optimal decay rates for the compressible fluid models of korteweg type abstract we consider the compressible navier stokes korteweg system that models the motions of the compressible isothermal viscous capillary fluids we prove the optimal l 2 and l p p 2 decay rates for the classical solutions and their spatial derivatives in particular the optimal l 2 decay rate of the second order spatial derivatives is obtained the proof is based on the detailed study of the linear decay estimates and nonlinear energy estimates

on a model of nonlocal continuum mechanics part i existence and regularity in this work we assume that the total stored energy functional for a body b depends not only on the local strain field but also on the spatial average of the strain field over the body weighted with an influence kernel we investigate the problem of minimizing the total stored energy subject to a given bulk displacement delta geq 0 after the general setup for this problem is reviewed we give sufficient conditions for an energy minimizing strain field e cdot to satisfy an integro differential euler lagrange equation the result is general and applies to material energies that display a wide variety of singular behavior through analysis of this euler lagrange equation for a special class of influence kernels we arrive at a regularity theorem which ensures that energy minimizing strain fields must be periodic piecewise smooth and possess a finite number of simple discontinuities we then combine this with a well known existence result for relaxed minimization problems to arrive at a genera

interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains abstract we investigate interface problems in nonlocal diffusion and demonstrate how to reformulate and generalize the classical treatment of interface problems in the presence of nonlocal interactions through formal derivations we show that nonlocal diffusion interface problems converge to their classical local counterparts in the limit of vanishing nonlocality a central focus of this paper is local nonlocal interface problems or interface problems with sharp transitions between local and nonlocal domains such problems can be cast as instances of a nonlocal interface problem with a finite horizon in certain regions and a vanishing horizon in other regions we derive a local nonlocal interface problem and utilize conservation principles to obtain local nonlocal interface conditions comparisons between nonlocal local and local nonlocal interface problems are presented analytically and numerically with a focus on multiscale aspects of nonlocal models induced by their inherent length scales

homoclinic solutions of an integral equation existence and stability abstract we study existence and stability of homoclinic type solutions of a bistable integral equation these are stationary solutions of an integrodifferential equation which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity

blowing up boundary conditions for a nonlocal nonlinear diffusion equation in several space dimensions abstract we analyze boundary value problems prescribing dirichlet or neumann boundary conditions for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation in a bounded smooth domain r n with n 1 first we prove existence and uniqueness of solutions and the validity of a comparison principle for these problems next we impose boundary data that blow up in finite time and study the behavior of the solutions

vanishing capillarity limit of the compressible non isentropic navier stokes korteweg system to navier stokes system abstract in this paper we consider the three dimensional compressible nonisentropic fluid model of korteweg type we mainly present the vanishing capillarity limit of smooth solution to the initial value problem based on the uniform estimates with respect to the capillary coefficient we show the unique smooth solutions of the three dimensional korteweg model converges globally in time to the smooth solution of the three dimensional compressible nonisentropic navier stokes system as tends to zero also we give the convergence rate estimates in h 2 norm with respect to

existence and uniqueness of travelling waves for a neural network a one dimensional scalar neural network with two stable steady states is analysed it is shown that there exists a unique monotone travelling wave front which joins the two stable states some additional properties of the wave such as the direction of its velocity are discussed

the fractional laplacian operator on bounded domains as a special case of the nonlocal diffusion operator we analyze a nonlocal diffusion operator having as special cases the fractional laplacian and fractional differential operators that arise in several applications in our analysis a nonlocal vector calculus is exploited to define a weak formulation of the nonlocal problem we demonstrate that when sufficient conditions on certain kernel functions hold the solution of the nonlocal equation converges to the solution of the fractional laplacian equation on bounded domains as the nonlocal interactions become infinite we also introduce a continuous galerkin finite element discretization of the nonlocal weak formulation and we derive a priori error estimates through several numerical examples we illustrate the theoretical results and we show that by solving the nonlocal problem it is possible to obtain accurate approximations of the solutions of fractional differential equations circumventing the problem of treating infinite volume constraints

motion by curvature by scaling nonlocal evolution equations we prove convergence to a motion by mean curvature by scaling diffusively a nonlinear nonlocal evolution equation this equation was introduced earlier to describe the macroscopic behavior of a ferromagnetic spin system with kac interaction which evolves with glauber dynamics the convergence is proven in any time interval in which the limiting motion is regular

