on the hughes model for pedestrian flow the one dimensional case in this paper we investigate the mathematical theory of hughes model for the ow of pedestrians cf 17 consisting of a nonlinear conservation law for the density of pedestrians coupled with an eikonal equation for a potential modelling the common sense of the task we rst consider an approximation of the original model in which the eikonal equation is replaced by an elliptic approximation for such an approximated system we prove existence and uniqueness of entropy solutions in one space dimension in the sense of kru zkov 21 in which the boundary conditions are posed following the approach of bardos et al 7 we use bv estimates on the density and stability estimates on the potential in order to prove uniqueness furthermore we analyse the evolution of characteristics for the original hughes model in one space dimension and study the behaviour of simple solutions in order to reproduce interesting phenomena related to the formation of shocks and rarefaction waves the characteristic calculus is supported by numerical simulations

convergence analysis for operator splitting methods applied to conservation laws with stiff source terms we analyze the order of convergence for operator splitting methods applied to conservation laws with stiff source terms we suppose that the source term q u is dissipative it is proved that the l 1 error introduced by the time splitting can be bounded by o d t vert q u 0 vert l 1 which is an improvement of the o q d t upper bound where d t is the splitting time step q is the lipschitz constant of q or q max u vert q u vert in case q is smooth a generic model with a special form of stiff source is also investigated we propose a nonuniform temporal mesh to eliminate the effect of the initial layer introduced by the stiff source term our results are derived by using parabolic regularizations rather than using kuznetsov s approximation theory which has been employed as a standard approach for error analysis to the dimensional or time splitting methods numerical examples are presented to illustrate the theoretical results

a non local elliptic hyperbolic system related to the short pulse equation abstract in this paper we prove the well posedness of a non local elliptic hyperbolic system related to the short pulse equation it is a model which describes the evolution of the electrical field of linearly polarized continuum spectrum pulses in optical waveguides including fused silica telecommunication type or photonic crystal fibers as well as hollow capillaries filled with transparent gases

cubic correspondences arising from g 2 we use the automorphic realization of the minimal representation the theta representation of the three fold cover of g 2 to construct a correspondence of automorphic forms between members of dual pairs inside g 2 the dual pairs under consideration are sl2 sl2 and sl3 z 3 in the first case we obtain the shimura correspondence between the three fold cover of sl2 and sl2 through explicit integral formulae in the second case we decompose theta over the three fold cover of sl3 both locally and globally the constituents are parametrized by their central characters and are cuspidal except essentially when the central character is trivial we thus obtain cuspidal theta representations and the unramified theta representation of the three fold cover of sl3

a fast galerkin finite element method for a space time fractional allen cahn equation abstract in this paper a space time fractional allen cahn equation is investigated to account for the memory effect of certain materials or the anomalous diffusion processes in heterogeneous distributed media for the mixture of two immiscible phases due to the nonlocal features of fractional derivatives conventional solvers such as the gauss elimination method require o m 2 m n memory units and o m n 2 m 3 n operations to solve the space time fractional allen cahn equation where m is the number of spatial unknowns and n is the number of time steps by exploring the special structure of the stiffness matrix and utilizing the invariant energy quadratization approach we developed an efficient linearized divide and conquer galerkin finite element method without resorting to any lossy compression this new method is efficient as it significantly reduces the memory requirement to o m n and the operation count to o m n log 2 n log m we have developed a fast galerkin finite element method with local refinement that allows accurate capture of the coarsening dynamics at the interface furthermore we have investigated the modeling capacities and energy dissipation properties of the space time fractional allen cahn equation numerical experiments are presented to demonstrate the efficiency of the new methods and the flexibility of turnable sharpness and decay behavior of the space time fractional allen cahn equation

a stable and conservative finite difference scheme for the cahn hilliard equation we propose a stable and conservative finite difference scheme to solve numerically the cahn hilliard equation which describes a phase separation phenomenon numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill posed problem we design a new difference scheme based on a general strategy proposed recently by furihata and mori the new scheme inherits characteristic properties the conservation of mass and the decrease of the total energy from the equation the decrease of the total energy implies boundedness of discretized sobolev norm of the solution this in turn implies by discretized sobolev s lemma boundedness of max norm of the solution and hence the stability of the solution an error estimate for the solution is obtained and the order is o delta x 2 delta t 2 numerical examples demonstrate the effectiveness of the proposed scheme

phase field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing this paper describes a novel numerical algorithm for simulating interfacial dynamics of non newtonian fluids the interface between two immiscible fluids is treated as a thin mixing layer across which physical properties vary steeply but continuously the property and evolution of the interfacial layer is governed by a phase field variable that obeys a cahn hilliard type of convection diffusion equation this circumvents the task of directly tracking the interface and produces the correct interfacial tension from the free energy stored in the mixing layer viscoelasticity and other types of constitutive equations can be incorporated easily into the variational phase field framework the greatest challenge of this approach is in resolving the sharp gradients at the interface this is achieved by using an efficient adaptive meshing scheme governed by the phase field variable the finite element scheme easily accommodates complex flow geometry and the adaptive meshing makes it possible to simulate large scale two phase systems of complex fluids in two dimensional and axisymmetric three dimensional implementations the numerical toolkit is applied here to drop deformation in shear and elongational flows rise of drops and retraction of drops and torii some of these solutions serve as validation of the method and illustrate its key features while others explore novel physics of viscoelasticity in the deformation and evolution of interfaces

