the green s function and large time behavior of solutions for the one dimensional boltzmann equation we study the pointwise behavior of the green s function of the boltzmann equation our results reveal the particle and fluid aspects of the equation the particle aspect is represented by singular waves these waves are carried by transport equations and dominate the short time behavior of the solution we devise a picard type iteration for constructing the increasingly regular particlelike waves the fluidlike waves reveal the dissipative behavior of the type of navier stokes equations as usually seen by the chapman enskog expansion these waves are constructed as part of the long wave expansion in the spectrum of the fourier mode for the space variable the fluidlike waves represent the long time behavior of the solution as an application we obtain the pointwise description of the large time behavior of the convergence to the global maxwellian when the initial perturbation is not necessarily smooth in our analysis of the exchanges of the microscopic velocity decay and space decay we make essential use of the hard sphere models c 2004 wiley periodicals inc contents

sobolev preconditioning for the poisson boltzmann equation abstract this paper presents an overview of least squares steepest descent using sobolev gradients for several prototype differential equations in the linear case the method is viewed as a very effective preconditioning strategy for the basic iterative method which arises from steepest descent in particular it acts to smooth the euclidean gradient results are given for the one dimensional possion boltzmann equation from semiconductor device modeling

pointwise a posteriori error estimates for elliptic problems on highly graded meshes pointwise a posteriori error estimates are derived for linear second order elliptic problems over general polygonal domains in 2d the analysis carries over regardless of convexity accounting even for slit domains and applies to highly graded unstructured meshes as well a key ingredient is a new asymptotic a priori estimate for regularized green s functions the estimators lead always to upper bounds for the error in the maximum norm along with lower bounds under very mild regularity and nondegeneracy assumptions the effect of both point and line singularities is examined three popular local estimators for the energy norm are shown to be equivalent when suitably interpreted to those introduced here

the initial value problem for the equation of motion of irrotational inviscid and heat conductive fluids abstract the cauchy problem of the equation of motion of irrotational inviscid and heat conductive fluids is considered it is proved that the heat diffusion prevents the development of singularities in small amplitude classical solutions using an equivalent reformulation of the cauchy problem to obtain effective energy estimates the full solution converges to their equilibrium state at the rate 1 t 3 4 in the l 2 norm as the non isentropic compressible navier stokes system

thermal non equilibrium flows in three space dimensions we study the equations describing the motion of a thermal non equilibrium gas in three space dimensions it is a hyperbolic system of six equations with a relaxation term the dissipation mechanism induced by the relaxation is weak in the sense that the shizuta kawashima criterion is violated this implies that a perturbation of a constant equilibrium state consists of two parts one decays in time while the other stays in fact the entropy wave grows weakly along the particle path as the process is irreversible we study thermal properties related to the well posedness of the nonlinear system we also obtain a detailed pointwise estimate on the green s function for the cauchy problem when the system is linearized around an equilibrium constant state the green s function provides a complete picture of the wave pattern with an exact and explicit leading term comparing with existing results for one dimensional flows our results reveal a new feature of three dimensional flows not only does the entropy wave not decay but the velocity also contains a non decaying part strongly coupled with its decaying one the new feature is supported by the second order approximation via the chapman enskog expansions which are the navier stokes equations with vanished shear viscosity and heat conductivity

generalized huygens principle for bipolar non isentropic compressible navier stokes poisson system in dimension three abstract the generalized huygens principle for the non isentropic bipolar navier stokes poisson system in dimension three is given which is different from the unipolar case all of the previous estimates containing huygens waves on wave behaviors for compressible flow models rely heavily on the conservative structure for instance the navier stokes system and even the isentropic bipolar navier stokes poisson system the reason is that the huygens waves decay slower than diffusion waves in l p r 3 space when 1 p 2 however the non isentropic case here is not conservative due to two energy equations and moreover it is not yet conservative even if one reformulates the original system as the isentropic case in 26 the main contribution is that we first solve the nonlinear problem when there exists the huygens wave in the green s function but the nonlinear system is not conservative which is achieved by establishing refined pointwise estimates for the green s function and nonlinear convolution estimates

maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions a semilinear second order parabolic equation is considered in a regular and a singularly perturbed regime for this equation we give computable a posteriori error estimates in the maximum norm semidiscrete and fully discrete versions of the backward euler crank nicolson and discontinuous galerkin dg r methods are addressed for their full discretizations we employ elliptic reconstructions that are respectively piecewise constant piecewise linear and piecewise quadratic for r 1 in time we also use certain bounds for the green s function of the parabolic operator

