initial boundary value problem for the system of compressible adiabatic flow through porous media abstract we study the initial value problem for the system of compressible adiabatic flow through porous media in the one space dimension with fixed boundary condition under the restriction on the oscillations in the initial data we establish the global existence and large time behavior for the classical solutions via the combination of characteristic analysis and energy estimate methods the time asymptotic states for the solution are found and the exponential convergence to the asymptotic states is proved it is also shown that this problem can be approximated very well time asymptotically by an initial boundary problem for the related diffusion equations obtained from the original hyperbolic system by darcy s law provided that the initial total mass is the same the difference between these two solutions tends to zero exponentially fast as time goes to infinity in the sense of h 1 the diffusive phenomena caused by damping mechanism with boundary effects are thus observed

boundary effect on asymptotic behaviour of solutions to the p system with linear damping we consider the asymptotic behaviour of solutions to the p system with linear damping on the half line r 0 vt ux 0 ut p v x u with the dirichlet boundary condition u x 0 0 or the neumann boundary condition ux x 0 0 the initial date v0 u0 x has the constant state v u at x l hsiao and t p liu commun math phys 143 1992 599 605 have shown that the solution to the corresponding cauchy problem behaves like diffusion wave and k nishihara j differential equations131 1996 171 188 137 1997 384 395 has proved its optimal convergence rate our main concern in this paper is the boundary effect in the case of null dirichlet boundary condition on u the solution v u is proved to tend to v 0 as t tends to infinity its optimal convergence rate is also obtained by using the green function of the diffusion equation with constant coefficients in the case of null neumann boundary condition on u v 0 t is conservative and v 0 t v0 0 by virtue of the first equation so that v x t is expected to tend to the diffusion wave v x t connecting v0 0 and v in fact the solution v u x t is proved to tend to v x t 0 in the special case v0 0 v the optimal convergence rate is also obtained however this is not known in the case v0 0 v

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

convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping we consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time asymptotically the hyperbolic model may be viewed as isentropic euler equations with friction term added to the momentum equation to model gas flow through a porous media in this case our result justifies darcy s law time asymptotically our model may also be viewed as an elastic model with damping

initial boundary value problem of a parabolic hyperbolic system arising from tumor angiogenesis abstract in this paper we consider the half space problem for a parabolic hyperbolic system arising from tumor angiogenesis under a mixed type boundary condition we construct the green s function for half space problem by relating it with fundamental solution for an initial value problem after differential equation method for the boundary operator and proper estimates for additional exponential factors generated by the asymmetry of characteristics one reduces the green s function for initial boundary value problem into fundamental solution for cauchy problem which could be estimated by singularity removal long wave short wave decomposition and weighted energy method outside cone we finally obtain the pointwise estimate for green s function which results in the pointwise convergence rate for nonlinear problem the green s function constructed is precise enough to avoid a priori estimate for derivatives by energy method the half space problem studied here is a main ingredient for future study of shock profile with presence of boundary

existence and asymptotic behavior of global smooth solution for p system with damping and boundary effect abstract in this paper we study the existence and asymptotic behavior of global smooth solution for p system with damping on the strip 0 1 0 v t u x 0 u t p v x u with the null dirichlet boundary condition u 0 t u 1 t 0 first we prove the existence of the global smooth solution under the assumption that only the c 0 norm of the first derivatives of the initial data is sufficiently small while the c 0 norm of the initial data is not necessarily small second when the initial data is in c 2 and the c 0 norm of the first derivatives of the initial data is large enough then the first derivatives of the corresponding c 2 smooth solution must blow up in finite time finally we consider the asymptotic behavior of the global smooth solution precisely we show that the solution of this problem tends to the corresponding nonlinear diffusive wave governed by the classical darcy s law exponentially fast as time goes to infinity in the sense of h 1 norm provided the initial total mass is the same and thus justifies the validity of darcy s law in large time

