a source identification problem in a time fractional wave equation with a dynamical boundary condition abstract we focus on a inverse source problem in a partial differential equation containing a fractional derivative in time of the order 1 2 the equation is accompanied with a non standard boundary condition which consists of the classic flux term and the dynamical time fractional derivative term on the one part of the boundary to determine both the solution of the equation and the source term a measurement in a form of a integral over space domain is considered using a time discretization and rothe s method we prove an existence of the strong solution in the suitable functional spaces and the error estimate is established moreover the uniqueness is addressed the results are supported by some numerical experiments

the riemann problem for a class of nonlinear degenerate equations in the present paper we study the riemann problem for degenerate equations in which the pressure is constant over some range of stains by the liu entropy condition the riemann solutions of twelve regions in the v u plane are completely constructed moreover we obtain the invariant regions for the riemann solutions c 2012 elsevier inc all rights reserved

convergence of the fractional step lax friedrichs scheme and godunov scheme for the isentropic system of gas dynamics a convergence theorem of the fractional step lax friedrichs scheme and godunov scheme for an inhomogeneous system of isentropic gas dynamics 1 5 3 is established by using the framework of compensated compactness meanwhile a corresponding existence theorem of global solutions with large data containing the vacuum is obtained

a semi discrete scheme for solving nonlinear hyperbolic type partial integro differential equations using radial basis functions in this paper we propose an effective numerical method for solving nonlinear volterra partial integro differential equations these equations include the partial differentiations of an unknown function and the integral term containing the unknown function as the memory of system radial basis functions and finite difference method as the main techniques play the important role to reduce a nonlinear partial integro differential equation to a linear system of equations some examples are demonstrated to describe the method numerical results confirm the validity and efficiency of the presented method

role of fundamental solutions for optimal lipschitz extensions on hyperbolic space in this paper we consider the problem of finding the relation between absolutely minimizing lipschitz extension of a given function defined over a subset of the hyperbolic space and the viscosity solution of the pde that appears from the associated variational problem here we have shown that the absolute minimizers can be fully characterized by a comparison principle comparison with cones with the fundamental solutions of the associated pde we have finally proved that the three properties i comparison with cones ii absolutely minimizing lipschitz extension and iii viscosity solution of associated pde are equivalent

a finite volume method for a two phase multicomponent polymer flooding multicomponent polymer flooding used in enhanced oil recovery is governed by a system of coupled non strictly hyperbolic conservation laws in the presence of gravity the flux functions need not be monotone and hence designing godunov type upwind schemes is difficult and computationally expensive to overcome this difficulty we use the basic idea of discontinuous flux to reduce the coupled system into an uncoupled system of scalar conservation laws with discontinuous coefficients for these scalar equations we use the dflu flux developed in 5 to construct a second order scheme the scheme is shown to satisfy a maximum principle and the performance of the scheme is shown on both one and two dimensional test problems

global bv entropy solutions and uniqueness for hyperbolic systems of balance laws we consider the cauchy problem for n n strictly hyperbolic systems of nonresonant balance laws

on the initial boundary value problem for the bipolar hydrodynamic model for semiconductors abstract the global existence and zero relaxation limit results of weak solutions of the initial boundary value problem to the bipolar hydrodynamic model for semiconductors are established by the theory of compensated compactness the boundary conditions of weak solutions in the sense of traces are discussed

l 1 convergence to the barenblatt solution for compressible euler equations with damping we study the asymptotic behavior of compressible isentropic flow through a porous medium when the initial mass is finite the model system is the compressible euler equation with frictional damping as t the density is conjectured to obey the well known porous medium equation and the momentum is expected to be formulated by darcy s law in this paper we prove that any l weak entropy solution to the cauchy problem of damped euler equations with finite initial mass converges strongly in the natural l1 topology with decay rates to the barenblatt profile of the porous medium equation the density function tends to the barenblatt solution of the porous medium equation while the momentum is described by darcy s law the results are achieved through a comprehensive entropy analysis capturing the dissipative character of the problem

decay estimates for time fractional and other non local in time subdiffusion equations in mathbb r d we prove optimal estimates for the decay in time of solutions to a rather general class of non local in time subdiffusion equations in mathbb r d an important special case is the time fractional diffusion equation which has seen much interest during the last years mostly due to its applications in the modeling of anomalous diffusion processes we follow three different approaches and techniques to study this particular case a estimates based on the fundamental solution and young s inequality b fourier multiplier methods and c the energy method it turns out that the decay behaviour is markedly different from the heat equation case in particular there occurs a critical dimension phenomenon the general subdiffusion case is treated by method b and relies on a careful estimation of the underlying relaxation function several examples of kernels including the ultraslow diffusion case illustrate our results