maximum norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter

the large time behavior of solutions in the critical l p framework for compressible viscous and heat conductive gas flows in this paper we are concerned with non isentropic navier stokes equations governing compressible viscous and heat conductive gases we formulate an additional regularity assumption of low frequencies in rd d 3 and then establish the sharp time weighted inequality which allows us to get the time decay estimates of global strong solutions in the lp critical besov spaces precisely it is shown that if the initial data belong to some besov space b2 s1 with s1 1 d2 s0 s0 2dp d2 then the lp norm of the critical global solutions admits the time decay t s12 d2 12 1p in particular t d2p if s1 s0 which coincides with that of the heat kernel in the lp framework in comparison with the work of danchin and xu j math fluid mech 20 1641 1665 2018 the low frequency assumption is improved such that the regularity s1 belongs to the whole range 1 d2 s0

asymptotics of solutions of nonlinear parabolic equations abstract this paper is concerned with the large time behaviour of solutions to the cauchy problem of the following nonlinear parabolic equations u t u f u d x u d x 2 u u r n u t x t 0 u 0 x x r n n 1 under the optimal growth conditions on the smooth nonlinear function f u dxu dx2u we obtain the global existence results to the above cauchy problem the influence of the nonlinear function f u dxu dx2u on the large time behaviour of the global smooth solution u t x is also studied

l1 asymptotic behavior of compressible isentropic viscous 1 d flow we study the large time behavior in l1 of the compressible isentropic viscous 1 d flow under the assumption that the initial data are smooth and small we show that the solutions are approximated by the solutions of a parabolic system and in turn by diffusion waves which are solutions of burgers equations decay rates in l1 are obtained our method is based on the study of pointwise properties in the physical space of the fundamental solution to the linearized system 1994 john wiley sons inc

a low frequency assumption for optimal time decay estimates to the compressible navier stokes equations the global existence issue in critical spaces for compressible navier stokes equations was addressed by danchin invent math 141 579 614 2000 and then developed by charve and danchin arch rational mech anal 198 233 271 2010 chen et al commun pure appl math 63 1173 1224 2010 and haspot arch rational mech anal 202 427 460 2011 in more general lp setting the main aim of this paper is to exhibit more precisely time decay estimates of solutions constructed in the critical regularity framework to the best of our knowledge the low frequency assumption usually plays a key role in the large time asymptotics of solutions which was firstly observed by matsumura and nishida j math kyoto univ 20 67 104 1980 in the l 1 mathbb r d space we now claim a new low frequency assumption for barotropic compressible navier stokes equations which may be of interest in the mathematical analysis of viscous fluids precisely if the initial density and velocity additionally belong to some besov space dot b sigma 1 2 infty mathbb r d with the regularity sigma 1 in 1 d 2 2d p d 2 then a sharp time weighted inequality including enough time decay information can be available where optimal decay exponents for the high frequencies are exhibited the proof mainly depends on some non standard besov product estimates as a by product those optimal time decay rates of lq lr type are also captured in the critical framework

maximum norm error estimators for three dimensional elliptic problems in this paper we define an a posteriori error estimator for finite element approximations of 3 d elliptic problems we prove that the estimator is equivalent up to logarithmic factors of the meshsize to the maximum norm of the error the results are valid for an arbitrary polyhedral domain and rather general meshes we also obtain analogous results for the nonconforming method of crouzeix raviart finally we present some numerical results comparing adaptive procedures based on controlling the error in different norms

solving the non isentropic navier stokes equations in odd space dimensions the green function method in this paper we investigate the wave structure of the solution for the compressible navier stokes equations in both one space dimension and three space dimensions first we give the pointwise estimate on the green function for the linearized system by dividing the physical domain into two parts which are the finite mach number region and outside finite mach number region we use long wave short wave decomposition and a weighted energy estimate method in each region separately the green function provides a complete picture of the wave pattern with explicit leading terms then with the help of the duhamel principle we give the pointwise estimate of the solution for the nonlinear problems 1 1 and 1 2 specially we find that the decay rate in lp r3 2 p for the energy is faster than those for the density and the momentum m

sharpl error estimates for semilinear elliptic problems with free boundaries a numerical scheme to approximate a semilinear pde involving a singular maximal monotone graph is analyzed inl a preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute the discrete maximum principle is thus avoided sharp pointwise error estimates are derived for both the smoothing and the discretization procedures an optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence these error estimates for solutions in conjunction with nondegeneracy properties of continuous problems provide sharp interface error estimates two model examples are discussed the obstacle problem and a combustion equation