blow up for a non local diffusion problem with neumann boundary conditions and a reaction term abstract in this paper we study the blow up problem for a non local diffusion equation with a reaction term u t x t j x y u y t u x t d y u p x t we prove that non negative and non trivial solutions blow up in finite time if and only if p 1 moreover we find that the blow up rate is the same as the one that holds for the ode u t u p that is lim t t t t 1 p 1 u t 1 p 1 1 p 1 next we deal with the blow up set we prove single point blow up for radially symmetric solutions with a single maximum at the origin as well as the localization of the blow up set near any prescribed point for certain initial conditions in a general domain with p 2 finally we show some numerical experiments which illustrate our results

optimal distributed control of nonlocal steady diffusion problems a control problem constrained by a nonlocal steady diffusion equation that arises in several applications is studied the control is the right hand side forcing function and the objective of control is a standard matching functional a recently developed nonlocal vector calculus is exploited to define a weak formulation of the state system when sufficient conditions on certain kernel functions and the volume constraints hold the existence and uniqueness of the optimal state and control is demonstrated and an optimality system is derived we demonstrate the convergence as the nonlocal interactions vanish of the optimal nonlocal state to the optimal state of a local pde constrained control problem we also define continuous and discontinuous galerkin finite element discretizations of the optimality system for which we derive a priori error estimates numerical examples are provided illustrating these convergence results and also illustrating the differences between optimal controls and states obtained f

on the maximal lp lq regularity for a compressible fluid model of korteweg type on general domains abstract the aim of this paper is to show the maximal l p l q regularity for a compressible fluid model of korteweg type on general domains of the n dimensional euclidean space for n 2 e g bounded domains exterior domains half spaces layers tubes and their perturbed domains our approach is based on the theory of the r boundedness for a generalized resolvent problem associated with the korteweg type model

metastability and stability of patterns in a convolution model for phase transitions where e is a small positive parameter and f is bistable having only three zeros 1 a 1 and is balanced i e r 1 1 f u du 1 4 0 see fig 1 for a typical initial datum j which crosses a several times at the beginning the solution evolves as if the diffusion term euxx were not there and in the time scale o ej n ej the solution gets close to 1 or 1 in intervals separated by thin layers of thickness at the order of o ej n ej 17 28 at this moment a spatial pattern is generated it persists on a time scale o ece but is eventually destroyed typically u x t 1 u x t 1 uniformly for x on 1 20 1 as t 1 it is possible that u x t converges to a or a nonconstant steady state as t 1 but this is an unlikely event because all these steady states are unstable 16 27 research supported in part by the national science foundation and bor leqsf

how to approximate the heat equation with neumann boundary conditions by nonlocal diffusion problems we present a model for nonlocal diffusion with neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary we study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero we prove that the solutions of this family of problems converge to a solution of the heat equation with neumann boundary conditions

regularity and separation from potential barriers for a non local phase field system abstract we show that solutions of a two phase model involving a non local interactive term become more regular immediately after the moment they separate from the pure phases this result allows us to prove stronger convergence to equilibria a new proof of the separation property is also given

existence of solution for a nonlocal dispersal model with nonlocal term via bifurcation theory abstract in this paper we study the existence of solution for the following class of nonlocal problems l 0 u u q x y u y p d y in where r n n 1 is a smooth bounded domain p 0 is a real parameter q r is a nonnegative function and l 0 c c is a nonlocal dispersal operator the existence of solution is obtained via bifurcation theory

on the principal eigenvalue of some nonlocal diffusion problems abstract in this paper we analyze some properties of the principal eigenvalue 1 of the nonlocal dirichlet problem j u x u x u x in with u x 0 in r n here is a smooth bounded domain of r n and the kernel j is assumed to be a c 1 compactly supported even nonnegative function with unit integral among other properties we show that 1 is continuous or even differentiable with respect to continuous differentiable perturbations of the domain we also provide an explicit formula for the derivative finally we analyze the asymptotic behavior of the decreasing function 1 when the dilatation parameter 0 tends to zero or to infinity

fock representations of the affine lie algebra a 1 1 the aim of this note is to show that the affine lie algebraa1 1 has a natural family v of fock representations on the spacec xi yj i andj parametrized by v c2 by corresponding the highest weight of to each the parameter spacec2 forms a double cover of the weight spacec 0 c 1 with singularities at linear forms of level 2 this number is 1 times the dual coxeter number our results contain explicit realizations of irreducible non integrable highest wieghta1 1 modules for generic v

global existence and asymptotic behaviour for a nonlocal phase field model for non isothermal phase transitions in this paper a nonlocal phase field model for non isothermal phase transitions with a non conserved order parameter is studied the paper extends recent investigations to the non isothermal situation complementing results obtained by h gajewski for the non isothermal case for conserved order parameters in phase separation phenomena the resulting field equations studied in this paper form a system of integro partial differential equations which are highly nonlinearly coupled for this system results concerning global existence uniqueness and large time asymptotic behaviour are derived the main results are proved using techniques that have been recently developed by p krejci and the authors for phase field systems involving hysteresis operators