the relaxation of the hydrodynamic model for semiconductors to the drift diffusion equations abstract we establish the convergence and consistency of approximate solutions derived by the modified godunov scheme for the initial boundary value problem to a simplified one dimensional hydrodynamic model for semiconductors using the compensated compactness method the traces of weak solutions are introduced and then the weak solutions are proved to satisfy the natural boundary conditions the zero relaxation limit of the hydrodynamic model to the drift diffusion model is proved when the momentum relaxation time tends to zero

regularity and stability analysis for a class of semilinear nonlocal differential equations in hilbert spaces abstract we deal with a class of semilinear nonlocal differential equations in hilbert spaces which is a general model for some anomalous diffusion equations by using the theory of integral equations with completely positive kernel together with local estimates some existence regularity and stability results are established an application to nonlocal partial differential equations is shown to demonstrate our abstract results

godunov type methods for conservation laws with a flux function discontinuous in space scalar conservation laws with a flux function discontinuous in space are approximated using a godunov type method for which a convergence theorem is proved the case where the flux functions at the interface intersect is emphasized a very simple formula is given for the interface flux a numerical comparison between the godunov numerical flux and the upstream mobility flux is presented for two phase flow in porous media a consequence of the convergence theorem is an existence theorem for the solution of the scalar conservation laws under consideration furthermore for regular solutions uniqueness has been shown

completely positive measures and feller semigroups on donne d abord une approche unifiee aux mesures completement positives qui jouent un role important dans la theorie des equations de volterra a travers le theoreme de bernstein puis on etudie la solution d une equation de transport liee par une mesure non negative qu elle definit a l equation qui caracterise les noyaux completement positifs on obtient une desorption complete des generateurs de semigroupes de feller induits par de telles mesures on donne une application aux equations de flux thermique lineaire dans des materiaux avec memoire

global structure of riemann solutions to a system of two dimensional hyperbolic conservation laws abstract the riemann problem for a two dimensional nonstrictly hyperbolic system of conservation laws is considered without the restriction that each jump of the initial data projects one planar elementary wave ten topologically distinct solutions are obtained by applying the method of generalized characteristic analysis some of these solutions involve the nonclassical waves i e the delta shock wave and the delta contact discontinuity for which we explicitly give the expressions of their strengths locations and propagation speeds moreover we demonstrate that the nature of our solutions is identical with that of solutions to the corresponding one dimensional cauchy problem which provides a verification that our construction produces the correct unique global solutions

quasilinear evolutionary equations and continuous interpolation spaces in this paper we analyze the abstract parabolic evolutionary equations dt u x a u u f u h t u 0 x in continuous interpolation spaces allowing a singularity as t 0 here dt denotes the time derivative of order 0 2 we first give a treatment of fractional derivatives in the spaces lp 0 t x and then consider these derivatives in spaces of continuous functions having at most a prescribed singularity as t 0 the corresponding trace spaces are characterized and the dependence on is demonstrated via maximal regularity results on the linear equation dt u x au f u 0 x we arrive at results on existence uniqueness and continuation on the quasilinear equation finally an example is presented

convergence of shock capturing schemes for the compressible euler poisson equations we are concerned with approximate methods to construct global solutions with geometrical structure to the compressible euler poisson equations in several space variables a shock capturing numerical scheme is introduced to overcome the new difficulties from the nonlinear resonance of the system and the nonlocal behavior of the source terms the convergence and consistency of the shock capturing scheme for the equations is proved with the aid of the compensated compactness method then new existence results of the global solutions with geometrical structure are obtained the traces of the weak solutions are defined and then the weak solutions are proved to satisfy the boundary conditions the initial data are arbitrarily large withl bounds

global entropy solutions to exothermically reacting compressible euler equations abstract the global existence of entropy solutions is established for the compressible euler equations for one dimensional or plane wave flow of an ideal gas which undergoes a one step exothermic chemical reaction under arrhenius type kinetics we assume that the reaction rate is bounded away from zero and the total variation of the initial data is bounded by a parameter that grows arbitrarily large as the equation of state converges to that of an isothermal gas the heat released by the reaction causes the spatial total variation of the solution to increase however the increase in total variation is proved to be bounded in t 0 as a result of the uniform and exponential decay of the reactant to zero as t approaches infinity