a space time fractional phase field model with tunable sharpness and decay behavior and its efficient numerical simulation abstract we present a space time fractional allen cahn phase field model that describes the transport of the fluid mixture of two immiscible fluid phases the space and time fractional order parameters control the sharpness and the decay behavior of the interface via a seamless transition of the parameters although they are shown to provide more accurate description of anomalous diffusion processes and sharper interfaces than traditional integer order phase field models do fractional models yield numerical methods with dense stiffness matrices consequently the resulting numerical schemes have significantly increased computational work and memory requirement we develop a lossless fast numerical method for the accurate and efficient numerical simulation of the space time fractional phase field model numerical experiments shows the utility of the fractional phase field model and the corresponding fast numerical method

existence uniqueness and regularity results on nonlocal balance laws abstract we consider a class of nonlocal balance laws as initial value problems on a finite time horizon and show existence and uniqueness of the corresponding weak solutions the description nonlocal refers to the velocity of the balance law that depends on the weighted integral over an area in space at any given time existence of a weak solution for initial data and right hand side data in l 1 l in l and in special cases in l 1 is shown via the method of characteristics resulting in a fixed point problem in the nonlocal term the uniqueness of a weak solution with relatively weak assumptions on the flux function and the nonlocal term is established so that the uniqueness result does not require the well known kru kov entropy condition as it is typical for local balance laws and was up to now used in the available literature also for nonlocal balance laws

erosion profile by a global model for granular flow in this paper we study an integro differential equation that models the erosion of a mountain profile caused by small avalanches the equation is in conservative form with a non local flux involving an integral of the mountain slope under suitable assumptions on the erosion rate the mountain profile develops several types of singularities which we call kinks shocks and hyper kinks we study the formation of these singularities and derive admissibility conditions furthermore entropy weak solutions to the cauchy problem are constructed globally in time taking limits of piecewise affine approximate solutions entropy and entropy flux functions are introduced and a lax entropy condition is established for the weak solutions

propagation of ultra short optical pulses in cubic nonlinear media abstract we derive a partial differential equation that approximates solutions of maxwell s equations describing the propagation of ultra short optical pulses in nonlinear media and which extends the prior analysis of alterman and rauch phys lett a 264 2000 390 diffractive nonlinear geometric optics for short pulses preprint 2002 we discuss non rigorously conditions under which this approximation should be valid but the main contributions of this paper are 1 an emphasis on the fact that the model equation for short pulse propagation may depend on the details of the optical susceptibility in the wavelength regime under consideration 2 a numerical comparison of solutions of this model equation with solutions of the full nonlinear partial differential equation 3 a local well posedness result for the model equation and 4 a proof that in contrast to the nonlinear schrodinger equation which models slowly varying wavetrains this equation has no smooth pulse solutions which propagate with fixed shape and speed

characterization of minkowski measurability in terms of surface area abstract the r parallel set to a set a in euclidean space consists of all points with distance at most r from a recently the asymptotic behaviour of volume and surface area of the parallel sets as r tends to 0 has been studied and some general results regarding their relations have been established here we complete the picture regarding the resulting notions of minkowski content and s content in particular we show that a set is minkowski measurable if and only if it is s measurable i e if and only if its s content is positive and finite and that positivity and finiteness of the lower and upper minkowski contents imply the same for the s contents and vice versa the results are formulated in the more general setting of kneser functions furthermore the relations between minkowski and s contents are studied for more general gauge functions the results are applied to simplify the proof of the modified weyl berry conjecture in dimension one

on the singular local limit for conservation laws with nonlocal fluxes we give an answer to a question posed in amorim et al esaim math model numer anal 49 1 19 37 2015 which can loosely speaking be formulated as follows consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel in the singular limit where the convolution kernel is replaced by a dirac delta one formally recovers a conservation law can we rigorously justify this formal limit we exhibit counterexamples showing that despite numerical evidence suggesting a positive answer one does not in general have convergence of the solutions we also show that the answer is positive if we consider viscous perturbations of the nonlocal equations in this case in the singular local limit the solutions converge to the solution of the viscous conservation law

numerical methods for hyperbolic conservation laws with stiff relaxation ii higher order godunov methods a higher order godunov method is presented for hyperbolic systems of conservation laws with stiff relaxing source terms the goal is to develop a godunov method that produces higher order accurate

analysis and control of a scalar conservation law modeling a highly re entrant manufacturing system abstract in this paper we study a scalar conservation law that models a highly re entrant manufacturing system as encountered in semi conductor production as a generalization of coron et al 2010 14 the velocity function possesses both the local and nonlocal character we prove the existence and uniqueness of the weak solution to the cauchy problem with initial and boundary data in l we also obtain the stability continuous dependence of both the solution and the out flux with respect to the initial and boundary data finally we prove the existence of an optimal control that minimizes in the l p sense with 1 p the difference between the actual out flux and a forecast demand over a fixed time period