blow up for a semilinear non local diffusion system abstract in this paper we study the blow up phenomena for the system u t j x y u y d y u u v p x t 0 t v t j x y v y d y v u q v x t 0 t under certain hypothesis we prove that the solution blows up infinite time and we classify in terms of the parameters when both components blow up together simultaneous blow up or when only one component blows up non simultaneous blow up we also prove the blow up rate for a particular case

existence uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal this paper is concerned with a nonlocal evolution equation which is used to model the spatial dispersal of organisms we study the existence uniqueness and stability of the positive steady solution for this nonlocal evolution equation under general conditions the global dynamics are also investigated and a trichotomy of the global asymptotics is established

global solutions of a high dimensional system for korteweg materials the global existence of solutions to a high dimensional system of korteweg materials is established when the initial data are small the complete model was proposed by dunn and serrin and the local existence of the solution was obtained in an earlier paper by the authors

chiral de rham complex we define natural sheaves of vertex algebras over smooth manifolds which may be regarded as semi infinite de rham complexes of certain d modules over the loop spaces for calabi yau manifolds they admit n 2 supersymmetry connection with wakimoto modules is discussed

cauchy problem of a nonlocal p laplacian evolution equation with nonlocal convection abstract this paper is concerned with the cauchy problem of a nonlocal equation that takes into account convective and p laplacian diffusive effects u t x t r n j x y u y t u x t p 2 u y t u x t d y g f u f u x t with j radially symmetric and g not necessarily symmetric first we prove the existence and uniqueness of solutions and if the convolution kernels j and g are rescaled appropriately we show that solutions of the nonlocal problem converge to the solution of the usual p laplacian diffusion equation with convection finally as a supplementary result we study the asymptotic behavior of solutions as t and give the decay estimate

fast and accurate implementation of fourier spectral approximations of nonlocal diffusion operators and its applications abstract this work is concerned with the fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions for radially symmetric kernels the nonlocal operators under consideration are diagonalizable in the fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or fourier symbols consisting of possibly singular and highly oscillatory integrals for a large class of fractional power like kernels we propose a new approach based on reformulating the fourier symbols both as coefficients of a series expansion and solutions of some simple ode models we then propose a hybrid algorithm that utilizes both truncated series expansions and high order runge kutta ode solvers to provide fast evaluation of fourier symbols in both one and higher dimensional spaces it is shown that this hybrid algorithm is robust efficient and accurate as applications we combine this hybrid spectral discretization in the spatial variables and the fourth order exponential time differencing runge kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal allen cahn equations nonlocal cahn hilliard equations and nonlocal phase field crystal models numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models

elliptic genera of toric varieties and applications to mirror symmetry the paper contains a proof that elliptic genus of a calabi yau manifold is a jacobi form finds in which dimensions the elliptic genus is determined by the hodge numbers and shows that elliptic genera of a calabi yau hypersurface in a toric variety and its mirror coincide up to sign the proof of the mirror property is based on the extension of elliptic genus to calabi yau hypersurfaces in toric varieties with gorenstein singularities

a chiral borel weil bott theorem abstract we compute the cohomology of modules over the algebra of twisted chiral differential operators over the flag manifold this is applied to 1 finding the character of g integrable irreducible highest weight modules over the affine lie algebra at the critical level and 2 computing a certain elliptic genus of the flag manifold the main tool is a result that interprets the drinfeld sokolov reduction as a derived functor

an integrodifferential model for phase transitions stationary solutions in higher space dimensions we study the existence and stability of stationary solutions of an integrodifferential model for phase transitions which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity as such this model is a nonlocal extension of the allencahn equation which incorporates long range interactions we find that the set of stationary solutions for this model is much larger than that of the allencahn equation

a non local hydrodynamic model for the shear viscosity of confined fluids analysis of a homogeneous kernel we present a detailed analysis of a hydrodynamic constitutive model recently applied to study the non local viscosity of highly confined inhomogeneous fluids zhang et al 2004 j chem phys 121 10778 zhang et al 2005 j chem phys 122 219901 this model makes the assumption that for pore widths significantly greater than the width of the viscosity kernel one can approximate the true inhomogeneous kernel with a homogeneous kernel we test this assumption with a simple mathematical model that mimics the simulation scheme of zhang et al our analysis shows that the assumption of a homogeneous non local viscosity kernel is reasonable for low density fluids confined to relatively large pore widths channel separation distances but becomes less reliable for higher density fluids and smaller channel separations we find that the extraction of the viscosity kernel is not simple because of the lack of information about the stress response function at the fluid wall boundaries this lack of information can lead to singularities in the kernel and can also distort its shape as a concluding remark we also suggest a methodology which avoids these problems

a symptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions abstract in this paper we study the asymptotic behaviour of a semidiscrete numerical approximation for u t u xx u p in a bounded interval 0 1 with dirichlet boundary conditions we focus in the behaviour of blowing up solutions we find that the blow up rate for the numerical scheme is the same as for the continuous problem also we find the blow up set for the numerical approximations and prove that it is contained in a neighbourhood of the blow up set of the continuous problem when the mesh parameter is small enough

analysis and approximation of a fractional laplacian based closure model for turbulent flows and its connection to richardson pair dispersion abstract we study a turbulence closure model in which the fractional laplacian of the velocity field represents the turbulence diffusivity we investigate the energy spectrum of the model by applying pao s energy transfer theory for the case 1 3 the corresponding power law of the energy spectrum in the inertial range has a correction exponent on the regular kolmogorov 5 3 scaling exponent for this case this model represents richardson s particle pair distance superdiffusion of a fully developed homogeneous turbulent flow as well as levy jumps that lead to the superdiffusion for other values of the power law of the energy spectrum is consistent with the regular kolmogorov 5 3 scaling exponent we also propose and study a modular time stepping algorithm in semi discretized form the algorithm is minimally intrusive to a given legacy code for solving navier stokes equations by decoupling the local part and nonlocal part of the equations for the unknowns we prove the algorithm is first order accurate and unconditionally stable we also derive error estimates for full discretizations of the model which in addition to the time stepping algorithm involves a finite element spatial discretization and a domain truncation approximation to the range of the fractional laplacian

multiple layered solutions of the nonlocal bistable equation abstract the nonlocal bistable equation is a model proposed recently to study materials whose constitutive relations among the variables that describe their states are nonlocal it resembles the local bistable equation the allen cahn equation in some way but contains a much richer set of solutions in this paper we consider two types of solutions the first are the periodic solutions on a finite interval these solutions are observed in materials like elastic crystals undergoing martensitic phase transitions and diblock copolymers at low temperatures they are constructed by a variational method known as the limit technique the second are solutions on the entire real line with transition layers which are found by the formal matched asymptotics argument we construct them to compare with the single layer heteroclinic and traveling wave solutions of the local bistable equation the existence of multiple layered solutions depends on a unique nonlocal feature the presence of two properly balanced competing effects of the constitutive relation the oscillation inhibiting effect and the oscillation forcing effect which coexist at two different length scales

a mean field equation of motion for the dynamic ising model a mean field type of approximation is used to derive two differential equations one approximately representing the average behavior of the ising model with glauber spin flip stochastic dynamics and the other doing the same for kawasaki spin exchange dynamics the proposed new equations are compared with the cahn allen and cahn hilliard equations representing the same systems and with information about the exact behavior of the microscopic models

a nonlocal nonlinear diffusion equation with blowing up boundary conditions we deal with boundary value problems prescribing dirichlet or neumann boundary conditions for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation first we prove existence uniqueness and the validity of a comparison principle for these problems next we impose boundary data that blow up in finite time and study the behavior of the solutions

maximum and antimaximum principles for some nonlocal diffusion operators abstract in this work we consider the maximum and antimaximum principles for the nonlocal dirichlet problem j u u u h r n j x y u y d y u x u x h x 0 in a bounded domain with u x 0 in r n the kernel j in the convolution is assumed to be a continuous compactly supported nonnegative function with unit integral we prove that for 1 the solution verifies u 0 in if h l 2 h 0 while for 1 and close to 1 the solution verifies u 0 in provided h x x d x 0 h l this last assumption is also shown to be optimal the neumann version of the problem is also analyzed

phase segregation dynamics in particle systems with long range interactions i macroscopic limits we present and discuss the derivation of a nonlinear nonlocal integrodifferential equation for the macroscopic time evolution of the conserved order parameter r t of a binary alloy undergoing phase segregation our model is ad dimensional lattice gas evolving via kawasaki exchange with respect to the gibbs measure for a hamiltonian which includes both short range local and long range nonlocal interactions the nonlocal part is given by a pair potential dj x y 0 x and y in d in the limit 0 the macroscopic evolution is observed on the spatial scale 1 and time scale 2 i e the density r t is the empirical average of the occupation numbers over a small macroscopic volume element centered atr x a rigorous derivation is presented in the case in which there is no local interaction in a subsequent paper part ii we discuss the phase segregation phenomena in the model in particular we argue that the phase boundary evolutions arising as sharp interface limits of the family of equations derived in this paper are the same as the ones obtained from the corresponding limits for the cahn hilliard equation