pointwise a posteriori error control for elliptic obstacle problems we consider a finite element method for the elliptic obstacle problem over polyhedral domains in d which enforces the unilateral constraint solely at the nodes we derive novel optimal upper and lower a posteriori error bounds in the maximum norm irrespective of mesh fineness and the regularity of the obstacle which is just assumed to be holder continuous they exhibit optimal order and localization to the non contact set we illustrate these results with simulations in 2d and 3d showing the impact of localization in mesh grading within the contact set along with quasi optimal meshes

pointwise a posteriori error estimates for monotone semi linear equations we derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi linear equations the estimates hold for lagrange elements of any fixed order non smooth nonlinearities and take numerical integration into account the proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately and then applying the continuous maximum principle no geometric mesh constraints are thus required numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature

pointwise green function bounds for shock profiles of systems with real viscosity following the pointwise semigroup approach of zh mz 1 we establish sharp pointwise green function bounds and consequent linearized stability for viscous shock profiles of general hyperbolic parabolic systems of conservation laws of dissipative type under the necessary assumptions z 1 z 3 z 4 of spectral stability i e stable point spectrum of the linearized operator about the wave transversality of the connecting profile and hyperbolic stability of the corresponding ideal shock of the associated inviscid system with no additional assumptions on the structure or strength of the shock these bounds are used in a companion paper mz 2 to establish nonlinear stability of small amplitude lax shocks of symmetrizable hyperbolic parabolic systems

the finite element approximation of the nonlinear poisson boltzmann equation a widely used electrostatics model in the biomolecular modeling community the nonlinear poisson boltzmann equation along with its finite element approximation are analyzed in this paper a regularized poisson boltzmann equation is introduced as an auxiliary problem making it possible to study the original nonlinear equation with delta distribution sources a priori error estimates for the finite element approximation are obtained for the regularized poisson boltzmann equation based on certain quasi uniform grids in two and three dimensions adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge the poisson boltzmann equation does not appear to have been previously studied in detail theoretically and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the poisson boltzmann equation note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear poisson boltzmann equation with delta distribution sources and it also introduces the first provably convergent adaptive method for the equation this last result is currently one of only a handful of existing convergence results of this type for nonlinear problems

fully adaptive newton galerkin methods for semilinear elliptic partial differential equations in this paper we develop an adaptive procedure for the numerical solution of general semilinear elliptic problems with possible singular perturbations our approach combines both prediction type adaptive newton methods and a linear adaptive finite element discretization based on a robust a posteriori error analysis thereby leading to a fully adaptive newton galerkin scheme numerical experiments underline the robustness and reliability of the proposed approach for various examples

a posteriori error estimator and error control for contact problems in this paper we consider two error estimators for one body contact problems the first error estimator is defined in terms of h div conforming stress approximations and equilibrated fluxes while the second is a standard edge based residual error estimator without any modification with respect to the contact we show reliability and efficiency for both estimators more over the error is bounded by the first estimator with a constant one plus a higher order data oscillation term plus a term arising from the contact that is shown numerically to be of higher order the second estimator is used in a control based afem refinement strategy and the decay of the error in the energy is shown several numerical tests demonstrate the performance of both estimators

quasi optimal and robust a posteriori error estimates in for the approximation of allen cahn equations past singularities quasi optimal a posteriori error estimates in l 0 t l 2 are derived for the finite element approximation of allen cahn equations the estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface the error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming nonconforming mixed and discontinuous galerkin methods numerical experiments illustrate the theoretical results

convergence to diffusion waves of solutions for viscous conservation laws we study the large time behavior of solutions of viscous conservation laws it is shown that solutions tend to diffusion waves which are constructed based on the heat equation and burgers equation the convergence is in thelp 1 p sense and is obtained as a consequence of thel2 decay of the difference between the solution and its asymptotic state of diffusion waves

stability of small amplitude shock profiles of symmetric hyperbolic parabolic systems combining pointwise green s function bounds obtained in a companion paper 36 with earlier spectral stability results obtained in 16 we establish nonlinear orbital stability of small amplitude lax type viscous shock profiles for the class of dissipative symmetric hyperbolic parabolic systems identified by kawashima 20 notably including compressible navier stokes equations and the equations of magnetohydrodynamics obtaining sharp rates of decay in lp with respect to small l1 h3 perturbations 2 p our analysis extends and somewhat refines the approach introduced in 35 to treat stability of relaxation profiles 2004 wiley periodicals inc

asymptotic behavior of solutions to a system of viscous conservation laws related to a phase transition problem we study the asymptotic behavior of solutions for a system of viscous conservation laws with discontinuous initial data we discuss mainly the case where the system without the viscosity term is of hyperbolic elliptic mixed type this problem is related to a phase transition problem we study the initial value problem and show the decay rates of solutions to piecewise constant states where two phases coexist the modification necessary for the hyperbolic case is also discussed