numerical approximation of entropy solutions for hyperbolic integro differential equations scalar hyperbolic integro differential equations arise as models for e g radiating or self gravitating fluid flow we present finite volume schemes on unstructured grids applied to the cauchy problem for such equations for a rather general class of integral operators we show convergence of the approximate solutions to a possibly discontinuous entropy solution of the problem for a specific model problem in radiative hydrodynamics we introduce a convergent fully discrete finite volume scheme under the assumption of sufficiently fast spatial decay of the entropy solution we can even establish the convergence rate h1 4 ln h where h denotes the grid parameter the convergence proofs rely on appropriate variants of the classical kruzhkov method for local balance laws together with a truncation technique to cope with the nonlocal character of the integral operator

scaling behaviour of the time fractional kardar parisi zhang equation the scaling behaviour of the time fractional kardar parisi zhang tfkpz equation in 1 1 dimensions is investigated by scaling analysis and numerical simulations the surface morphology and critical exponents with different fractional orders are obtained the analytical results are consistent with the corresponding numerical solutions based on a caputo type fractional derivative we find that similar to the normal kardar parisi zhang equation anomalous behaviour does not appear in the tfkpz model according to the scaling idea of local slope and numerical evidence however there exists significant finite time effect of local scaling exponents in the tfkpz system our results also imply that memory effects can affect the scaling behaviour of evolving fractional surface growth

a general phase transition model for vehicular traffic an extension of the colombo phase transition model is proposed the congestion phase is described by a two dimensional zone defined around an equilibrium flux known as the classical fundamental diagram general criteria to build such a set valued fundamental diagram are enumerated and instantiated on several equilibrium fluxes with different concavity properties the solution of the riemann problem in the presence of phase transitions is obtained through the construction of a riemann solver which enables the definition of the solution of the cauchy problem using wavefront tracking the free flow phase is described using a newell daganzo fundamental diagram which allows for a more tractable definition of phase transition compared to the original colombo phase transition model the accuracy of the numerical solution obtained by a modified godunov scheme is assessed on benchmark scenarios for the different flux functions constructed

time fractional allen cahn and cahn hilliard phase field models and their numerical investigation abstract we study time fractional allen cahn and cahn hilliard phase field models to account for the anomalously subdiffusive transport behavior in heterogeneous porous materials or memory effect of certain materials we develop an efficient finite difference scheme and a fourier spectral scheme to effectively treat the significantly increased memory requirement and computational complexity which arise due to the nonlocal behavior of the time fractional models for time fractional cahn hilliard model we observe from the numerical results that the bigger the fractional order is the faster the energy decays however for time fractional allen cahn model we derived an opposite conclusion moreover we also study the coarsening dynamics for time fractional cahn hilliard model numerical results reveal that the scaling law for the energy decays as o t 3 which is consistent with the well known result o t 1 3 for integer order cahn hilliard model

a phase field model and its numerical approximation for two phase incompressible flows with different densities and viscosities modeling and numerical approximation of two phase incompressible flows with different densities and viscosities are considered a physically consistent phase field model that admits an energy law is proposed and several energy stable efficient and accurate time discretization schemes for the coupled nonlinear phase field model are constructed and analyzed ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for problems with large density and viscosity ratios

existence uniqueness and regularity of multi dimensional nonlocal balance laws with damping abstract in this article we generalize some existence and uniqueness results in 22 for scalar nonlocal balance laws to multi dimensional nonlocal balance laws also in the multi dimensional case it is possible to get rid of the entropy condition which is usually postulated to guarantee a unique weak solution of local balance laws but which we prove to be redundant in the nonlocal case we start with a well known existence theorem for linear balance laws and the explicit formula for a solution to obtain a fixed point problem in the nonlocal term using banach s fixed point theorem we prove existence and uniqueness of a fixed point this existence and uniqueness result is used for constructing and proving a unique solution of the multi dimensional nonlocal balance law on a sufficiently small time horizon while in many publications the nonlocal term will be considered over the entire space domain we present an existence and uniqueness result which allows rather general integration areae for the nonlocal term eventually we obtain a unique weak solution on every finite time horizon existence and uniqueness are complemented by stability results and an exhaustive study on more regular weak solutions in case the data is more regular

rankin selberg integrals for the group g2 introduction the problem of determining the analytic continuation and functional equation of general l functions associated to automorphic representations of reductive lie groups and representations of the l groups has been pursued by 2 different techniques the first is the interpretation of the l function as arising in the constant term of eisenstein series the langlands shahidi method the second is the generalized rankin selberg integral representation of the l function we refer the reader to gs for an excellent description of both methods and a comparison of the various techniques used in any event the langlands shahidi method determines a prescribed list of l functions this list does not include all the possible cases of l functions that arise the rankin selberg method has so far yielded only some of the l functions that come about by the first method one of the major questions thus is whether the rankin selberg method will give new examples of l functions not possible by langlands shahidi method we show that in fact this is the case we construct here via the rankin selberg method the l function associated to the tensor product of the standard 7 dimensional representation of g2 with the standard 2 dimensional representation of gl2 in fact as a bonus of our method we also represent the l function of the standard 7 dimensional representation of g2