gerbes of chiral differential operators ii vertex algebroids introduction 605 0 recollections and notation 612 1 vertex algebroids 615 2 from vertex algebras to vertex algebroids 622 3 category of vertex algebroids 628 4 cofibered structure 633 5 computation of a cocycle 637 6 characteristic classes of vector bundles 643 7 torseurs and gerbes 647 8 gerbes of vertex algebroids 653 9 chiral differential operators on curves and sheaves of virasoro algebras 657 10 enveloping algebra of a vertex algebroid 664 11 calabi yau structures 668 12 virasoro and calabi yau structures 671 references 680

a nonlocal dispersal equation arising from a selection migration model in genetics this paper is concerned with the existence uniqueness and asymptotic stability of positive steady states for a nonlocal dispersal equation arising from selection migration models in genetics due to the lack of compactness and regularity of the nonlocal operators many classical methods cannot be used directly to the nonlocal dispersal problems this motivates us to find new techniques we first establish a criterion on the stability and instability of steady states this result is effective to get a necessary condition to guarantee a positive steady state it also gives the uniqueness then we prove the existence of nontrivial solutions by the corresponding auxiliary equations and maximum principle finally we consider the dynamic behavior of the initial value problem

on a nonlocal diffusion model with neumann boundary conditions abstract we study a nonlocal diffusion model analogous to heat equation with neumann boundary conditions we prove the existence and uniqueness of solutions and a comparison principle furthermore we analyze the asymptotic behavior of the solutions as the temporal variable goes to infinity and the boundary datum depends only on a spacial variable

a nonlocal convection diffusion equation in this paper we study a nonlocal equation that takes into account convective and diffusive effects ut j u u g f u f u in rd with j radially symmetric and g not necessarily symmetric first we prove existence uniqueness and continuous dependence with respect to the initial condition of solutions this problem is the nonlocal analogous to the usual local convection diffusion equation ut u b f u in fact we prove that solutions of the nonlocal equation converge to the solution of the usual convection diffusion equation when we rescale the convolution kernels j and g appropriately finally we study the asymptotic behaviour of solutions as t when f u u q 1u with q 1 we find the decay rate and the first order term in the asymptotic regime

a non smooth version of the lojasiewicz simon theorem with applications to non local phase field systems abstract we consider a simple system modelling phase transition phenomena with long term interactions it is shown that any solution converges with growing time to a single stationary state to this end a non smooth version of the celebrated simon lojasiewicz theorem is proved

localized radial basis functions based pseudo spectral method lrbf psm for nonlocal diffusion problems abstract spectral pseudo spectral methods based on high order polynomials have been successfully used for solving partial differential and integral equations in this paper we will present the use of a localized radial basis functions based pseudo spectral method lrbf psm for solving 2d nonlocal problems with radial nonlocal kernels the basic idea of the lrbf psm is to construct a set of orthogonal functions by rbfs on each overlapping sub domain from which the global solution can be obtained by extending the approximation on each sub domain to the entire domain numerical implementation indicates that the proposed lrbf psm is simple to use efficient and robust to solve various nonlocal problems

generalized motion by mean curvature as a macroscopic limit of stochastic ising models with long range interactions and glauber dynamics we study themacroscopic limit of an appropriately rescaledstochastic ising model withlong range interactions evolving withglauber dynamics as well as the correspondingmean field equation which is nonlinear and nonlocal in the limit we obtain an interface evolving with normal velocity k wherek isthe mean curvature and thetransport coefficient is identified by aneffective green kubo type formula the above assertions are valid for all positive times the motion of the interface being interpreted in theviscosity sense after the onset of the geometric singularities

semi infinite induction and wakimoto modules the purpose of this paper is to suggest the construction and study properties of semi infinite induction which relates to semi infinite cohomology the same way induction relates to homology and coinduction to cohomology we prove a version of the shapiro lemma showing that the semi infinite cohomology of a module is isomorphic to that of the semi infinitely induced module a practical outcome of our construction is a simple construction of the wakimoto modules highest weight modules used in double sided bgg resolutions of irreducible modules

robust a posteriori stress analysis for quadrature collocation approximations of nonlocal models via nonlocal gradients abstract as alternatives to partial differential equations pdes nonlocal continuum models given in integral forms avoid the explicit use of conventional spatial derivatives and allow solutions to exhibit desired singular behavior it is of practical interest to develop robust numerical schemes not only for the numerical solution of nonlocal models but also for the evaluation of suitably defined derivatives of solutions the latter motivates the development of a nonlocal analog of gradient recovery for numerical solution of pdes for structure mechanical models this leads to a posteriori nonlocal stress analysis we illustrate that when smooth solutions are found in nonlocal models one may compute local gradients of nonlocal solutions using conventional techniques like that for pdes more generically however we present a framework for stress analysis of nonlocal solutions based on nonlocal gradient operators and their asymptotically compatible discretization we demonstrate that the nonlocal gradient recovery is consistent in the local limit and is more advantageous than using local gradients of nonlocal solutions superconvergence properties of some special nonlocal gradient operators are identified for nonlocal continuum models moreover methods are presented to preserve such features in the numerical discretization both computational observations and theoretical insights are provided to substantiate our findings