pointwise convergence to knudsen layers of the boltzmann equation in this paper green s functions for the boltzmann equation around a global maxwellian are used to construct the non characteristic nonlinear knudsen layers as well as their time asymptotic stability furthermore the detailed pointwise structures nonlinear wave couplings and wave interactions with boundary are studied

initial boundary value problem for one dimensional wave solutions of the boltzmann equation tai ping liu and his collaborator shih hsien yu solve the initial boundary value problem for the boltzmann equation because of the inclusion of microscopic velocity in the kinetic theory the boltzmann equation can model richer and physically more realistic boundary effects more so than the fluid equations such as the navier stokes this motivates our study the reflection of fluid waves off the boundary is indicated in the figure for the case only one of the euler wave velocities is positive

how accurate are finite elements on anisotropic triangulations in the maximum norm abstract in kopteva 2014 a counterexample of an anisotropic triangulation was given on which the exact solution has a second order error of linear interpolation while the computed solution obtained using linear finite elements is only first order pointwise accurate this example was given in the context of a singularly perturbed reaction diffusion equation in this paper we present further examples of unanticipated pointwise convergence behaviour of lagrange finite elements on anisotropic triangulations in particular we show that linear finite elements may exhibit lower than expected orders of convergence for the laplace equation as well as for certain singular equations and their accuracy depends not only on the linear interpolation error but also on the mesh topology furthermore we demonstrate that pointwise convergence rates which are worse than one might expect are also observed when higher order finite elements are employed on anisotropic meshes a theoretical justification will be given for some of the observed numerical phenomena

a free boundary problem for semilinear elliptic equations etude de la nature de la frontiere libre u 0 pour un probleme de dirichlet semilineaire

fully localized a posteriori error estimators and barrier sets for contact problems we derive novel pointwise a posteriori error estimators for elliptic obstacle problems which except for obstacle resolution completely vanish within the full contact set localization we then construct a posteriori barrier sets for free boundaries under a natural stability or nondegeneracy condition we illustrate localization properties as well as reliability and efficiency for both solutions and free boundaries via several simulations in 2 and 3 dimensions

interval routing in reliability networks in this paper we consider routing with compact tables in reliability networks more precisely we study interval routing on random graphs g b p obtained from a base graph b by independently removing each edge with a failure probability 1 p we focus on additive stretched routing for n node random graphs for which the base b is a square mesh and p 0 5 that is the percolation model at the critical phase we show a lower bound of log n 2 on the number of intervals required per edge for every additive stretch 0 on the other side our experimental results show that the size of the largest biconnected components is n0 827 and thus that there exists a trivial shortest path routing scheme using at most o n0 827 intervals per edge the results are extended to random meshes of higher dimension we show that asymptotically almost surely the number of intervals per edge for a random r dimensional mesh with n nodes is 16 r 2 1 r r 4 log n 1 1 r for every additive stretch 0 and for every integral dimension r 1 log2n

pointwise estimates on the green s function for a scalar linear convection diffusion equation abstract pointwise estimates are found on green s functions for the scalar linear convection diffusion equations that arise when a scalar conservation law with non constant diffusion is linearized about a viscous shock profile of arbitrary strength the estimates take the form of gaussian kernels centered around paths determined by the typically different asymptotic states of the convection function the analysis extends the spectral transform method to the non constant coefficient case

an a posteriori error analysis of an elliptic optimal control problem in measure space abstract we propose an a posteriori error estimator for an optimal control problem with sparsity promoting cost the control variable lies in the space of regular borel measures we consider a solution technique that relies on the discretization of the control variable as a linear combination of dirac measures the proposed a posteriori error estimator can be decomposed into the sum of two contributions an error estimator in the maximum norm for the discretization of the adjoint equation and an estimator in the l 2 norm that accounts for the approximation of the state equation we prove that the designed error estimator is locally efficient and we explore its reliability properties the analysis is valid for two and three dimensional domains we illustrate the theory with numerical examples

initial and shock layers for boltzmann equation we study an initial value problem of the boltzmann equation with a euler shock wave as initial data our analysis exhibits the presence of three singular layers the initial layer formation layer and the shock layer an approximate solution is constructed based on a solution of the burgers type equation for the formation layer time scale the macroscopic conservation laws are preserved for the approximate solution the green s function of the linearized boltzmann equation around the approximate solution is constructed by a modification of the mathbb t mathbb c scheme introduced by yu nonlinear wave propagation over a boltzmann shock profile 2013 with the green s function approach and a wave tracing method one shows that the error of the approximate solution converges to zero with the convergent rate o 1 log varepsilon varepsilon 1 t 1 sigma 0 in pointwise norm cdot l infty xi 3 around the shock profile for sigma 0 in 0 frac 1 2 and with the rate o 1 varepsilon 1 t 1 2 outside the shock zone where varepsilon is the strength of the weak hyperbolic shock wave