vanishing viscosity limit of the navier stokes equations to the euler equations for compressible fluid flow we establish the vanishing viscosity limit of the navier stokes equations to the isentropic euler equations for one dimensional compressible fluid flow for the navier stokes equations there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup norm of solutions with respect to the physical viscosity coefficient may not be directly controllable furthermore convex entropy entropy flux pairs may not produce signed entropy dissipation measures to overcome these difficulties we first develop uniform energy type estimates with respect to the viscosity coefficient for solutions of the navier stokes equations and establish the existence of measure valued solutions of the isentropic euler equations generated by the navier stokes equations based on the uniform energy type estimates and the features of the isentropic euler equations we establish that the entropy dissipation measures of the solutions of the navier stokes equations for weak entropy entropy flux pairs generated by compactly supported c2 test functions are confined in a compact set in h 1 which leads to the existence of measure valued solutions that are confined by the tartar murat commutator relation a careful characterization of the unbounded support of the measure valued solution confined by the commutator relation yields the reduction of the measurevalued solution to a dirac mass which leads to the convergence of solutions of the navier stokes equations to a finite energy entropy solution of the isentropic euler equations with finite energy initial data relative to the different end states at infinity 2010 wiley periodicals inc

convergence of a difference scheme for conservation laws with a discontinuous flux convergence is established for a scalar finite difference scheme based on the godunov or engquist osher eo flux for scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient other works in this direction have established convergence for methods employing the solution of 2 2 riemann problems the algorithm discussed here uses only scalar riemann solvers satisfaction of a set of kruzkov type entropy inequalities is established for the limit solution from which geometric entropy conditions follow assuming a piecewise constant coefficient it is shown that these conditions imply l1 contractiveness for piecewise c1 solutions thus extending a well known theorem

uniform boundedness in time and large time behavior of weak solutions to a kind of dissipative system abstract this paper deals with the uniform boundedness as well as the existence and large time behavior of the weak entropy solutions to a kind of compressible euler equation with dissipation effect the existence and uniform boundedness in time of weak solutions are proved by using the lax friedrichs scheme and compensate compactness time asymptotically the density is showed to satisfy a kind of nonlinear fokker planck equation and the momentum obeys to the darcy s law as a by product the exponentially decay rate is obtained

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

a model of continuous sedimentation of flocculated suspensions in clarifier thickener units the chief purpose of this paper is to formulate and partly analyze a new mathematical model for continuous sedimentation consolidation processes of flocculated suspensions in clarifier thickener units this model appears in two variants for cylindrical and variable cross sectional area units respectively models 1 and 2 in both cases the governing equation is a scalar strongly degenerate parabolic equation in which both the convective and diffusion fluxes depend on parameters that are discontinuous functions of the depth variable the initial value problem for this equation is analyzed for model 1 we introduce a simple finite difference scheme and prove its convergence to a weak solution that satisfies an entropy condition a limited analysis of steady states as desired stationary modes of operation is performed numerical examples illustrate that the model realistically describes the dynamics of flocculated suspensions in clarifier thickeners

l1 convergence rates to the barenblatt solution for the damped compressible euler equations abstract in our previous work 17 it is shown that any l weak entropy solution of damped compressible euler equation converges to the barrenblatt s solution with finite mass in l 1 norm with a convergence rate for 1 3 in this paper by introducing an elaborate iterative method and using the intensive entropy analysis we prove the l 1 convergence to the barrenblatt s solution for any large 3 furthermore we get better l 1 convergence rate for 2 3

diffusive relaxation limit of classical solutions to the damped compressible euler equations abstract we construct uniform global classical solutions to the damped compressible euler equations on the framework of general besov spaces which includes both the usual sobolev spaces h s r d s 1 d 2 and the critical besov space b 2 1 1 d 2 r d such extension heavily depends on a revision of commutator estimates and an elementary fact that indicates the connection between homogeneous and inhomogeneous chemin lerner spaces furthermore we obtain the diffusive relaxation limit of euler equations towards the porous medium equation by means of aubin lions compactness argument

convergence of upwind finite difference schemes for a scalar conservation law with indefinite discontinuities in the flux function we consider the scalar conservation law with flux function discontinuous in the space variable i e begin eqnarray label eq1 u t h x f u 1 h x g u x 0 quad mbox in r times r nonumber u 0 x u 0 x quad mbox in r label 0 1 end eqnarray where h is the heaviside function and f and g are smooth with the assumptions that either f is convex and g is concave or f is concave and g is convex the existence of a weak solution of ref eq1 is proved by showing that upwind finite difference schemes of godunov and enquist osher type converge to a weak solution uniqueness follows from a kruzkhov type entropy condition we also provide explicit solutions to the riemann problem for ref eq1 at the level of numerics we give easy to implement numerical schemes of godunov and enquist osher type the central feature of this paper is the modification of the singular mapping technique the main analytical tool for these types of equations which allows us to show that the numerical schemes converge equations of type ref eq1 with the above hypothesis on the flux may occur when considering the following scalar conservation law with discontinuous flux begin equation label eq2 begin array email protected l u t k x f u x 0 u 0 x u 0 x end array end equation with f convex and k of indefinite sign