structure of entropy solutions for multi dimensional scalar conservation laws an entropy solution u of a multi dimensional scalar conservation law is not necessarily in bv even if the conservation law is genuinely nonlinear we show that u nevertheless has the structure of a bv function in the sense that the shock location is codimension one rectifiable this result highlights the regularizing effect of genuine nonlinearity in a qualitative way it is based on the locally finite rate of entropy dissipation the proof relies on the geometric classification of blow ups in the framework of the kinetic formulation

analysis and approximation of a fractional cahn hilliard equation we derive a fractional cahn hilliard equation fche by considering a gradient flow in the negative order sobolev space h alpha alpha in 0 1 where the choice alpha 1 corresponds to the classical cahn hilliard equation while the choice alpha 0 recovers the allen cahn equation the existence of a unique solution is established and it is shown that the equation preserves mass for all positive values of fractional order alpha and that it indeed reduces the free energy we then turn to the delicate question of the l infty boundedness of the solution and establish an l infty bound for the fche in the case where the nonlinearity is a quartic polynomial as a consequence of the estimates we are able to show that the fourier galerkin method delivers a spectral rate of convergence for the fche in the case of a semidiscrete approximation scheme finally we present results obtained using computational simulation of the fche for a variety of choices of fractional order alpha it is

a rankin selberg integral for the adjoint representation of gl3 in this paper we construct steps 1 4 and par t of 2 as formulated at g sh p 19 for the funct ion l s r here is a cusp form on g l a and r is the eight dimensional adjoint representa t ion of g l a the l group of g l 3 to do so we construct a rankin selberg integral using the embedding of sl 3 in gz let us ment ion tha t the langland s conjucture for this l function can not be obta ined by the langlands shahid i method the paper is organized as follows in the first section we collect some informat ion on the s t ructure of gz and compute the space of double cosets p gz sl3 where p is a maximal parabol ic subgroup of g2 in sect two the global integral is const ructed and is shown to be euler ian with whi t t ake r model finally in sect three we carry out the unramif ied computa t ion using the local computa t ion of the integral for gl3 x gl3 as const ructed in i j s for simplicity we shall assume tha t rc has a trivial central character

on the global well posedness of bv weak solutions to the kuramoto sakaguchi equation abstract the kuramoto model is a prototype phase model describing the synchronous behavior of weakly coupled limit cycle oscillators when the number of oscillators is sufficiently large the dynamics of kuramoto ensemble can be effectively approximated by the corresponding mean field equation namely the kuramoto sakaguchi ks equation this ks equation is a kind of scalar conservation law with a nonlocal flux function due to the mean field interactions among oscillators in this paper we provide a unique global solvability of bounded variation bv weak solutions to the kinetic ks equation for identical oscillators using the method of front tracking in hyperbolic conservation laws moreover we also show that our bv weak solutions satisfy local in time l 1 stability with respect to bv initial data for the ensemble of identical kuramoto oscillators we explicitly construct an exponentially growing bv weak solution generated from bv perturbation of incoherent state for any positive coupling strength this implies the nonlinear instability of incoherent state in a positive coupling strength regime we provide several numerical examples and compare them with our analytical results

an error estimate for finite volume methods for multidimensional conservation laws in this paper an l ll error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained these methods can be formally high order accurate and are defined on general triangulations the error is proven to be of order ft 4 where h represents the size of the mesh via an extension of kuznetsov approximation theory for which no estimate of the total variation and of the modulus of con tinuity in time are needed the result is new even for the finite volume method constructed from monotone numerical flux functions

convergence of the finite volume method for multidimensional conservation laws we establish the convergence of the finite volume method applied to multidimensional hyperbolic conservation laws and based on monotone numerical flux functions our technique applies with a fairly unrestrictive assumption on the triangulations flat elements are allowed and to lipschitz continuous flux functions we treat the initial and boundary value problem and obtain the strong convergence of the scheme to the unique entropy discontinuous solution in the sense of kruzkov the proof of convergence is based on a convergence framework coquel and lefloch math comp 57 1991 pp 169 210 and j numer anal 30 1993 pp 675 700 from a convex decomposition of the scheme we derive a new estimate for the rate of entropy dissipation and a new formulation of the discrete entropy inequalities these estimates are shown to be sufficient for the passage to the limit in the discrete equation convergence follows from diperna s uniqueness result in the class of entropy measure valued solutions

first and second order error estimates for the upwind source at interface method the upwind source at interface method for hyperbolic conservation laws with source term introduced in b perthame c simeoni convergence of the upwind interface source method for hyperbolic conservation laws hyperbolic problems theory numerics applications t hou and e tadmor eds springer 2003 is essentially first order accurate under appropriate hypotheses of consistency on the finite volume discretization of the source term we prove l p error estimates 1 le p infty in the case of a uniform spatial mesh for which an optimal result can be obtained we thus conclude that the same convergence rates hold as for the corresponding homogeneous problem to improve the numerical accuracy we develop two different approaches of dealing with the source term and we discuss the question of deriving second order error estimates numerical evidence shows that those techniques produce high resolution schemes compatible with the upwind source at interface method