a posteriori error analysis for a class of integral equations and variational inequalities we consider elliptic and parabolic variational equations and inequalities governed by integro differential operators of order 2s in 0 2 our main motivation is the pricing of european or american options under levy processes in particular pure jump processes or jump diffusion processes with tempered stable processes the problem is discretized using piecewise linear finite elements in space and the implicit euler method in time we construct a residual type a posteriori error estimator which gives a computable upper bound for the actual error in h s norm the estimator is localized in the sense that the residuals are restricted to the discrete non contact region numerical experiments illustrate the accuracy of the space and time estimators and show that they can be used to measure local errors and drive adaptive algorithms

optimal time decay estimates for the compressible navier stokes equations in the critical l p framework the global existence issue for the isentropic compressible navier stokes equations in the critical regularity framework was addressed in danchin invent math 141 3 579 614 2000 more than 15 years ago however whether optimal time decay rates could be shown in critical spaces has remained an open question here we give a positive answer to that issue not only in the l2 critical framework of danchin invent math 141 3 579 614 2000 but also in the general lp critical framework of charve and danchin arch ration mech anal 198 1 233 271 2010 chen et al commun pure appl math 63 9 1173 1224 2010 haspot arch ration mech anal 202 2 427 460 2011 we show that under a mild additional decay assumption that is satisfied if for example the low frequencies of the initial data are in l p 2 mathbb r d the lp norm the slightly stronger dot b 0 p 1 norm in fact of the critical global solutions decays like t d frac 1p frac14 for t to infty exactly as firstly observed by matsumura and nishida in proc jpn acad ser a 55 337 342 1979 in the case p 2 and d 3 for solutions with high sobolev regularity

compressible navier stokes equations with zero heat conductivity abstract we study the large time behavior of solutions of the compressible navier stokes equations without heat conductivity acoustic waves are dissipative due to the viscosity entropy waves are nondissipative due to lack of heat conductivity thus the system is of composite type the simplest prototype of this class of nonlinear partial differential equations we study the nonlinear interactions of waves using a combination of energy method weighted energy method and pointwise estimates

a posteriori error estimates for optimal control problems with state and control constraints we discuss the full discretization of an elliptic optimal control problem with pointwise control and state constraints we provide the first reliable a posteriori error estimator that contains only computable quantities for this class of problems moreover we show that the error estimator converges to zero if one has convergence of the discrete solutions to the solution of the original problem the theory is illustrated by numerical tests

energy norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the energy norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter to deal with anisotropic triangulations a special quasi interpolation operator is employed that may be of independent interest

maximum norm a posteriori error estimates for singularly perturbed elliptic reaction diffusion problems residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polyhedral domains standard finite element approximations are considered the error constants are independent of the diameters of mesh elements and the small perturbation parameter in our analysis we employ sharp bounds on the green s function of the linearized differential operator numerical results are presented that support our theoretical findings

galerkin and runge kutta methods unified formulation a posteriori error estimates and nodal superconvergence we unify the formulation and analysis of galerkin and runge kutta methods for the time discretization of parabolic equations this together with the concept of reconstruction of the approximate solutions allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods

stability of large amplitude viscous shock profiles of hyperbolic parabolic systems we establish nonlinear l1 h3 lp orbital stability 2 p with sharp rates of decay of large amplitude lax type shock profiles for a class of symmetric hyperbolic parabolic systems including compressible gas dynamics and magnetohydrodynamics mhd under the necessary conditions of strong spectral stability i e a stable point spectrum of the linearized operator about the wave transversality of the profile and hyperbolic stability of the associated ideal shock this yields in particular together with the spectral stability results of 50 the nonlinear stability of arbitrarily large amplitude shock profiles of isentropic navier stokes equations for a gamma law gas as 1 the first complete large amplitude stability result for a shock profile of a system with real i e partial viscosity a corresponding small amplitude result was established in 53 54 for general systems of kawashima class by a combination of kawashima type energy estimates and pointwise green function bounds where the small amplitude assumption was used only to close the energy estimates here under the mild additional assumption that hyperbolic characteristic speeds relative to the shock are not only nonzero but of a common sign we close the estimates instead by use of a goodman type weighted norm 25 26 designed to control estimates in the crucial hyperbolic modes