a consistent finite element formulation for linear buckling analysis of spatial beams a consistent co rotational finite element formulation and numerical procedure for the linear buckling analysis of three dimensional elastic euler beam is presented a mechanism for generating configuration dependent conservative moment is proposed and the corresponding load stiffness matrix is derived it is assumed that the prebuckling displacements and rotations of the structure and the corresponding deformations of the elements are linearly proportional to the external loading the prebuckling rotations of the structure are fixed axis rotations or small rotations and the effect of the prebuckling displacement on transformation matrix for the coordinates transformation can be ignored all coupling among bending twisting and stretching deformations for beam element is considered by consistent linearization of the fully geometrically nonlinear beam theory an inverse power method for the solution of the generalized eigenvalue problem is used to find the buckling load and buckling mode numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method

on the well posedness for nls in hs abstract in this article we prove the local well posedness of the cauchy problem for the nonlinear schrodinger equation i t u u c u u in h s r n with 0 4 n 2 s and 1 s min n 2 2

strichartz type estimates in mixed besov spaces with application to critical nonlinear schrodinger equations abstract we study the cauchy problem for the nonlinear schrodinger equation with power nonlinearity in the fractional order sobolev space h s r n we consider the case where the nonlinear term is h s critical and the differentiability thereof is less than s we prove the existence of time global solutions of the cauchy problem for small initial data to this end we prove strichartz type estimates for linear inhomogeneous schrodinger equations in mixed besov spaces

uniform convergence to equilibrium for granular media we study the long time asymptotics of a nonlinear nonlocal equation used in the modelling of granular media we prove a uniform exponential convergence to equilibrium for degenerately convex and non convex interaction or confinement potentials improving in particular results by j a carrillo r j mccann and c villani the method is based on studying the dissipation of the wasserstein distance between a solution and the steady state

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

global well posedness and scattering for the defocusing l 2 critical nonlinear schrodinger equation when d 2 in this article we prove that the defocusing cubic nonlinear schrodinger initial value problem is globally well posed and scattering for u0 l2 r2 the proof uses the bilinear estimates of planchon and vega and a frequency localized interaction morawetz estimate similar to the high frequency estimate of colliander keel staffilani takaoka and tao and especially the low frequency estimate of dodson

an integro differential equation model for alignment and orientational aggregation abstract we study an integro differential equation modeling angular alignment of interacting bundles of cells or filaments a bifurcation analysis of the related stationary problem was done by geigant and stoll in e geigant m stoll bifurcation analysis of an orientational aggregation model j math biol 46 6 2003 537 563 here we analyze the time dependent problem and prove that the type of alignment one or multi directional depends on the initial distribution the interaction potential and the preferred optimal orientation of the bundles of cells or filaments our main technical tool is the analysis of the evolution of suitable functionals for the cell density which allows to also specify the direction s where the final alignment takes place

a co rotational formulation for thin walled beams with monosymmetric open section abstract a consistent co rotational total lagrangian finite element formulation and numerical procedure for the geometric nonlinear buckling and postbuckling analysis of thin walled beams with monosymmetric open section is presented the element developed here has two nodes with seven degrees of freedom per node the element nodes are chosen to be located at the shear centers of the end cross sections of the beam element and the shear center axis is chosen to be the reference axis the deformations of the beam element are described in the current element coordinate system which is constructed at the current configuration of the beam element in element nodal forces all coupling among bending twisting and stretching deformations of the beam element is considered by consistent second order linearization of the fully geometrically nonlinear beam theory however the third order term of the twist rate of the beam axis is considered in element nodal forces an incremental iterative method based on the newton raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations the zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state a parabolic interpolation method of the arc length is used to find the buckling load numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method

the cauchy problem for heat equations with exponential nonlinearity abstract the cauchy problem for the semilinear heat equations is studied in the orlicz space exp l 2 r n where any power behavior of interaction works as a subcritical nonlinearity we prove the existence of global solutions for the semilinear heat equations with the exponential nonlinearity under the smallness condition on the initial data in exp l 2 r n