non selfsimilar solutions for a hyperbolic system of conservation laws in two space dimensions the non selfsimilar riemann problem for a two dimensional nonstrictly hyperbolic system of conservation laws is considered where the initial data are two constant states separated by a smooth curve without dimension reduction or coordinate transformation the two dimensional global solutions are constructed for six cases according to the normal velocities on both sides of the initial discontinuity moreover the interactions of non selfsimilar elementary waves are discussed by respectively taking the initial discontinuity as a parabola and a circle which enable us to provide the global structures of the entropy solutions explicitly c 2012 elsevier inc all rights reserved

semilinear subdiffusion with memory in the one dimensional case abstract for 0 1 we consider the semilinear integro differential equation on the one dimensional domain a b in the unknown u u x t d t u l 1 u 0 t k t s l 2 u s d s f x t u g x t where d t is the caputo fractional derivative and l 1 and l 2 are uniform elliptic operators with time dependent smooth coefficients under certain structural conditions on the nonlinearity f the global existence and uniqueness of classical solutions to the related initial boundary value problems are established via the so called continuation argument approach the key point is looking for suitable a priori estimates of the solution in the fractional holder spaces

global solution and large time behavior of the 3d compressible euler equations with damping national natural science foundation of china nsaf 10976026 national natural science foundation of china 11271305

a difference scheme for conservation laws with a discontinuous flux the nonconvex case in a previous work by the author convergence was established for a simple difference scheme approximating a scalar conservation law where the flux was concave and had a discontinuous spatially varying coefficient j d towers siam j numer anal 38 2000 pp 681 698 the main result of this paper is an extension of that convergence theorem to the situation where the flux may have any finite number of critical points additionally spatially varying source terms are allowed the spatially varying numerical flux is also shown to satisfy maximum and minimum principles and to be total variation decreasing tvd in time

harnack s inequality for a space time fractional diffusion equation and applications to an inverse source problem abstract in this paper we focus on a space time fractional diffusion equation with the generalized caputo s fractional derivative operator and a general space nonlocal operator with the fractional laplace operator as a special case a weak harnack s inequality has been established by using a special test function and some properties of the space nonlocal operator based on the weak harnack s inequality a strong maximum principle has been obtained which is an important characterization of fractional parabolic equations with these tools we establish a uniqueness result of an inverse source problem on the determination of the temporal component of the inhomogeneous term which seems to be the first theoretical result of the inverse problem for such a general fractional diffusion model

existence of global solutions for unsteady isentropic gas flow in a laval nozzle in this paper we study the motion of isentropic gas in the laval nozzle the laval nozzle is the most important type of nozzle utilized in some turbines in particular we consider unsteady flows including transonic gas flows and prove the existence of global solutions for the cauchy problem in spite of its importance this problem has received little attention until now the most difficult point is to obtain bounded estimates for approximate solutions to overcome this we introduce a modified godunov scheme the corresponding approximate solutions consist of piecewise steady state solutions of an auxiliary equation and yield a sharper bounded estimate as a result we find an invariant region for our solutions finally in order to prove their convergence we use the compensated compactness framework

representation of solutions and large time behavior for fully nonlocal diffusion equations abstract we study the cauchy problem for a nonlocal heat equation which is of fractional order both in space and time we prove four main theorems i a representation formula for classical solutions ii a quantitative decay rate at which the solution tends to the fundamental solution iii optimal l 2 decay of mild solutions in all dimensions iv l 2 decay of weak solutions via energy methods the first result relies on a delicate analysis of the definition of classical solutions after proving the representation formula we carefully analyze the integral representation to obtain the quantitative decay rates of ii next we use fourier analysis techniques to obtain the optimal decay rate for mild solutions here we encounter the critical dimension phenomenon where the decay rate attains the decay rate of that in a bounded domain for large enough dimensions consequently the decay rate does not anymore improve when the dimension increases the theory is markedly different from that of the standard caloric functions and this substantially complicates the analysis finally we use energy estimates and a comparison principle to prove a quantitative decay rate for weak solutions defined via a variational formulation our main idea is to show that the l 2 norm is actually a subsolution to a purely time fractional problem which allows us to use the known theory to obtain the result

the comparison of the riemann solutions in gas dynamics in this paper we compare two riemann solution constructions of isentropic and nonisentropic gas dynamic equations with the same initial values we obtain that when rarefaction waves occur in the non isentropic solution the same family rarefaction waves occur in the isentropic solution however when advancing or reflected shock waves occur in the isentropic case the same kind shock waves occur in the nonisentropic case moreover by numerical calculation it can be found that the wave intensities of the two solutions are quite close our conclusion reflects that almost non isentropic gas dynamic process can be simulated by isentropic gas dynamic process in practice c 2015 elsevier inc all rights reserved