a novel linear second order unconditionally energy stable scheme for a hydrodynamic q tensor model of liquid crystals abstract the hydrodynamic q tensor model has been used for studying flows of liquid crystals and liquid crystal polymers it can be derived from a variational approach together with the generalized onsager principle in which the total energy decreases in time in this paper we develop a novel linear second order semi discrete scheme in time to solve the governing system of equations in the model the scheme is developed following the novel energy quadratization strategy so that it is linear and unconditionally energy stable at the semi discrete level this scheme is further discretized in space using a second order finite difference method and implemented on a gpu for high performance computing the convergence rate in time is established using a mesh refinement test several numerical examples are presented to demonstrate the usefulness of the model and the effectiveness of the numerical scheme in simulating defect dynamics in flows of liquid crystals

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

deterministic particle approximation for nonlocal transport equations with nonlinear mobility abstract we construct a deterministic lagrangian many particle approximation to a class of nonlocal transport pdes with nonlinear mobility arising in many contexts in biology and social sciences the approximating particle system is a nonlocal version of the follow the leader scheme we rigorously prove that a suitable discrete piece wise density reconstructed from the particle scheme converges strongly in l l o c 1 towards the unique entropy solution to the target pde as the number of particles tends to infinity the proof is based on uniform bv estimates on the approximating sequence and on the verification of an approximated version of the entropy condition for large number of particles as part of the proof we also prove uniqueness of entropy solutions we further provide a specific example of non uniqueness of weak solutions and discuss the interplay of the entropy condition with the steady states finally we produce numerical simulations supporting the need of a concept of entropy solution in order to get a well posed semigroup in the continuum limit and showing the behaviour of solutions for large times

well posedness of a conservation law with non local flux arising in traffic flow modeling we prove the well posedness of entropy weak solutions of a scalar conservation law with non local flux arising in traffic flow modeling the result is obtained providing accurate mathbf l infty l bv and mathbf l 1 l1 estimates for the sequence of approximate solutions constructed by an adapted lax friedrichs scheme

error bounds for fractional step methods for conservation laws with source terms fractional step methods have been used to approximate solutions of scalar conservation laws with source terms in this paper the stability and accuracy of the basic fractional step algorithms are analyzed when these algorithms are used to compute discontinuous solutions of nonhomogeneous scalar conservation laws the authors show that time splitting methods for conservation laws with source terms always converge to the unique weak solution satisfying the entropy condition in particular it is proved that the l 1 errors in the splitting methods are bounded by o sqrt delta t where delta t is the splitting time step the l 1 convergence rate of a class of fully discrete splitting methods is also investigated

nonlocal systems of balance laws in several space dimensions with applications to laser technology for a class of systems of nonlinear and nonlocal balance laws in several space dimensions we prove the local in time existence of solutions and their continuous dependence on the initial datum the choice of this class is motivated by a new model devoted to the description of a metal plate being cut by a laser beam using realistic parameters solutions to this model obtained through numerical integrations meet qualitative properties of real cuts moreover the class of equations considered comprises a model describing the dynamics of solid particles along a conveyor belt

controllability for a scalar conservation law with nonlocal velocity abstract in this paper we study the state controllability and nodal profile controllability for a scalar conservation law with a nonlocal velocity that models a highly re entrant manufacturing system as encountered in semi conductor production we first prove a local state controllability result i e there exists a control that drives the solution from any given initial data to any desired final data in a certain time period provided that the initial and final data are both close to a given equilibrium 0 we also obtain a global state controllability result for the same system where there is no limitation on the distance between the initial and final data finally we prove a nodal profile controllability result i e there exists a control under which the solution starts from any initial data reaches exactly any given out flux over a fixed time period

asymptotic behaviour of nonlocal p laplacian reaction diffusion problems abstract in this paper we focus on studying the existence of attractors in the phase spaces l 2 and l p among others for time dependent p laplacian equations with nonlocal diffusion and nonlinearities of reaction diffusion type firstly we prove the existence of weak solutions making use of a change of variable which allows us to get rid of the nonlocal operator in the diffusion term thereupon the regularising effect of the equation is shown applying an argument of a posteriori regularity since under the assumptions made we cannot guarantee the uniqueness of weak solutions in addition this argument allows to ensure the existence of an absorbing family in w 0 1 p this leads to the existence of the minimal pullback attractors in l 2 l p and some other spaces as l p relationships between these families are also established

coarsening kinetics from a variable mobility cahn hilliard equation application of a semi implicit fourier spectral method an efficient semi implicit fourier spectral method is implemented to solve the cahn hilliard equation with a variable mobility the method is orders of magnitude more efficient than the conventional forward euler finite difference method thus allowing us to simulate large systems for longer times we studied the coarsening kinetics of interconnected two phase mixtures using a cahn hilliard equation with its mobility depending on local compositions in particular we compared the kinetics of bulk diffusion dominated and interface diffusion dominated coarsening in two phase systems results are compared with existing theories and previous computer simulations

the initial boundary value problem for general non local scalar conservation laws in one space dimension we prove global well posedness results for weak entropy solutions of bounded variation bv of scalar conservation laws with non local flux on bounded domains under suitable regularity assumptions on the flux function in particular existence is obtained by proving the convergence of an adapted lax friedrichs algorithm lipschitz continuos dependence from initial and boundary data is derived applying kru hkov s doubling of variable technique