co rotational finite element formulation for thin walled beams with generic open section a consistent co rotational total lagrangian finite element formulation for the geometric nonlinear buckling and postbuckling analysis of thin walled beams with generic open section is presented the element developed here has two nodes with seven degrees of freedom per node the element nodes are chosen to be located at the shear centers of the end cross sections of the beam element and the shear center axis is chosen to be the reference axis the deformations of the beam element are described in the current element coordinate system constructed at the current configuration of the beam element the element nodal forces are derived using the virtual work principle the virtual rigid body motion corresponding to the virtual nodal displacements is excluded in the derivation of the element nodal forces a procedure is proposed to determine the virtual rigid body motion the way used to determine the element coordinate system and element nodal deformations corresponding to the virtual nodal displacements and that corresponding to the incremental nodal displacement are consistent in element nodal forces all coupling among bending twisting and stretching deformations of the beam element is considered by consistent second order linearization of the fully geometrically nonlinear beam theory in the derivation of the element tangent stiffness matrix the change of element nodal forces induced by the element rigid body rotations should be considered for the present method thus a stability matrix is included in the element tangent stiffness matrix an incremental iterative method based on the newton raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations the zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state numerical examples are presented to investigate the accuracy and efficiency of the proposed method the effect of the terms in the element nodal force and tangent stiffness matrix which will converge to zero with the decrease of element size on the convergence rate of solution and accuracy for the buckling load and nonlinear behavior of three dimensional beam structures are also investigated through numerical examples 2005 elsevier b v all rights reserved

finite time blow up of solutions of an aggregation equation in rn we consider the aggregation equation u t nabla cdot u nabla k u 0 in rn n 2 where k is a rotationally symmetric nonnegative decaying kernel with a lipschitz point at the origin e g k x e x we prove finite time blow up of solutions from specific smooth initial data for which the problem is known to have short time existence of smooth solutions

myopic random walkers and exclusion processes single and multispecies abstract a motility mechanism based on a simple exclusion process where the probability of movement of an agent depends on the number of unoccupied nearest neighbor sites is considered such interacting agents are termed myopic this problem is an extension of the famous blind or myopic ant in a labyrinth problem for the interacting agent models considered here each agent plays the role of an ant in a labyrinth where the paths of allowed sites though the labyrinth consist of sites not occupied by other agents we derive a nonlinear diffusion equation for the average occupancy of the discrete agent based model for myopic agents in contrast interacting blind agents have a constant probability of movement to each of their nearest neighbor sites giving rise to a linear diffusion equation insight into the various terms in the nonlinear diffusion coefficient is obtained from a study of multiple subpopulations of interacting myopic agents where an advection diffusion equation for each subpopulation is derived and from tracking an individual agent within the crowd where a motility coefficient is extracted averaged discrete simulation data compares very well with the solution to the continuum models we also compare the behavior of myopic and blind agents the myopic motility mechanism is biologically motivated to emulate information an individual cell gathers from environment cues the multispecies model developed and investigated here assists with the interpretation of experimental data involving the tracking subpopulations of cells within a total cell population

a consistent co rotational formulation for non linear three dimensional beam elements abstract this paper describes a co rotational formulation for three dimensional beams in which both the internal force vector and tangent stiffness matrix are consistently derived from the adopted strain measures the latter relate to standard beam theory but are embedded in a continuously rotating frame a set of numerical examples show that the element provides an excellent numerical performance