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

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

best asymptotic profile for linear damped p system with boundary effect abstract this paper is devoted to the study of the 2 2 linear damped p system with boundary effect by a heuristic analysis we realize that the best asymptotic profile for the original solution is the parabolic solution of the ibvp for the corresponding porous media equation with a specified initial data in particular we further show the convergence rates of the original solution to its best asymptotic profile which are much better than the existing rates obtained in the previous works the approach adopted in the paper is the elementary weighted energy method with green function method together

homological properties of modules over semigroup algebras abstract let s be a semigroup in this paper we investigate the injectivity of l 1 s as a banach right module over l 1 s for weakly cancellative s this is the same as studying the flatness of the predual left module c 0 s for such semigroups s we also investigate the projectivity of c 0 s we prove that for many semigroups s for which the banach algebra l 1 s is non amenable the l 1 s module l 1 s is not injective the main result about the projectivity of c 0 s states that for a weakly cancellative inverse semigroup s c 0 s is projective if and only if s is finite

existence and stability of planar diffusion waves for 2 d euler equations with damping abstract to study the non linear stability of a non trivial profile for a multi dimensional systems of gas dynamics the combination of the green function on estimating the lower order derivatives and the energy method for the higher order derivatives is shown to be not only useful but sometimes maybe also essential in this paper we study the stability of a planar diffusion wave for the isentropic euler equations with damping in two dimensional space by introducing an approximate green function for the linearized equations around the planar diffusion wave and by applying the energy method we prove the global existence and the l 2 convergence rate of the solution when the initial data is a small perturbation of the planar diffusion wave the decay rates of the perturbation and its lower order spatial derivatives obtained are optimal in the l 2 norm furthermore the constructed approximate green function in this paper can be used for the pointwise and the l p estimates of the solutions concerned in fact the approach by combining of the green function and energy method can be applied to other system especially when the derivatives of the coefficients in the system have certain time decay properties

convergence to strong nonlinear diffusion waves for solutions to p system with damping in this paper we consider the so called p system with linear damping and we will prove an optimal decay estimates without any smallness conditions on the initial error more precisely if we restrict the initial data v0 u0 in the space h3 r l1 r h2 r l1 r then we can derive faster decay estimates than those given in p marcati m mei b rubino optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping j math fluid mech 7 2 2005 224 240 h zhao convergence to strong nonlinear diffusion waves for solutions of p system with damping j differential equations 174 1 2001 200 236 and m jian c zhu convergence to strong nonlinear diffusion waves for solutions to p system with damping on quadrant j differential equations 246 1 2009 50 77

convergence to nonlinear diffusion waves for solutions of euler equations with time depending damping abstract in this paper we are concerned with the asymptotic behavior of solutions to the system of euler equations with time depending damping in particular include the constant coefficient damping we rigorously prove that the solutions time asymptotically converge to the diffusion wave whose profile is self similar solution to the corresponding parabolic equation which justifies darcy s law compared with previous results about euler equations with constant coefficient damping obtained by hsiao and liu 1992 2 and nishihara 1996 9 we obtain a general result when the initial perturbation belongs to the same space i e h 3 r h 2 r our proof is based on the classical energy method

convergence to nonlinear diffusion waves for solutions of p system with time dependent damping abstract in this paper we study the cauchy problem for the hyperbolic p system with time gradually degenerate damping term 1 1 t u for 0 1 and show that the damped p system has a couple of global solutions uniquely and such solutions tend time asymptotically to the shifted nonlinear diffusion waves which are the solutions of the corresponding nonlinear parabolic equation governed by the darcy s law we further derive the convergence rates when the initial perturbations are in l 2 the approach adopted is the technical time weighted energy method