convergence to equilibrium in wasserstein distance for damped euler equations with interaction forces we develop tools to construct lyapunov functionals on the space of probability measures in order to investigate the convergence to global equilibrium of a damped euler system under the influence of external and interaction potential forces with respect to the 2 wasserstein distance we also discuss the overdamped limit to a nonlocal equation used in the modelling of granular media with respect to the 2 wasserstein distance and provide rigorous proofs for particular examples in one spatial dimension

well balanced schemes for conservation laws with source terms based on a local discontinuous flux formulation we propose and analyze a finite volume scheme of the godunov type for conservation laws with source terms that preserve discrete steady states the scheme works in the resonant regime as well as for problems with discontinuous flux moreover an additional modification of the scheme is not required to resolve transients and solutions of non linear algebraic equations are not involved our well balanced scheme is based on modifying the flux function locally to account for the source term and to use a numerical scheme especially designed for conservation laws with discontinuous flux due to the difficulty of obtaining bv estimates we use the compensated compactness method to prove that the scheme converges to the unique entropy solution as the discretization parameter tends to zero we include numerical experiments in order to show the features of the scheme and how it compares with a wellbalanced scheme from the literature

hyperbolic to parabolic relaxation theory for quasilinear first order systems abstract in this paper we study the limiting behavior of nonhomogeneous hyperbolic systems of balance laws when the relaxed equilibria are described by means of systems of parabolic type in particular we obtain a complete theory for the 2 2 systems of genuinely nonlinear hyperbolic balance laws in 1 d with a strong dissipative term a different method which combines the div curl lemma with accretive operators is then applied to study the limiting profiles in the case of nonhomogeneous isentropic gas dynamics we also investigate relaxation results for some 2 d cases which include the cattaneo model for nonlinear heat conduction and the compressible euler flow moreover convergence result is also obtained for general semilinear systems in 1 d

boundedness of weak solutions to evolutionary partial integro differential equations with discontinuous coefficients we investigate linear and quasilinear evolutionary partial integro differential equations of second order which include time fractional evolution equations of time order less than one by means of suitable energy estimates and de giorgi s iteration technique we establish results asserting the global boundedness of appropriately defined weak solutions of these problems we also show that a maximum principle holds for such equations

cauchy problem for fractional diffusion equations abstract we consider an evolution equation with the regularized fractional derivative of an order 0 1 with respect to the time variable and a uniformly elliptic operator with variable coefficients acting in the spatial variables such equations describe diffusion on inhomogeneous fractals a fundamental solution of the cauchy problem is constructed and investigated

weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift diffusion equation we investigate the relaxation problem for the hydrodynamic isentropic euler poisson system when the momentum relaxation time tends to zero very sharp estimates on the solutions independent of the relaxation time are obtained and used to establish compactness

the low energy tqft of the generalized double semion model the generalized double semion gds model introduced by freedman and hastings is a lattice system similar to the toric code with a gapped hamiltonian whose definition depends on a triangulation of the ambient manifold m but whose space of ground states does not depend on the triangulation but only on the underlying manifold in this paper we use topological quantum field theory tqft to investigate the low energy limit of the gds model we define and study a functorial tqft z mathrm gds in every dimension n such that for every closed n 1 manifold m z mathrm gds m is isomorphic to the space of ground states of the gds model on m the isomorphism can be chosen to intertwine the actions of the mapping class group of m that arise on both sides throughout this paper we compare our constructions and results with their known analogues for the toric code

convergence to the barenblatt solution for the compressible euler equations with damping and vacuum we study the asymptotic behavior of a compressible isentropic flow through a porous medium when the initial mass is finite the model system is the compressible euler equation with frictional damping as t the density is conjectured to obey the well known porous medium equation and the momentum is expected to be formulated by darcy s law in this paper we give a definite answer to this conjecture without any assumption on smallness or regularity for the initial data we prove that any l weak entropy solution to the cauchy problem of damped euler equations with finite initial mass converges strongly in lp with decay rates to matching barenblatt s profile of the porous medium equation the density function tends to the barenblatt s solution of the porous medium equation while the momentum is described by darcy s law

existence of global weak solutions to a special system of euler equation with a source ii general case abstract in this paper we apply the maximum principle and the theory of compensated compactness to establish an existence theorem for global weak solutions to the cauchy problem of the non strictly hyperbolic system a special system of euler equation with a general source term

on a weak solution for a transonic flow problem on essaie d etablir l existence de solutions faibles pour un ecoulement stationnaire bidimensionnel irrotationnel compressible non visqueux dans un canal de longueur infinie ou autour d un profil aerodynamique la vitesse de l ecoulement etant donnee a l infini