formation of clumps and patches in self aggregation of finite size particles abstract new model equations are derived for the dynamics of self aggregation of finite size particles differences from standard debye huckel p debye e huckel zur theorie der elektrolyte 2 das grenzgesetz fur die elektrische leiftfahrigkeit on the theory of electrolytes 2 limiting law of electrical conductivity phys z 24 1923 305 325 and keller segel e f keller l a segel j theoret biol 26 1970 399 415 e f keller l a segel j theoret biol 30 1971 225 234 models are a the mobility of particles depends on the locally averaged particle density and b linear diffusion acts on that locally averaged particle density the cases both with and without diffusion are considered here surprisingly these simple modifications of standard models allow progress in the analytical description of evolution as well as the complete analysis of stationary states when remains positive the evolution of collapsed states in our model reduces exactly to finite dimensional dynamics of interacting particle clumps simulations show these collapsed clumped states emerging from smooth initial conditions even in one spatial dimension if vanishes for some averaged density the evolution leads to the spontaneous formation of jammed patches a weak solution with the density having compact support simulations confirm that a combination of these patches forms the final state for the system

remark on the scattering operator for the cubic nonlinear dirac equation in three space dimensions abstract this paper is concerned with the scattering operator s for the three dimensional dirac equation with a cubic nonlinearity it follows from known results that s is well defined on a neighborhood of 0 in the sobolev space h r 3 c 4 for any 1 in the present paper we prove that for any m n and s max m there exists some neighborhood u of 0 in the weighted sobolev space h s m r 3 c 4 such that s u h s m r 3 c 4

building macroscale models from microscale probabilistic models a general probabilistic approach for nonlinear diffusion and multispecies phenomena a discrete agent based model on a periodic lattice of arbitrary dimension is considered agents move to nearest neighbor sites by a motility mechanism accounting for general interactions which may include volume exclusion the partial differential equation describing the average occupancy of the agent population is derived systematically a diffusion equation arises for all types of interactions and is nonlinear except for the simplest interactions in addition multiple species of interacting subpopulations give rise to an advection diffusion equation for each subpopulation this work extends and generalizes previous specific results providing a construction method for determining the transport coefficients in terms of a single conditional transition probability which depends on the occupancy of sites in an influence region these coefficients characterize the diffusion of agents in a crowded environment in biological and physical processes

a geometrical nonlinear eccentric 3d beam element with arbitrary cross sections abstract in this paper a finite element formulation of eccentric space curved beams with arbitrary cross sections is derived based on a timoshenko beam kinematic the strain measures are derived by exploitation of the green lagrangean strain tensor thus the formulation is conformed with existing nonlinear shell theories finite rotations are described by orthogonal transformations of the basis systems from the initial to the current configuration since for arbitrary cross sections the centroid and shear center do not coincide torsion bending coupling occurs in the linear as well as in the finite deformation case the linearization of the boundary value formulation leads to a symmetric bilinear form for conservative loads the resulting finite element model is characterized by 6 degrees of freedom at the nodes and therefore is fully compatible with existing shell elements since the reference curve lies arbitrarily to the line of centroids the element can be used to model eccentric stiffener of shells with arbitrary cross sections

a co rotational finite element formulation for buckling and postbuckling analyses of spatial beams abstract a consistent co rotational finite element formulation and numerical procedure for the buckling and postbuckling analyses of three dimensional elastic euler beam is presented all coupling among bending twisting and stretching deformations for a beam element is considered by consistent second order linearization of the fully geometrically nonlinear beam theory however the third order terms which are relevant to the twist rate and curvature of the beam axis are also considered an incremental iterative method based on the newton raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations the zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state a bisection method of the arc length is proposed to find the buckling load numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method and to investigate the effect of third order terms on the buckling load and postbuckling behavior of three dimensional beams

generalized mixed finite element model for pre and post quasistatic buckling response of thin walled framed structures adopting an updated lagrangian approach the general framework for the fully non linear analysis of thin walled framed structures is developed using a simple two node c0 model hmb2 the governing equations are derived based on a consistent linearization of an incremental mixed variational principle of modified hellinger reissner type with independent assumptions for displacement and strain fields all coupled significant modes of deformations i e stretching bending shear torsion and warping are accounted for in the generalized beam theory employed emphasis is placed on devising effective solution procedures to deal with finite rotations in space particularly with regard to their effect on the derivations of load correction matrices corresponding to configuration dependent externally applied forces moments the effectiveness and practical usefulness of present model are demonstrated through a number of test problems involving beam assemblages undergoing large displacements and rotations in space