initial boundary value problem of euler equations with damping in r n abstract the asymptotic wave behavior of solution for the euler equations with damping in r n is investigated around a given constant equilibrium in this paper three classical boundary condition cases are considered here new approaches and techniques are introduced to deal with the multi dimensional case for the system for the linearized problem by comparing symbols in the transformed tangential spatial and time space we show that its green s function can be described in terms of the fundamental solution for the cauchy problem and reflected fundamental solution coupled with a boundary operator our approaches help us to simplify the green s function to the essential part and benefit the follow up nonlinear analysis for the nonlinear problem we prove the pointwise decaying rate with the help of the duhamel s principle

the pointwise estimates of diffusion wave for the navier stokes systems in odd multi dimensions we study the dissipation of solutions of the isentropic navier stokes equations in odd multi dimensions pointwise estimates of the time asymptotic shape of the solutions are obtained and shown to exhibit the generalized huygen s principle our approach is based on the detailed analysis of the green function of the linearized system this is used to study the coupling of nonlinear diffusion waves

convergence to strong nonlinear diffusion waves for solutions to p system with damping on quadrant in this paper we consider the so called p system with linear damping on quadrant we show that for a certain class of given large initial data v0 x u0 x the corresponding initial boundary value problem admits a unique global smooth solution v x t u x t and such a solution tends time asymptotically at the lp 2 p optimal decay rates to the corresponding nonlinear diffusion wave v x t u x t which satisfies 1 9 provided the corresponding prescribed initial error function v0 x u0 x lies in h3 r l1 r h2 r l1 r

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

l p convergence rates of planar waves for multi dimensional euler equations with damping abstract in this paper the l p convergence rates of planar diffusion waves for multi dimensional euler equations with damping are considered the analysis relies on a newly introduced frequency decomposition and green function based energy method it is a combination of the l p estimate on the low frequency component by using an approximate green function and l 2 estimate on the high frequency component through the energy method by noticing that the low frequency component in the approximate green function has the algebraic decay which governs the large time behavior while the high frequency component has the exponential decay but with singularity their combination leads to a global algebraic decay estimate to use the decay property only of the low frequency component in the approximate green function avoids the singularity in the high frequency component so that it simplifies and improves the previous works on this system this new approach of the combination of the green function and energy method through the frequency decomposition can also be applied to the hyperbolic parabolic systems satisfying the kawashima condition and also the systems whose derivatives of the coefficients have suitable time decay properties

initial boundary value problem for p system with damping in half space abstract in this paper the initial boundary value problem for p system with damping in half space is studied with a posed mix boundary condition the fundamental solution green s function for cauchy problem is constructed with a sharp decaying structure and a precise capture of the singularity by a singularity removal process later the symbols of both fundamental solution and green s function for initial boundary value problem are obtained with transformed variable s and spatial variable x while in this level the green s function could be decomposed into fundamental solution and boundary operator thus with the study of boundary operator and previous precise information about the fundamental solution the pointwise structure as well as the singular structure of the green s function is established finally the nonlinear stability is obtained by the green s function and a priori estimate from energy method

well posedness of the ibvp for 2 d euler equations with damping abstract in this paper we focus on the initial boundary value problem of the 2 d isentropic euler equations with damping we prove the global in time existence of classical solution to the initial boundary value problem for small smooth initial data by the method of local existence of solution combined with a priori energy estimates where the appropriate boundary condition plays an important role

large time behavior of solutions for compressible euler equations with damping in r 3 we consider the long time behavior and optimal decay rates of global strong solutions for the isentropic compressible euler equations with damping in r3 in the present paper when the regular initial data belong to some sobolev space hl r3 b 1 s r3 with l 4 and s 0 1 we show that the density of the system converges to its equilibrium state at the rates 1 t 34 s2 in the l2 norm or 1 t 32 s2 in the l norm respectively the momentum of the system decays at the rates 1 t 54 s2 in the l2 norm or 1 t 2 s2 in the l norm respectively which are shown to be optimal for the compressible euler equations with damping