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

a phase field description of dynamic brittle fracture abstract in contrast to discrete descriptions of fracture phase field descriptions do not require numerical tracking of discontinuities in the displacement field this greatly reduces implementation complexity in this work we extend a phase field model for quasi static brittle fracture to the dynamic case we introduce a phase field approximation to the lagrangian for discrete fracture problems and derive the coupled system of equations that govern the motion of the body and evolution of the phase field we study the behavior of the model in one dimension and show how it influences material properties for the temporal discretization of the equations of motion we present both a monolithic and staggered time integration scheme we study the behavior of the dynamic model by performing a number of two and three dimensional numerical experiments we also introduce a local adaptive refinement strategy and study its performance in the context of locally refined t splines we show that the combination of the phase field model and local adaptive refinement provides an effective method for simulating fracture in three dimensions

a second order and nonuniform time stepping maximum principle preserving scheme for time fractional allen cahn equations abstract in this work we present a second order nonuniform time stepping scheme for the time fractional allen cahn equation we show that the proposed scheme preserves the discrete maximum principle and by using the convolution structure of consistency error we present sharp maximum norm error estimates which reflect the temporal regularity as our analysis is built on nonuniform time steps we may resolve the intrinsic initial singularity by using the graded meshes moreover we propose an adaptive time stepping strategy for large time simulations numerical experiments are presented to show the effectiveness of the proposed scheme this seems to be the first second order maximum principle preserving scheme for the time fractional allen cahn equation

a non uniform time stepping convex splitting scheme for the time fractional cahn hilliard equation abstract in this paper a non uniform time stepping convex splitting numerical algorithm for solving the widely used time fractional cahn hilliard equation is introduced the proposed numerical scheme employs the l 1 formula for discretizing the time fractional derivative and a second order convex splitting technique to deal with the non linear term semi implicitly then the pseudospectral method is utilized for spatial discretization as a result the fully discrete scheme has several advantages second order accurate in time spectrally accurate in space uniquely solvable mass preserving and unconditionally energy stable rigorous proofs are given along with several numerical results to verify the theoretical results and to show the accuracy and effectiveness of the proposed scheme also some interesting phase separation dynamics of the time fractional cahn hilliard equation has been investigated

on a rankin selberg integral of the l function for widetilde mathrm sl _2 times mathrm gl _2 sl 2 gl2 we present a rankin selberg integral on the exceptional group g 2 which represents the l function for generic cuspidal representations of widetilde mathrm sl 2 times mathrm gl 2 as an application we show that certain fourier jacobi type periods on g 2 are non vanishing

boundary conditions for nonlinear hyperbolic systems of conservation laws we propose two formulations of the boundary conditions for nonlinear hyperbolic systems of conservation laws a first approach is based on the vanishing viscosity method and a second one is related to the riemann problem the equivalence between these two conditions is studied the latter formulation is extended to treat numerically physically relevant boundary conditions monodimensional experiments are presented

a fractional phase field model for two phase flows with tunable sharpness algorithms and simulations abstract we develop a fractional extension of a mass conserving allen cahn phase field model that describes the mixture of two incompressible fluids the fractional order controls the sharpness of the interface which is typically diffusive in integer order phase field models the model is derived based on an energy variational formulation an additional constraint is employed to make the allen cahn formulation mass conserving and comparable to the cahn hilliard formulation but at reduced cost the spatial discretization is based on a petrov galerkin spectral method whereas the temporal discretization is based on a stabilized adi scheme both for the phase field equation and for the navier stokes equation we demonstrate the spectral accuracy of the method with fabricated smooth solutions and also the ability to control the interface thickness between two fluids with different viscosity and density in simulations of two phase flow in a pipe and of a rising bubble we also demonstrate that using a formulation with variable fractional order we can deal simultaneously with both erroneous boundary effects and sharpening of the interface at no extra resolution

convergence of the ostrovsky equation to the ostrovsky hunter one abstract we consider the ostrovsky equation which contains nonlinear dispersive effects we prove that as the diffusion parameter tends to zero the solutions of the dispersive equation converge to discontinuous weak solutions of the ostrovsky hunter equation the proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the l p setting

fractional cahn hilliard allen cahn and porous medium equations abstract we introduce a fractional variant of the cahn hilliard equation settled in a bounded domain r n and complemented with homogeneous dirichlet boundary conditions of solid type i e imposed in the whole of r n after setting a proper functional framework we prove existence and uniqueness of weak solutions to the related initial boundary value problem then we investigate some significant singular limits obtained as the order of either of the fractional laplacians appearing in the equation is let tend to 0 in particular we can rigorously prove that the fractional allen cahn fractional porous medium and fractional fast diffusion equations can be obtained in the limit finally in the last part of the paper we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits

computational studies of coarsening rates for the cahn hilliard equation with phase dependent diffusion mobility we study computationally coarsening rates of the cahn hilliard equation with a smooth double well potential and with phase dependent diffusion mobilities the latter is a feature of many materials systems and makes accurate numerical simulations challenging our numerical simulations confirm earlier theoretical predictions on the coarsening dynamics based on asymptotic analysis we demonstrate that the numerical solutions are consistent with the physical gibbs thomson effect even if the mobility is degenerate in one or both phases for the two sided degenerate mobility we report computational results showing that the coarsening rate is on the order of l c t 1 4 independent of the volume fraction of each phase for the one sided degenerate mobility that is non degenerate in the positive phase but degenerate in the negative phase we illustrate that the coarsening rate depends on the volume fraction of the positive phase for large positive volume fractions the coarsening rate is on the order of l c t 1 3 and for small positive volume fractions the coarsening rate becomes l c t 1 4

numerical methods for hyperbolic conservation laws with stiff relaxation i spurious solutions the author considers the numerical solution of hyperbolic systems of conservation laws with relaxation using a shock capturing finite difference scheme on a fixed uniform spatial grid it is conjectured that certain a priori criteria ensure that the numerical method does not produce spurious solutions as the relaxation time vanishes one criterion is that the limits of vanishing relaxation time and vanishing viscosity commute for the viscous regularization of the hyperbolic system a second criterion is that a certain subcharacteristic condition be satisfied by the hyperbolic system this conjecture is supported with analytical and numerical results for a specific example the solution of generalized riemann problems of a model system of equations with a fractional step scheme in which godunov s method is coupled with the backward euler method similar ideas are applied to the numerical solution of stiff detonation problems

on approximation of local conservation laws by nonlocal conservation laws abstract we show that for monotone initial datum the solution of nonlocal conservation laws converges to the entropy solution of the corresponding local conservation laws when the nonlocal reach tends to zero this particularly covers the principle cases of conservation laws shocks and rarefactions the considered problem is addressed by studying the entropy of the nonlocal conservation laws in the limit and by exploiting the semi explicit solution formula developed in 30

inpainting of binary images using the cahn hilliard equation image inpainting is the filling in of missing or damaged regions of images using information from surrounding areas we outline here the use of a model for binary inpainting based on the cahn hilliard equation which allows for fast efficient inpainting of degraded text as well as super resolution of high contrast images

nonlocal systems of conservation laws in several space dimensions we present a lax friedrichs type algorithm to numerically integrate a class of nonlocal and nonlinear systems of conservation laws in several space dimensions the convergence of the approximate solutions is proved also providing the existence of a solution in a slightly more general setting than in other results in the current literature an application to a crowd dynamics model is considered

regularity theory and adjoint based optimality conditions for a nonlinear transport equation with nonlocal velocity in this contribution the optimal boundary control problem for a first order nonlinear nonlocal hyperbolic pde is studied motivated by various applications ranging from re entrant manufacturing systems to particle synthesis processes we establish the regularity of solutions for w 1 p data based on a general l 2 tracking type cost functional the existence uniqueness and regularity of the adjoint system in w 1 p is derived using the special structure induced from the nonlocal flux function of the state equation the assumption of w 1 p and not l p regularity comes thereby due to the fact that the adjoint equation asks for more regularity to be well defined this problem is discussed in detail and we give a solution by defining a special type of cost functional such that the corresponding optimality system is well defined

convergence of finite volume monotone schemes for scalar conservation laws on bounded domains this paper is devoted to the study of the finite volume methods used in the discretization of conservation laws defined on bounded domains general assumptions are made on the data the initial condition and the boundary condition are supposed to be measurable bounded functions using a generalized notion of solution to the continuous problem namely the notion of entropy process solution see 9 and a uniqueness result on this solution we prove that the numerical solution converges to the entropy weak solution of the continuous problem in l p loc for every p in 1 infty this also yields a new proof of the existence of an entropy weak solution

oleinik type estimates for the ostrovsky hunter equation abstract the ostrovsky hunter equation provides a model of small amplitude long waves in a rotating fluid of finite depth this is a nonlinear evolution equation in this study we consider the well posedness of the cauchy problem associated with this equation within a class of bounded discontinuous solutions we show that we can replace the kruzkov type entropy inequalities with an oleinik type estimate and we prove the uniqueness via a nonlocal adjoint problem this implies that a shock wave in an entropy weak solution to the ostrovsky hunter equation is admissible only if it jumps down in value similar to the inviscid burgers equation

numerical algorithms for simulations of a traffic model on road networks we introduce a simulation algorithm based on a fluid dynamic model for traffic flows on road networks which are considered as graphs composed by arcs that meet at some junctions the approximation of scalar conservation laws along arcs is made by three velocities kinetic schemes with suitable boundary conditions at junctions here we describe the algorithm and we give an example

finite volume schemes for nonhomogeneous scalar conservation laws error estimate in this paper we study finite volume schemes for the nonhomogeneous scalar conservation law u t rm div f x t u q x t u with initial condition u x 0 u0 x the source term may be either stiff or nonstiff in both cases we prove error estimates between the approximate solution given by a finite volume scheme the scheme is totally explicit in the nonstiff case semi implicit in the stiff case and the entropy solution the order of these estimates is h frac 1 4 in space time l 1 norm h denotes the size of the mesh furthermore the error estimate does not depend on the stiffness of the source term in the stiff case