anomalous diffusion with linear reaction dynamics from continuous time random walks to fractional reaction diffusion equations we have revisited the problem of anomalously diffusing species modeled at the mesoscopic level using continuous time random walks to include linear reaction dynamics if a constant proportion of walkers are added or removed instantaneously at the start of each step then the long time asymptotic limit yields a fractional reaction diffusion equation with a fractional order temporal derivative operating on both the standard diffusion term and a linear reaction kinetics term if the walkers are added or removed at a constant per capita rate during the waiting time between steps then the long time asymptotic limit has a standard linear reaction kinetics term but a fractional order temporal derivative operating on a nonstandard diffusion term results from the above two models are compared with a phenomenological model with standard linear reaction kinetics and a fractional order temporal derivative operating on a standard diffusion term we have also developed further extensions of the ctrw model to include more general reaction dynamics

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

the global cauchy problem for the non linear klein gordon equation on demontre l unicite de solutions faibles et l existence et l unicite de solutions globales fortement continues dans l espace energie

a kinematically exact space finite strain beam model finite element formulation by generalized virtual work principle the present paper presents a novel finite element formulation for static analysis of linear elastic spatial frame structures extending the formulation given by simo and vu quoc a geometrically exact rod model incorporating shear and torsion warping deformation int j solids structures 27 3 1991 371 3931 along the lines of the work on the planar beam theory presented by saje a variational principle for finite planar deformation of straight slender elastic beams internat j solids and structures 26 1990 887 9001 we apply exact non linear kinematic relationships of the space finite strain beam theory assuming the bernoulli hypothesis and neglecting the warping deformations of the cross section finite displacements and rotations as well as finite extensional shear torsional and bending strains are accounted for in the formulation a deformed configuration of the beam is described by the displacement vector of the deformed centroid axis and an orthonormal moving frame rigidly attached to the cross section of the beam the position of the moving frame relative to a fixed reference frame is specified by an orthogonal matrix parametrized by the rotational vector which rotates the moving frame from an arbitrary position into the deformed configuration in one step also the incremental rotational vector is introduced which rotates the moving frame from the configuration obtained at the previous iteration step into the current configuration of the beam its components relative to the fixed global coordinate system are taken to be the rotational degrees of freedom at nodal points because in 3 d space both the axial and the follower moments are non conservative not the variational principle but the principle of virtual work has been introduced as a basis for the finite element discretization here we have proposed the generalized form of the principle of virtual work by including exact kinematic equations by means of a procedure similar to that of lagrangian multipliers this makes possible the elimination of the displacement vector field from the principle so that the three components of the incremental rotational vector field remain the only functions to be approximated in the finite element implementation of the principle other researchers on the other hand employ the three components of the incremental rotational vector field and the three components of the incremental displacement vector field as a result more accurate and efficient family of beam finite elements for the non linear analysis of space frames has been obtained a one field formulation results in the fact that in the present finite elements the locking never occurs any combination of deformation states is described equally precisely this is in contrast with the elements developed in literature where in order to avoid the locking a reduced numerical integration has to be applied which unfortunately diminishes the accuracy of the solution polynomials have been chosen for the approximation of the components of the rotational vector in this case the order of the numerical integration can rationally be estimated and the computer program can be coded in such a way that the degree of polynomials need not be limited to a particular value the newton method is used for the iterative solution of the non linear equilibrium equations in an non equilibrium configuration the tangent stiffness matrix obtained by the linearization of governing equations using the directional derivative is non symmetric even for conservative loadings only upon achieving an equilibrium state the tangent stiffness matrix becomes symmetric thus obtained tangent stiffness matrix can be symmetrized without affecting the rate of convergence of the newton method for non conservative loadings however the tangent stiffness matrix is always non symmetric the numerical examples demonstrate capability of the present formulation to determine accurately the non linear behaviour of space frames in numerical examples the out of plane buckling loads are determined and the whole pre and post critical load displacement paths of a cantilever and a right angle frame are traced these in the analysis of space beams standard verification example problems show excellent accuracy of the solution even when employing only one element to describe the displacements of the size of the structure itself the rotations of 2n and extensional strains much beyond the realistic values of linear elastic material