singularities of solutions to the compressible euler equations and euler poisson equations with damping in this paper we consider the formation of singularities of solutions to the compressible euler equations with damping and pressureless euler poisson equations with damping under certain assumptions of the initial data we can get the blow up result of the solutions in the finite time in this paper we consider the formation of singularities of solutions to the compressible euler equations with damping and pressureless euler poisson equations with damping under certain assumptions of the initial data we can get the blow up result of the solutions in the finite time

existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in eulerian and lagrangian coordinates we prove the existence and compactness stability of entropy solutions for the hyperbolic systems of conservation laws corresponding to the isentropic gas dynamics where the pressure and density are related by a law for any 1 our results considerably extend and simplify the program initiated by diperna and provide a complete existence proof our methods are based on the compensated compactness and the kinetic formulation of systems of conservation laws 1996 john wiley sons inc

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

asymptotic behaviors of solutions of quasilinear hyperbolic equations with linear damping ii this paper as a follow up to the recent work of k nishihara 1997 j differential equations133 384 395 is concerned with the asymptotic behaviors of the solution of quasilinear hyperbolic equation with linear damping vtt a vx x vt 0 a v 0forallv r which satisfies the following prescribed initial condition v t x vt t x t 0 v0 x v1 x x r compared with the results obtained by k nishihara the main novelties of our present paper lie in the following first we recover all the results obtained by k nishihara under some other smallness assumptions on the initial data v0 x v1 x which are much weaker than those needed in k nishihara s arguments second under some additional assumptions on the nonlinear function a v and on the initial data v0 x v1 x when 0 0 1 0 we succeed in proving that t x is still an asymptotic profile of v t x and this answers partially the open problem left by k nishihara

existence and uniqueness of solutions for a class of non newtonian fluids with vacuum and damping abstract in this paper we proved local existence and uniqueness of solutions for a class of non newtonian fluids with vacuum and damping in one dimensional bounded intervals the main difficulty is due to the strong nonlinearity of the system and initial vacuum

pointwise estimates and lp convergence rates to diffusion waves for p system with damping abstract by introducing a new approximate green function we obtain the pointwise estimates on the solutions of euler equations with linear frictional damping from which we can deduce the optimal l p 1 p convergence rates to the nonlinear diffusion waves the pointwise estimates and l p 1 p convergence rates given in this paper are new

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

global existence of classical solutions of full euler maxwell equations abstract we prove the global existence and large time behavior of classical solutions to the full euler maxwell equations in chemin lerner spaces with critical regularity which generalizes those results on the isentropic euler maxwell equations

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

maximal repetition and zero entropy rate maximal repetition of a string is the maximal length of a repeated substring this paper investigates maximal repetition of strings drawn from stochastic processes strengthening previous results two new bounds for the almost sure growth rate of maximal repetition are identified an upper bound in terms of conditional renyi entropy of order gamma 1 given a sufficiently long past and a lower bound in terms of unconditional shannon entropy gamma 1 both the upper and the lower bound can be proved using an inequality for the distribution of recurrence time we also supply an alternative proof of the lower bound which makes use of an inequality for the expectation of subword complexity in particular it is shown that a power law logarithmic growth of maximal repetition with respect to the string length recently observed for texts in natural language may hold only if the conditional renyi entropy rate given a sufficiently long past equals zero according to this observation natural language cannot be faithfully modeled by a typical hidden markov process which is a class of basic language models used in computational linguistics

half space problem for euler equations with damping in 3 d abstract in this paper we consider the half space problem for euler equations with damping in 3 d we restudy the fundamental solution for the cauchy problem to obtain an exponentially sharp pointwise structure and a clear decomposition of the singular regular components later both green s function for initial boundary value problem and fundamental solutions for cauchy problems are investigated in the transformed domain after laplace transform the symbols are obtained and a connection between green s function and fundamental solutions are established for the pointwise space time structure of green s function finally the sharp estimates for green s function together with a priori estimates from the energy method for high order derivatives result in the nonlinear stability of the solution and also the decaying rates