asymptotic behavior of the solutions to the damped compressible euler equations with vacuum abstract the asymptotic behavior of solutions of the damped compressible euler equations is conjectured to obey to the famous porous media equations pmes the previous works on this topic concern the case away from vacuum where the system is strictly hyperbolic in present paper we prove that the l entropy weak solution with vacuum obtained by the compensated compactness theory converges strongly in l loc p 1 p space to the unique similarity solution of the related pme as time goes to infinity

the dflu flux for systems of conservation laws the dflu numerical flux was introduced in order to solve hyperbolic scalar conservation laws with a flux function discontinuous in space we show how this flux can be used to solve a certain class of systems of conservation laws such as systems modeling polymer flooding in oil reservoir engineering furthermore these results are extended to the case where the flux function is discontinuous in the space variable such a situation arises for example while dealing with oil reservoirs which are heterogeneous numerical experiments are presented to illustrate the efficiency of this new scheme compared to other standard schemes like upstream mobility lax friedrichs and force schemes

convergence of the godunov scheme for a simplified one dimensional hydrodynamic model for semiconductor devices convergence of approximate solutions derived by the godunov scheme for a simplified one dimensional hydrodynamic model for semiconductor devices is established by using the compensated compactness method a global existence theorem is shown and a numerical method for the computation of the physical global solution of this model is provided by this approach

convergence rate for compressible euler equations with damping and vacuum we study the asymptotic behavior of l weak entropy solutions to the compressible euler equations with damping and vacuum previous works on this topic are mainly concerned with the case away from the vacuum and small initial data in the present paper we prove that the entropy weak solution strongly converges to the similarity solution of the porous media equations in lp r 2 p with decay rates the initial data can contain vacuum and can be arbitrary large a new approach is introduced to control the singularity near vacuum for the desired estimates

a de giorgi nash type theorem for time fractional diffusion equations we study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order alpha in 0 1 the coefficients are merely assumed to be bounded and measurable and they satisfy a uniform parabolicity condition our main result is a de giorgi nash type theorem which gives an interior holder estimate for bounded weak solutions in terms of the data and the l infty bound of the solution the proof relies on new a priori estimates for time fractional problems and uses de giorgi s technique and the method of non local growth lemmas which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional laplacian

on the cobordism classification of symmetry protected topological phases in the framework of atiyah s axioms of topological quantum field theory with unitarity we give a direct proof of the fact that symmetry protected topological phases without hall effects are classified by cobordism invariants we first show that the partition functions of those theories are cobordism invariants after a tuning of the euler term conversely for a given cobordism invariant we construct a unitary topological field theory whose partition function is given by the cobordism invariant assuming that a certain bordism group is finitely generated two theories having the same cobordism invariant partition functions are isomorphic

convergence of an engquist osher scheme for a multi dimensional triangular system of conservation laws we consider a multi dimensional triangular system of conservation laws this system arises as a model of three phase flow in porous media and includes multi dimensional conservation laws with discontinuous coefficients as a special case the system is neither strictly hyperbolic nor symmetric we propose an engquist osher type scheme for this system and show that the approximate solutions generated by the scheme converge to a weak solution numerical examples are also presented

operator spltting methods for systems of convection diffusion equations nonlinear error mechanisms and correction strategies abstract many numerical methods for systems of convection diffusion equations are based on an operator splitting formulation where convective and diffusive forces are accounted for in separate substeps we describe the nonlinear mechanism of the splitting error in such numerical methods in the one dimensional case a mechanism that is intimately linked to the local linearizations introduced implicitly in the hyperbolic convection steps by the use of an entropy condition for convection dominated flows we demonstrate that operator splitting methods typically generate a numerical widening of viscous fronts unless the splitting step is of the same magnitude as the diffusion scale to compensate for the potentially damaging splitting error we propose a corrected operator splitting cos method for general systems of convection diffusion equations with the ability of correctly resolving the nonlinear balance between the convective and diffusive forces in particular cos produces viscous shocks with a correct structure also when the splitting step is large a front tracking method for systems of conservation laws which in turn relies heavily on a riemann solver constitutes an important part of our cos strategy the proposed cos method is successfully applied to a system modeling two phase multicomponent flow in porous media and a triangular system modeling three phase flow

existence of global smooth solution to the relativistic euler equations abstract in this paper we prove an existence theorem of global smooth solutions to the cauchy problem for the one dimensional relativistic euler equations the analysis is based on a priori estimates which are obtained by the characteristic method and maximum principle