co rotational formulation for geometric nonlinear analysis of doubly symmetric thin walled beams abstract a doubly symmetric thin walled beam element with open section is derived using co rotational cr total lagrangian tl formulation the effects of deformation dependent third order terms of element nodal forces on the buckling load and post buckling behavior are investigated all coupling among bending twisting and stretching deformations for beam element is considered by consistent second order linearization of the fully geometrically nonlinear beam theory however all third order terms of nodal forces which are relevant to the twist rate rate of twist rate and curvature of the beam axis are also considered an incremental iterative method based on the newton raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations the zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state a parabolic interpolation method of the arc length is used to find the buckling load numerical examples are presented to demonstrate the accuracy and efficiency of the proposed element and to investigate the effect of third order terms of element nodal forces on the buckling load and post buckling behavior of doubly symmetric thin walled beams

multi species simple exclusion processes abstract a motility mechanism based on a simple exclusion process where the movement of discrete agents on a lattice is either unbiased symmetric or biased asymmetric is considered estimates of diffusivities from tracking data do not describe the population level response of the system this mismatch between the individual level and population level behaviour can be resolved by averaging the individual level mechanism in terms of an expected site occupancy new insight into simple exclusion processes is obtained by representing the system as a series of interacting subpopulations this formalism leads to a system of nonlinear advection diffusion equations which can be interpreted in terms of the agent fluxes these interactions have consequences for both agent based modelling and continuum modelling in cell biology such as tracking subpopulations of cells within a total cell population

when push comes to shove exclusion processes with nonlocal consequences abstract stochastic agent based models are useful for modelling collective movement of biological cells lattice based random walk models of interacting agents where each site can be occupied by at most one agent are called simple exclusion processes an alternative motility mechanism to simple exclusion is formulated in which agents are granted more freedom to move under the compromise that interactions are no longer necessarily local this mechanism is termed shoving a nonlinear diffusion equation is derived for a single population of shoving agents using mean field continuum approximations a continuum model is also derived for a multispecies problem with interacting subpopulations which either obey the shoving rules or the simple exclusion rules numerical solutions of the derived partial differential equations compare well with averaged simulation results for both the single species and multispecies processes in two dimensions while some issues arise in one dimension for the multispecies case

corotational total lagrangian formulation for three dimensional beamelement a corotational total lagrangian formulation of beam element is presented for the nonlinear analysis of three dimensional beam structures with large rotations but small strains the nonlinear coupling among the bending twisting and stretching deformations is considered all of the element deformations and equations are defined in body attached element coordinates three rotation parameters are proposed to determine the orientation of the element cross section two sets of element nodal parameters termed explicit nodal parameters and implicit nodal parameters are introduced the explicit nodal parameters are used in the assembly of the system equations from the element equations and chosen to be three components of the incremental translation vector and three components of the incremental rotation vector the implicit nodal parameters are used to determine the deformations of the beam element and chosen to be three components of the total displacement vector and nodal values of the three rotation parameters the element internal nodal forces corresponding to the implicit nodal parameters are obtained from the virtual work principle numerical examples are presented and compared with the numerical and experimental results reported in the literature to demonstrate the accuracy and efficiency of the proposed method hree dimensional beams are very important structural elements in all types of engineering systems in many applications these beam elements undergo finite rotations that require a nonlinear formulation to their structural analysis the development of new and efficient formulations for nonlinear analysis of beam structures has attracted the study of many researchers in recent years based on different kinematic assumptions different alternative formulation strategies and procedures to accommodate large rotation capability during the large deformation process have been presented 1 20 the kinematic assumptions used in refs 4 5 7 11 and 17 19 are based on timoshenko s hypothesis the effect of stretching bending torsion and transverse shear are taken into account