wakimoto modules opers and the center at the critical level abstract wakimoto modules are representations of affine kac moody algebras in fock modules over infinite dimensional heisenberg algebras in this paper we present the construction of the wakimoto modules from the point of view of the vertex algebra theory we then use wakimoto modules to identify the center of the completed universal enveloping algebra of an affine kac moody algebra at the critical level with the algebra of functions on the space of opers for the langlands dual group on the punctured disc giving another proof of the theorem of b feigin and the author

strong solutions of the navier stokes equations for a compressible fluid of allen cahn type in this work we study the navier stokes allen cahn system a combination of the compressible navier stokes equations with an allen cahn phase field description this model describes two phase patterns in a flowing liquid including phase transformations our purpose is to show the existence and uniqueness of local strong solutions for arbitrary initial data part of the proof is based on methods used in a companion paper which investigated the compressible navier stokes equations

linear non local diffusion problems in metric measure spaces the aim of this paper is to provide a comprehensive study of some linear nonlocal diusion problems in metric measure spaces these include for example open subsets in r n graphs manifolds multi structures or some fractal sets for this we study regularity compactness positiveness and the spectrum of the stationary nonlocal operator then we study the solutions of linear evolution nonlocal diusion problems with emphasis in similarities and dierences with the standard heat equation in smooth domains in particular prove weak and strong maximum principles and describe the asymptotic behaviour using spectral methods

existence and uniqueness of weak solutions for nonlocal parabolic problems via the galerkin method abstract fractional differential equations are becoming increasingly popular as a modeling tool to describe a wide range of non classical phenomena with spatial heterogeneities throughout the applied science and engineering a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal operators which include fractional laplacians on bounded domains in r n we develop the galerkin method to prove existence and uniqueness of weak solutions to nonlocal parabolic problems moreover we study the existence of orthonormal basis of eigenvectors associated to these nonlocal operators

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

existence of pulses in excitable media with nonlocal coupling abstract we prove the existence of fast traveling pulse solutions in excitable media with non local coupling existence results had been known until now in the case of local diffusive coupling and in the case of a discrete medium with finite range non local coupling our approach replaces methods from geometric singular perturbation theory that had been crucial in previous existence proofs by a pde oriented approach relying on exponential weights fredholm theory and commutator estimates

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

asymptotic behavior for a nonlocal diffusion equation in domains with holes the paper deals with the asymptotic behavior of solutions to a non local diffusion equation ut j u u lu in an exterior domain which excludes one or several holes and with zero dirichlet data on mathbb r n setminus omega when the space dimension is three or more this behavior is given by a multiple of the fundamental solution of the heat equation away from the holes on the other hand if the solution is scaled according to its decay factor close to the holes it behaves like a function that is l harmonic lu 0 in the exterior domain and vanishes in its complement the height of such a function at infinity is determined through a matching procedure with the multiple of the fundamental solution of the heat equation representing the outer behavior the inner and the outer behaviors can be presented in a unified way through a suitable global approximation

phase field and korteweg type models for the time dependent flow of compressible two phase fluids various versions of the navier stokes allen cahn nsac the navier stokes cahn hilliard nsch and the navier stokes korteweg nsk equations have been used in the literature to model the dynamics of two phase fluids one main purpose of this paper consists in re deriving nsac nsch and nsk from first principles in the spirit of rational mechanics for fluids of very general constitutive laws for nsac this deduction confirms and extends a proposal of blesgen regarding nsch it continues work of lowengrub and truskinovsky and provides the apparently first justified formulation in the non isothermal case for nsk it yields a most natural correction to the formulation by dunn and serrin the paper uniformly recovers as examples various classes of fluids distinguished according to whether none one or both of the phases are compressible and according to the nature of their co existence the latter is captured not only by the mixing energy but also by a mixing rule a constitutive law that characterizes the type of the mixing a second main purpose of the paper is to communicate the apparently new observation that in the case of two immiscible incompressible phases of different temperature independent specific volumes nsac reduces literally to nsk this finding may be considered as an independent justification of nsk an analogous fact is shown for nsch which under the same assumption reduces to a new non local version of nsk