entropy solutions to doubly nonlinear integro differential equations abstract we consider doubly nonlinear history dependent problems of the form t k b v b v 0 div a x v f the kernel k satisfies certain assumptions which are in particular satisfied by k t t 1 i e the case of fractional derivatives of order 0 1 is included we show existence of entropy solutions in the case of a nondecreasing b an existence result in the case of a strictly increasing b is used to get entropy solutions of approximate problems kruzhkov s method of doubling variables a comparison principle and the diagonal principle are used to obtain a e convergence for approximate solutions a uniqueness result has been shown in a previous work

convergence of finite volume schemes for triangular systems of conservation laws we consider non strictly hyperbolic systems of conservation laws in triangular form which arise in applications like three phase flows in porous media we device simple and efficient finite volume schemes of godunov type for these systems that exploit the triangular structure we prove that the finite volume schemes converge to weak solutions as the discretization parameters tend to zero some numerical examples are presented one of which is related to flows in porous media

well posedness in bv t and convergence of a difference scheme for continuous sedimentation in ideal clarifier thickener units we consider a scalar conservation law modeling the settling of particles in an ideal clarifier thickener unit the conservation law has a nonconvex flux which is spatially dependent on two discontinuous parameters we suggest to use a kru kov type notion of entropy solution for this conservation law and prove uniqueness l1 stability of the entropy solution in the bvt class functions w x t with tw being a finite measure the existence of a bvt entropy solution is established by proving convergence of a simple upwind finite difference scheme of the engquist osher type a few numerical examples are also presented

godunov type numerical methods for a model of granular flow we propose and analyze finite volume godunov type methods based on discontinuous flux for a 2 2 system of non linear partial differential equations proposed by hadeler and kuttler to model the dynamics of growing sandpiles generated by a vertical source on a flat bounded rectangular table the scheme is made well balanced by modifying the flux function locally by including source term as a part of the convection term its extension to multi dimensions is not straightforward for which an approach has been introduced here based on transport rays this approach is compared with another approach for inclusion of source term which uses the idea of inverting the divergence operator relying on the curl free component of the helmholtz decomposition of the source term numerical experiments are presented to illustrate the efficiency of the proposed scheme for both unsteady and steady state calculations and to make comparisons with the previously studied finite difference and semi lagrangian approaches by falcone and finzi vita

kinetic schemes on staggered grids for barotropic euler models entropy stability analysis we introduce in the one dimensional framework a new scheme of finite volume type for barotropic euler equations the numerical unknowns namely densities and velocities are defined on staggered grids the numerical fluxes are defined by using the framework of kinetic schemes we can consider general convex pressure laws we justify that the density remains non negative and the total physical entropy does not increase under suitable stability conditions performances of the scheme are illustrated through a set of numerical experiments

global solutions to the compressible euler equations with geometrical structure we prove the existence of global solutions to the euler equations of compressible isentropic gas dynamics with geometrical structure including transonic nozzle flow and spherically symmetric flow due to the presence of the geometrical source terms the existence results themselves are new especially as they pertain to radial flow in an unbounded region vec x geqq 1 and to transonic nozzle flow arbitrary data withl bounds are allowed in these results a shock capturing numerical scheme is introduced to compute such flows and to construct approximate solutions the convergence and consistency of the approximate solutions generated from this scheme to the global solutions are proved with the aid of a compensated compactness framework

conservation law with the flux function discontinuous in the space variable ii we deal with a single conservation law with discontinuous convex concave type fluxes which arise while considering sign changing flux coefficients the main difficulty is that a weak solution may not exist as the rankine hugoniot condition at the interface may not be satisfied for certain choice of the initial data we develop the concept of generalized entropy solutions for such equations by replacing the rankine hugoniot condition by a generalized rankine hugoniot condition the uniqueness of solutions is shown by proving that the generalized entropy solutions form a contractive semi group in l 1 existence follows by showing that a godunov type finite difference scheme converges to the generalized entropy solution the scheme is based on solutions of the associated riemann problem and is neither consistent nor conservative the analysis developed here enables to treat the cases of fluxes having at most one extrema in the domain of definition completely numerical results reporting the performance of the scheme are presented

an inverse source problem in a semilinear time fractional diffusion equation we study an inverse source problem for a semilinear time fractional diffusion equation of second order in a bounded domain in r d the missing solely time dependent source is recovered from an additional integral measurement the existence uniqueness and regularity of a weak solution is addressed we design a numerical algorithm based on rothe s method derive a priori estimates and prove convergence of iterates towards the exact solution theoretical results are supported by a numerical experiment

on the cauchy problems for polymer flooding with gravitation abstract we study two systems of conservation laws for polymer flooding in secondary oil recovery one with gravitation force and one without for each model we prove global existence of weak solutions for the cauchy problems under rather general assumptions on the flux functions approximate solutions are constructed through a front tracking algorithm and compactness is achieved through the bound on suitably defined wave strengths as the main technical novelty we introduce some new nonlinear functionals that yield a uniform bound on the total variation of the flux function