nonlinear scattering theory for a class of wave equations in hs abstract for the semi linear higher order wave equation and the nonlinear higher order schrodinger equation we show that the scattering operators map a band in h s into h s if the nonlinearities have sub critical powers in h s the smoothness of the scattering operators and the uniform boundedness of strong solutions for the defocusing nls equation are also shown provided that the nonlinearities have subcritical growth in h 1 moreover the spatial decaying behavior of solutions in energy space for the defocusing nls equation are obtained

an initial boundary value problem of 2d nonlinear schrodinger equation in a strip abstract the paper studies the initial boundary value problem ibvp of a nonlinear schrodinger equation posed on a strip domain r 0 1 with non homogeneous dirichlet boundary conditions for 0 s 1 2 if the initial condition x y is in sobolev space h s r 0 1 and the boundary condition h x t is in h s r 2 h x t l 2 r 2 1 1 2 1 2 s 2 h l 2 r 2 where h is the fourier transform of h with respect to t and x the local well posedness of the ibvp in c 0 t h s r 0 1 is proved for 1 2 s 5 2 because of compatibility conditions the same local well posedness result holds if an extra condition h x 0 h s 1 2 r is added moreover the global well posedness is obtained for s 1 the basic idea used here relies on the derivation of an integral operator for the non homogeneous boundary data and the proof of strichartz s estimates for this operator after the problem is transformed to finding a fixed point of an integral operator the contraction mapping argument then yields a fixed point using the strichartz s estimates for initial and boundary operators the global well posedness is proved using a priori estimates of the solutions

anomalous diffusion stable processes and generalized functions the evolution equations in real space and time corresponding to a class of anomalous diffusion processes arc examined as special cases evolution equations corresponding to stable processes are derived using the theory of generalized functions recovering some known results differently interpreted and an evolution law for stable processes of order unity

nonlinear diffusion and exclusion processes with contact interactions exclusion processes on a regular lattice are used to model many biological and physical systems at a discrete level the average properties of an exclusion process may be described by a continuum model given by a partial differential equation we combine a general class of contact interactions with an exclusion process we determine that many different types of contact interactions at the agent level always give rise to a nonlinear diffusion equation with a vast variety of diffusion functions d c we find that these functions may be dependent on the chosen lattice and the defined neighborhood of the contact interactions mild to moderate contact interaction strength generally results in good agreement between discrete and continuum models while strong interactions often show discrepancies between the two particularly when d c takes on negative values we present a measure to predict the goodness of fit between the discrete and continuous model and thus the validity of the continuum description of a motile contact interacting population of agents this work has implications for modeling cell motility and interpreting cell motility assays giving the ability to incorporate biologically realistic cell cell interactions and develop global measures of discrete microscopic data

limit of the solution of a nonlinear schrodinger equation at blow up time we consider the solution of the equation i v t and 0 where v is defined on 0 t rn and is a nonlinear complex valued function let us denote the function x v t x by v t we suppose that v blows up in the norm h1 at time t under suitable assumptions on and we prove the existence of a limit for v t when t t in appropriate spaces furthermore we specify the behavior of v near blow up time

almost sure local well posedness and scattering for the 4d cubic nonlinear schrodinger equation abstract we consider the cauchy problem for the defocusing cubic nonlinear schrodinger equation in four space dimensions and establish almost sure local well posedness and conditional almost sure scattering for random initial data in h x s r 4 with 1 3 s 1 the main ingredient in the proofs is the introduction of a functional framework for the study of the associated forced cubic nonlinear schrodinger equation which is inspired by certain function spaces used in the study of the schrodinger maps problem and is based on strichartz spaces as well as variants of local smoothing inhomogeneous local smoothing and maximal function spaces additionally we prove an almost sure scattering result for randomized radially symmetric initial data in h x s r 4 with 1 2 s 1

local well posedness for the h 2 critical nonlinear schrodinger equation in this paper we consider the nonlinear schrodinger equation iu t delta u lambda u frac 4 n 4 u in r n n ge 5 with lambda in c we prove local well posedness local existence unconditional uniqueness continuous dependence in the critical space dot h 2 r n

global and exploding solutions for nonlocal quadratic evolution problems nonlinear nonlocal parabolic equations modeling the evolution of density of mutually interacting particles are considered the inertial type nonlinearity is quadratic and nonlocal while the diffusive term also nonlocal is anomalous and fractal i e represented by a fractional power of the laplacian conditions for global in time existence versus finite time blow up are studied self similar solutions are constructed for certain homogeneous initial data

