior first exhibited in f merle s 1992 it comm pure appl math paper in the critical mass case

stability of the log log bound for blow up solutions to the critical non linear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation open image in new window with initial condition u0 h1 existence of such solutions is known but the complete blow up dynamic is not understood so far for initial data with negative energy finite time blow up with a universal sharp upper bound on the blow up rate corresponding to the so called log log law has been proved in 10 11 we focus in this paper onto the positive energy case where at least two blow up speeds are known to possibly occur we establish the stability in energy space h1 of the log log upper bound exhibited in the negative energy case and a sharp lower bound on blow up rate in the other regime which corresponds to known explicit blow up solutions

existence of blow up solutions in the energy space for the critical generalized kdv equation from these conservation laws h appears as an energy space so that it is a natural space in which to study the solutions note that p 2 is a special case for equation 2 indeed from the integrability theory see lax 14 we have for suitable u0 u0 and its derivatives with fast decay at infinity an infinite number of conservation laws the general question is to understand the dynamics induced by such equations local existence in time of solutions of 2 in the energy space is now well understood see kato 10 ginibre and tsutsumi 8 for the h theory s 32 kenig ponce and vega 11 for the l theory in the case of equation 1 and sharp h theory for 2 and bourgain 3 and 4 for the periodic case

generalized bethe ansatz solution of a one dimensional asymmetric exclusion process on a ring with blockage we present a model for a one dimensional anisotropic exclusion process describing particles moving deterministically on a ring of lengthl with a single defect across which they move with probability 0 p 1 this model is equivalent to a two dimensional six vertex model in an extreme anisotropic limit with a defect line interpolating between open and periodic boundary conditions we solve this model with a bethe ansatz generalized to this kind of boundary condition we discuss in detail the steady state and derive exact expressions for the currentj the density profilen x and the two point density correlation function in the thermodynamic limitl the phase diagram shows three phases a low density phase a coexistence phase and a high density phase related to the low density phase by a particle hole symmetry in the low density phase the density profile decays exponentially with the distance from the boundary to its bulk value on a length scale on the phase transition line diverges and the currentj approaches its critical valuejc p as a power law jc j 1 2 in the coexistence phase the width of the interface between the high density region and the low density region is proportional tol1 2 if the density f 1 2 and 0 independent ofl if 1 2 the connected two point correlation function turns out to be of a scaling form with a space dependent amplitude n x1 x2 a x2 ake r withr x2 x1 and a critical exponent 0

stability of blow up profile and lower bounds for blow up rate for the critical generalized kdv equation the generalized korteweg de vries equations are a class of hamiltonian systems in infinite dimension derived from the kdv equation where the quadratic term is replaced by a higher order power term these equations have two conservation laws in the energy space h 1 l 2 norm and energy we consider in this paper the critical generalized kdv equation which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in h 1 uniform in time for all h 1 solutions and thus global existence from 15 there do exist for this equation solutions u t such that u t h1 as t t where t we call them blow up solutions the question is to describe in a qualitative way how blow up occurs for solutions with l 2 mass close to the minimal mass allowing blow up and with decay in l 2 at the right we prove after rescaling and translation which leave invariant the l 2 norm that the solution converges to a universal profile locally in space at the blow up time t from the nature of this profile we improve the standard lower bound on the blow up rate for finite time blow up solutions

matrix product states of three families of one dimensional interacting particle systems abstract the steady states of three families of one dimensional non equilibrium models with open boundaries first proposed in n j phys 5 2003 145 1 are studied using a matrix product formalism it is shown that their associated quadratic algebras have two dimensional representations provided that the transition rates lie on specific manifolds of parameters exact expressions for the correlation functions of each model have also been obtained we have also studied the steady state properties of one of these models first introduced in j phys a math gen a 36 2003 74 with more details by introducing a canonical ensemble we calculate the canonical partition function of this model exactly using the yang lee theory of phase transitions we spot a second order phase transition from a power law to a jammed phase the density profile of particles in each phase has also been studied a simple generalization of this model in which both the left and the right boundaries are open has also been introduced it is shown that double shock structures may evolve in the system under certain conditions

finite size scaling and universality for the totally asymmetric simple exclusion process the applicability of the concepts of finite size scaling and universality to nonequilibrium phase transitions is considered in the framework of the one dimensional totally asymmetric simple exclusion process with open boundaries in the thermodynamic limit there are boundary induced transitions both of the first and second order between steady state phases of the model we derive finite size scaling expressions for the current near the continuous phase transition and for the local density near the first order transition under different stochastic dynamics and compare them to establish the existence of universal functions next we study numerically the finite size behavior of the lee yang zeros of the normalization factor for the different steady state probabilities

on universality of blow up profile for l 2 critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation iut u u 4 nu with initial condition u0 h1 existence of such solutions is known but the complete blow up dynamic is not understood so far for a specific set of initial data finite time blow up with a universal sharp upper bound on the blow up rate has been proved in 22 23

first order phase transition in a reaction diffusion model with open boundary the yang lee theory approach a coagulation decoagulation model is introduced on a chain of length l with open boundary the model consists of one species of particles which diffuse coagulate and decoagulate preferentially in the leftward direction they are also injected and extracted from the left boundary with different rates we will show that on a specific plane in the space of parameters the steady state weights can be calculated exactly using a matrix product method the model exhibits a first order phase transition between a low density and a high density phase the density profile of the particles in each phase is obtained both analytically and using the monte carlo simulation the two point density density correlation function in each phase has also been calculated by applying the yang lee theory we can predict the same phase diagram for the model this model is further evidence for the applicability of the yang lee theory in the non equilibrium statistical mechanics context

construction of solutions to the l 2 critical kdv equation with a given asymptotic behaviour we consider the critical korteweg de vries equation ut uxx u x 0 t x r let rj t x qcj x xj cjt j 1 n be n soliton solutions to this equation denote u t the kdv linear group and let v h be with sufficient decay on the right that is 1 x0 v l for some 0 0 we construct a solution u t to the critical korteweg de vries equation such that lim t u t u t v n x j 1 rj t h1 0

blow up for the critical generalized korteweg de vries equation i dynamics near the soliton we consider the quintic generalized korteweg de vries equation gkdv u t u xx u 5 x 0 which is a canonical mass critical problem for initial data in h1 close to the soliton in earlier works on this problem finite or infinite time blow up was proved for non positive energy solutions and the solitary wave was shown to be the universal blow up profile see 16 26 and 20 for well localized initial data finite time blow up with an upper bound on blow up rate was obtained in 18