numerical approximations for a nonlocal evolution equation in this paper we study numerical approximations of continuous solutions to the nonlocal p laplacian type diffusion equation u t t x int omega j x y u t y u t x p 2 u t y u t x dy first we find that a semidiscretization in space of this problem gives rise to an ode system whose solutions converge uniformly to the continuous one as the mesh size goes to zero moreover the semidiscrete approximation shares some properties of the continuous problem it preserves the total mass and the solution converges to the mean value of the initial condition as t goes to infinity next we also discretize the time variable and present a totally discrete method which also enjoys the above mentioned properties in addition we investigate the limit as p goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile finally we present some numerical experiments that illustrate our results

blowup in diffusion equations a survey this paper deals with quasilinear reaction diffusion equations for which a solution local in time exists if the solution ceases to exist for some finite time we say that it blows up in contrast to linear equations blowup can occur even if the data are smooth and well defined for all times depending on the equation either the solution or some of its derivatives become singular we shall concentrate on those cases where the solution becomes unbounded in finite time this can occur in quasilinear equations if the heat source is strong enough there exist many theoretical studies on the question on the occurrence of blowup in this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup the asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power a better knowledge would be useful also for their numerical treatment thus not surprisingly the numerical analysis of this type of problems is still at a rather early stage the goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems

analysis and approximation of nonlocal diffusion problems with volume constraints a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in mbrn the nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion including a notion of nonlocal flux the ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory in particular it is shown that fractional laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider the numerous applications elucidate different interpretations of the operator and the associated governing equations for example a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process sufficie

asymptotic behavior for a nonlocal diffusion equation in exterior domains the critical two dimensional case abstract we study the long time behavior of bounded integrable solutions to a nonlocal diffusion equation t u j u u where j is a smooth radially symmetric kernel with support b d 0 r 2 the problem is set in an exterior two dimensional domain which excludes a hole h and with zero dirichlet data on h in the far field scale 1 x t 1 2 2 with 1 2 0 the scaled function log t u x t behaves as a multiple of the fundamental solution for the local heat equation with a certain diffusivity determined by j the proportionality constant which characterizes the first non trivial term in the asymptotic behavior of the mass is given by means of the asymptotic logarithmic momentum of the solution lim t r 2 u x t log x d x this asymptotic quantity can be easily computed in terms of the initial data in the near field scale x t 1 2 h t with lim t h t 0 the scaled function t log t 2 u x t log x converges to a multiple of x log x where is the unique stationary solution of the problem that behaves as log x when x the proportionality constant is obtained through a matching procedure with the far field limit finally in the very far field x t 1 2 g t with g t the solution is proved to be of order o t log t 1

global solvability of the navier stokes equations with a free surface in the maximal lp lq regularity class abstract we consider the motion of incompressible viscous fluids bounded above by a free surface and below by a solid surface in the n dimensional euclidean space for n 2 the aim of this paper is to show the global solvability of the navier stokes equations with a free surface describing the above mentioned motion in the maximal l p l q regularity class our approach is based on the maximal l p l q regularity with exponential stability for the linearized equations and also it is proved that solutions to the original nonlinear problem are exponentially stable

a nonlocal nonlinear diffusion equation in higher space dimensions abstract we study the initial value problem for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation in the whole r n n 1 or in a bounded smooth domain with neumann or dirichlet boundary conditions first we prove the existence uniqueness and the validity of a comparison principle for solutions of these problems in r n we show that if initial data is bounded and compactly supported then the solutions is compactly supported for all positive time t this implies the existence of a free boundary concerning the neumann problem we prove that the asymptotic behavior of the solutions as t they converge to the mean value of the initial data for the dirichlet problem we prove that the asymptotic behavior of the solutions as t they converge to zero

blow up for the non local p laplacian equation with a reaction term abstract in this paper we study the blow up phenomenon for the non local p laplacian equation with a reaction term u t x t a j x y u y t u x t p 2 u y t u x t d y u q x t x t 0 t with dirichlet conditions a r n u 0 in r n or neumann conditions a those problems are the non local analogous to the equation v t p v v q with the corresponding conditions we determine in both cases which are the global existence exponents that coincide with the exponents of the corresponding local problem for neumann boundary conditions however we observe differences with respect to the global existence exponents of the local dirichlet problem moreover we show that the blow up rate is the same as the one that holds for the ode u t u q that is lim t t t t 1 q 1 u t 1 q 1 1 q 1 we also find some differences between the local and the non local models concerning the blow up sets precisely we show that regional blow up is not possible for non local problems finally we include some numerical experiments which illustrate our results