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

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

aggregation patterns from nonlocal interactions discrete stochastic and continuum modeling conservation equations governed by a nonlocal interaction potential generate aggregates from an initial uniform distribution of particles we address the evolution and formation of these aggregating steady states when the interaction potential has both attractive and repulsive singularities currently no existence theory for such potentials is available we develop and compare two complementary solution methods a continuous pseudoinverse method and a discrete stochastic lattice approach and formally show a connection between the two interesting aggregation patterns involving multiple peaks for a simple doubly singular attractive repulsive potential are determined for a swarming morse potential characteristic slow fast dynamics in the scaled inverse energy is observed in the evolution to steady state in both the continuous and discrete approaches the discrete approach is found to be remarkably robust to modifications in movement rules related to the potential function the comparable evolution dynamics and steady states of the discrete model with the continuum model suggest that the discrete stochastic approach is a promising way of probing aggregation patterns arising from two and three dimensional nonlocal interaction conservation equations

continuum models of cohesive stochastic swarms the effect of motility on aggregation patterns abstract mathematical models of swarms of moving agents with non local interactions have many applications and have been the subject of considerable recent interest for modest numbers of agents cellular automata or related algorithms can be used to study such systems but in the present work instead of considering discrete agents we discuss a class of one dimensional continuum models in which the agents possess a density x t at location x at time t the agents are subject to a stochastic motility mechanism and to a global cohesive inter agent force the motility mechanisms covered include classical diffusion nonlinear diffusion which may be used to model in a phenomenological way volume exclusion or other short range local interactions and a family of linear redistribution operators related to fractional diffusion equations a variety of exact analytic results are discussed including equilibrium solutions and criteria for unimodality of equilibrium distributions full time dependent solutions and transitions between asymptotic collapse and asymptotic escape we address the behaviour of the system for diffusive motility in the low diffusivity limit for both smooth and singular interaction potentials and show how this elucidates puzzling behaviour in fully deterministic non local particle interaction models we conclude with speculative remarks about extensions and applications of the models

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

the role of non linear theories in transient dynamic analysis of flexible structures abstract transient dynamic analysis of flexible structures undergoing large motions is considered for rotating structures it is explicitly shown that appropriate account of the influence of centrifugal force on the bending stiffness requires the use of a geometrically non linear at least second order beam theory use of a first order linearized linear beam theory results in a spurious loss of bending stiffness for a rotating plane beam a set of linear partial differential equations of motion that includes all inertia effects coriolis centrifugal acceleration of revolution and coupling between extensional and flexural deformations is derived from the fully non linear beam theory by consistent linearization the analysis is subsequently extended to the more general case of a plate accomodating shear deformation and undergoing a general three dimensional rotating motion the discretization process of the resulting linear equations of motion for the beam and the plate is also discussed

continuous dependence of cauchy problem for nonlinear schrodinger equation in hs abstract we consider the cauchy problem for the nonlinear schrodinger equation i t u u 0 u 1 u u in r n where 0 1 c in h s subcritical and critical case 0 4 n 2 s when 1 s n 2 and 0 when s n 2 we show that the solution depends continuously on the initial value in the standard sense in h s r n if satisfies certain assumptions

hs global well posedness for semilinear wave equations abstract we consider the cauchy problem for semilinear wave equations in r n with n 3 making use of bourgain s method in conjunction with the endpoint strichartz estimates of keel and tao we establish the hs global well posedness with s