existence of global solutions for isentropic gas flow in a divergent nozzle with friction abstract we study isentropic gas flow in a divergent nozzle with friction it is physically known that the flow is influenced by friction in this paper we investigate the mathematical effect of the friction on solutions we first construct approximate solutions by using the godunov scheme and the fractional step procedure we obtain the bounded estimate by the invariant region this is the difficult point of the present paper we use the compensated compactness to prove their convergence finally we prove the existence of global solutions with arbitrary l data

lyapunov functions and convergence to steady state for differential equations of fractional order we study the asymptotic behaviour as t of bounded solutions to certain integro differential equations in finite dimensions which include differential equations of fractional order between 0 and 2 we derive appropriate lyapunov functions for these equations and prove that any global bounded solution converges to a steady state of a related equation if the nonlinear potential mathcal e occurring in the equation satisfies the ojasiewicz inequality

higher dimensional algebra and topological quantum field theory the study of topological quantum field theories increasingly relies upon concepts from higher dimensional algebra such as n categories and n vector spaces we review progress towards a definition of n category suited for this purpose and outline a program in which n dimensional topological quantum field theories tqfts are to be described as n category representations first we describe a suspension operation on n categories and hypothesize that the k fold suspension of a weak n category stabilizes for k n 2 we give evidence for this hypothesis and describe its relation to stable homotopy theory we then propose a description of n dimensional unitary extended tqfts as weak n functors from the free stable weak n category with duals on one object to the n category of n hilbert spaces we conclude by describing n categorical generalizations of deformation quantization and the quantum double construction

setting the quantum integrand of m theory in anomaly free quantum field theories the integrand in the bosonic functional integral the exponential of the effective action after integrating out fermions is often defined only up to a phase without an additional choice we term this choice setting the quantum integrand in the low energy approximation to m theory the e8 model for the c field allows us to set the quantum integrand using geometric index theory we derive mathematical results of independent interest about pfaffians of dirac operators in 8k 3 dimensions both on closed manifolds and manifolds with boundary these theorems are used to set the quantum integrand of m theory for closed manifolds and for compact manifolds with either temporal global or spatial local boundary conditions in particular we show that m theory makes sense on arbitrary 11 manifolds with spatial boundary generalizing the construction of heterotic m theory on cylinders

solutions of hamilton jacobi equations and scalar conservation laws with discontinuous space time dependence abstract we establish a unique stable solution to the hamilton jacobi equation u t h k x t u x 0 x t 0 with lipschitz initial condition where k x t is allowed to be discontinuous in the x t plane along a finite number of possibly intersecting curves parameterized by t we assume that for fixed k h k p is convex in p and lim p h k p p the solution is determined by showing that if k is made smooth by convolving k in the x direction with the standard mollifier then the control theory representation of the viscosity solution to the resulting hamilton jacobi equation must converge uniformly as the mollification decreases to a lipschitz continuous solution with an explicit control theory representation this also defines the unique stable solution to the corresponding scalar conservation law u t f k x t u x 0 x t 0 with k discontinuous

a priori bounds for degenerate and singular evolutionary partial integro differential equations we study quasilinear evolutionary partial integro differential equations of second order which include time fractional p laplace equations of time order less than one by means of suitable energy estimates and de giorgi s iteration technique we establish results asserting the global boundedness of appropriately defined weak solutions of these problems we also show that a maximum principle is valid for such equations

the 3d compressible euler equations with damping in a bounded domain abstract we proved global existence and uniqueness of classical solutions to the initial boundary value problem for the 3d damped compressible euler equations on bounded domain with slip boundary condition when the initial data is near its equilibrium time asymptotically the density is conjectured to satisfy the porous medium equation and the momentum obeys to the classical darcy s law based on energy estimate we showed that the classical solution converges to steady state exponentially fast in time we also proved that the same is true for the related initial boundary value problem of porous medium equation and thus justified the validity of darcy s law in large time

the atiyah patodi singer index and domain wall fermion dirac operators we introduce a mathematician friendly formulation of the physicist friendly derivation of the atiyah patodi singer index in a previous paper motivated by the study of lattice gauge theory the physicist half of the authors derived a formula expressing the atiyah patodi singer index in terms of the eta invariant of domain wall fermion dirac operators when the base manifold is a flat 4 dimensional torus in this paper we generalise this formula to any even dimensional closed riemannian manifolds and prove it mathematically rigorously our proof uses a witten localisation argument combined with a devised embedding into a cylinder of one dimension higher our viewpoint sheds some new light on the interplay among the atiyah patodi singer boundary condition domain wall fermions and edge modes