global existence and optimal decay rate of the compressible navier stokes korteweg equations with external force abstract in this paper we consider the three dimensional compressible fluid models of korteweg type the equation takes the form of the compressible navier stokes equation with the cauchy stress tensor in the momentum equations under the smallness assumption of the external force in some sobolev space we first show the existence of stationary solution by standard iterative arguments next we present that the global in time solution of the initial value problem for the three dimensional compressible navier stokes korteweg equations exists uniquely and approaches the stationary solution as t provided the prescribed initial data and the external force are sufficiently small finally based on the elaborate energy estimates of the solution for the nonlinear system and l 2 decay estimates of semigroup generated by the corresponding linearized equation we show the optimal l 2 convergence rates of the solution towards the stationary solution as far as we know this is the first result about the global existence and l 2 decay rate of smooth solutions for the compressible navier stokes korteweg equations with the external force

a nonlocal quasilinear multi phase system with nonconstant specific heat and heat conductivity in this paper we prove the existence and global boundedness from above for a solution to an integrodifferential model for nonisothermal multi phase transitions under nonhomogeneous third type boundary conditions the system couples a quasilinear internal energy balance ruling the evolution of the absolute temperature with a vectorial integro differential inclusion governing the vectorial phase parameter dynamics the specific heat and the heat conductivity k are allowed to depend both on the order parameter and on the absolute temperature of the system and the convex component of the free energy may or may not be singular uniqueness and continuous data dependence are also proved under additional assumptions

localization of affine w algebras we introduce the notion of an asymptotic algebra of chiral differential operators we then construct via a chiral hamiltonian reduction one such algebra over a resolution of the intersection of the slodowy slice with the nilpotent cone we compute the space of global sections of this algebra thereby proving a localization theorem for affine w algebras at the critical level

stabilized linear semi implicit schemes for the nonlocal cahn hilliard equation abstract comparing with the well known classic cahn hilliard equation the nonlocal cahn hilliard equation is equipped with a nonlocal diffusion operator and can describe more practical phenomena for modeling phase transitions of microstructures in materials on the other hand it evidently brings more computational costs in numerical simulations thus efficient and accurate time integration schemes are highly desired in this paper we propose two energy stable linear semi implicit methods with first and second order temporal accuracies respectively for solving the nonlocal cahn hilliard equation the temporal discretization is done by using the stabilization technique with the nonlocal diffusion term treated implicitly while the spatial discretization is carried out by the fourier collocation method with fft based fast implementations the energy stabilities are rigorously established for both methods in the fully discrete sense numerical experiments are conducted for a typical case involving gaussian kernels we test the temporal convergence rates of the proposed schemes and make a comparison of the nonlocal phase transition process with the corresponding local one in addition long time simulations of the coarsening dynamics are also performed to predict the power law of the energy decay

a spectral collocation method for nonlocal diffusion equations with volume constrained boundary conditions abstract the nonlocal diffusion model describes the diffusion process of solutes in complex media properly while the classical theory of partial differential equations can not provide an appropriate description the purpose of this paper is to provide illustrations from both theoretical and numerical perspectives of the computation of nonlocal diffusion models by legendre collocation methods compared to local numerical methods legendre collocation methods can achieve a fixed accuracy with much fewer unknowns whenever the computational domain is regular and the solutions are sufficiently smooth this paper is a groundwork towards efficient high order methods including spectral and spectral element methods for nonlocal diffusion equations with volume constrained boundary conditions

d modules on the affine grassmannian and representations of affine kac moody algebras let mathfrak g be a simple lie algebra for a level thought of as a symmetric mathfrak g invariant form of mathfrak g let hat mathfrak g kappa be the corresponding affine kac moody algebra let grg be the affine grassmannian of mathfrak g and let d grg mod be the category of twisted right d modules on grg by taking global sections of a d module we obtain a functor d grg mod mathfrak g mod it is known that this functor is exact and faithful when is less than critical or irrational in this paper we show that the functor is also exact and faithful when is the critical level

boundary fluxes for nonlocal diffusion abstract we study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary this problem may be seen as a generalization of the usual neumann problem for the heat equation first we prove existence uniqueness and a comparison principle next we study the behavior of solutions for some prescribed boundary data including blowing up ones finally we look at a nonlinear flux boundary condition

on a simple criterion for the existence of a principal eigenfunction of some nonlocal operators abstract in this paper we are interested in the existence of a principal eigenfunction of a nonlocal operator which appears in the description of various phenomena ranging from population dynamics to micro magnetism more precisely we study the following eigenvalue problem j x y g y y g n y d y a x where r n is an open connected set j a non negative kernel and g a positive function first we establish a criterion for the existence of a principal eigenpair p p we also explore the relation between the sign of the largest element of the spectrum with a strong maximum property satisfied by the operator as an application of these results we construct and characterise the solutions of some nonlinear nonlocal reaction diffusion equations