global well posedness and relaxation limits of a model for radiating gas abstract we study the initial value problem for a hyperbolic elliptic coupled system with arbitrary large discontinuous initial data we prove existence and uniqueness for that model by means of l1 contraction and comparison properties moreover after suitable scalings we study both the hyperbolic parabolic and the hyperbolic hyperbolic relaxation limits for that system

well posedness for a slow erosion model abstract we improve in two ways the well posedness results of 2 for a slow erosion model proposed in 11 firstly we study the asymptotic profile when u 0 1 u 0 l where u 0 is the initial datum moreover using a compensated compactness based argument we prove the existence of solutions when u 0 1 u 0 l 3

a continuum model for a re entrant factory high volume multistage continuous production flow through a re entrant factory is modeled through a conservation law for a continuous density variable on a continuous production line augmented by a state equation for the speed of the production along the production line the resulting nonlinear nonlocal hyperbolic conservation law allows fast and accurate simulations little s law is built into the model it is argued that the state equation for a re entrant factory should be nonlinear comparisons of simulations of the partial differential equation pde model and discrete event simulations are presented a general analysis of the model shows that for any nonlinear state equation there exist two steady states of production below a critical start rate a high volume high throughput time state and a low volume low throughput time state the stability of the low volume state is proved output is controlled by adjusting the start rate to a changed demand rate two linear factories and a re entrant factory each one modeled by a hyperbolic conservation law are linked to provide proof of concept for efficient supply chain simulations instantaneous density and flux through the supply chain as well as work in progress wip and output as a function of time are presented extensions to include multiple product flows and preference rules for products and dispatch rules for re entrant choices are discussed

on convergence of numerical schemes for hyperbolic conservation laws with stiff source terms we deal in this study with the convergence of a class of numerical schemes for scalar conservation laws including stiff source terms we suppose that the source term is dissipative but it is not necessarily a lipschitzian function the convergence of the approximate solution towards the entropy solution is established for first and second order accurate muscl and for splitting semi implicit methods

first order quasilinear equations with boundary conditions in the l framework abstract we study a class of first order quasilinear equations on bounded domains in the l framework using the semi kru kov entropy flux pairs we define a weak entropy solution state an existence and uniqueness result and a maximum principle

continuity properties of the solution map for the generalized reduced ostrovsky equation abstract it is shown that the data to solution map for the generalized reduced ostrovsky gro equation is not uniformly continuous on bounded sets in sobolev spaces on the circle with exponent s 3 2 considering that for this range of exponents the gro equation is well posed with continuous dependence on initial data this result makes the continuity of the solution map an optimal property however if a weaker h r topology is used then it is shown that the solution map becomes holder continuous in h s

well posedness and small data scattering for the generalized ostrovsky equation abstract we consider the generalized ostrovsky equation u t x u u p x x we show that the equation is locally well posed in h s s 3 2 for all integer values of p 2 for p 4 we show that the equation is globally well posed for small data in h 5 w 3 1 and moreover it scatters small data the latter results are corroborated by numerical computations which confirm the heuristically expected decay of u l r t r 2 2 r

the numerical solutions for the energy dissipative and mass conservative allen cahn equation abstract there are three criteria when we model with phase field equations the first one is the evolution of phase separation the second one is energy dissipation law the last one is whether we have the law of conservation of mass or not if we minimize the ginzburg landau free energy functional the kinetics of phase separation inherently follows however the other properties such as energy dissipation and mass conservation are more difficult to model hitherto there have been a few proposals to conserve mass using the allen cahn ac equation but we do not have the energy dissipation law in these models when considered in light of local effects of mass correction for the reasons certain researchers still prefer to use the cahn hilliard equation e g biharmonic equation instead of using the allen cahn equation when we consider a mass conserving case this paper is concerned with numerical solutions of the ac equation in the mass conserving space where the solution is corrected at each iteration for mass conservation we introduce two different methods for solving the conservative ac equation with periodic boundary conditions in this work we have the ac equation in common but use two different mass correction operators note that both of mass correction operators satisfy the discrete energy dissipation property first we project the solution of the ac equation to the mass conserving space and then recover the linear convex splitting scheme for the conventional mass conserving equation the linear convex splitting scheme is well known but the convergence of numerical solutions is elaborately studied in terms of gradient projection next we propose the other energy minimizing method on the mass conserving space it translates the solution of the ac equation to the mass conserving space via an energy dissipation operator the dissipation of discrete energy functional for the proposed method is also established as far as the author knows no report has been presented for the numerical scheme having discrete energy dissipation property of the mass conserving ac equation without using the nonlocal and constant multiplier to verify the common properties of mass projection and proposed methods numerically we simulate the evolutions for phase separation dissipation of energy and mass conservation we moreover consider the short and long time stages of evolutions to see the difference between two methods these several tests are provided for its applications of the proposed method for modeling natural phenomena