banach algebra structure and amenability of a class of matrix algebras with applications abstract we introduce a new category of banach algebras l 1 munn algebras which we use as a tool in the study of semigroup algebras then we characterize amenable l 1 munn algebras and also semisimple ones in this category applying these results to the semigroup algebras provides some characterizations of amenable semigroup algebras we also provide a counter example to a conjecture of duncan and paterson

l p convergence rates of solutions for n dimensional quasilinear damped wave equation we realize the asymptotic profile for the solutions to the n dimensional quasilinear damped wave equation in the divergence form is the solution of the corresponding linear parabolic equation with special initial data and we further show the lp 1 p infinity in particular p 1 convergence rates to this linear asymptotic profile c 2013 elsevier inc all rights reserved

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

large time behavior for the system of compressible adiabatic flow through porous media in r3 abstract the cauchy problem of the system of compressible adiabatic flow through porous media in r 3 is considered we obtain the global existence and large time behavior of the solution when the initial data is close to its equilibrium in h 3 norm this is a continuing work of 31 where due to non dissipative property of the entropy s an additional assumption that the initial data is bounded in l 1 norm plays a key role in closing the a priori energy estimates and hence establishing the global existence of the solution in this paper to remove this assumption we first decompose the solution u into high frequency part u h and low frequency part u l then by constructing delicate energy functionals for u h and optimal decay rates for u l we can get the desired energy estimates to close the a priori assumption furthermore we get the optimal l 2 l 2 decay rate of the solution and show that the velocity decays faster than the density

relaxation time limit of the multidimensional bipolar hydrodynamic model in besov space abstract in this paper we study a multidimensional bipolar hydrodynamic model for semiconductors or plasmas this system takes the form of the bipolar euler poisson model with electric field and frictional damping added to the momentum equations in the framework of the besov space theory we establish the global existence of smooth solutions for cauchy problems when the initial data are sufficiently close to the constant equilibrium next based on the special structure of the nonlinear system we also show the uniform estimate of solutions with respect to the relaxation time by the high and low frequency decomposition methods finally we discuss the relaxation time limit by compact arguments that is it is shown that the scaled classical solution strongly converges towards that of the corresponding bipolar drift diffusion model as the relaxation time tends to zero

lp convergence rate to nonlinear diffusion waves for p system with damping abstract in this paper we study the p system with frictional damping and show that the solutions time asymptotically tend to the nonlinear diffusion waves governed by the classical darcy s law by introducing an approximate green function we obtain the optimal l p 2 p convergence rate of the solution which is a perturbation of the nonlinear diffusion wave to the hyperbolic system

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

convergence to strong nonlinear diffusion waves for solutions of p system with damping this paper is concerned with the p system with frictional damping and our main purpose is two fold first we show that for a certain class of given large initial data nu 0 x u 0 x the cauchy problem 1 1 1 2 admits a unique global smooth solution nu t x u t x and such a solution tends time asymptotically at the optimal l p 2 less than or equal to p less than or equal to infinity decay rates to the corresponding nonlinear diffusion wave over bar t x u over bar t x governed by the classical darcy s law provided that the corresponding prescribed initial error function v 0 x u 0 x ties in h 3 x h 2 r boolean and l 1 x l 1 r compared with former results in this direction obtained by l hsiao and t p liu 1992 comm math phys 143 599 605 k nishihara 1996 j differential equations 131 171 188 and k nishihara w k wang and t yang 2000 j differential equations 161 191 218 our main novelty lies in the facts that the nonlinear diffusion wave over bar t x u over bar t x need not to be weak and v 0 x u 0 x can be chosen arbitrarily large secondly we show that the nonlinear diffusion waves over bar t x u over bar t x are nonlinear stable provided that the strength or the nonlinear diffusion waves is weak and that the initial disturbance v 0 x u 0 x satisfies the assumption that parallel tov 0xx x parallel to l infinity parallel tou 0x x parallel to l infinity is sufficiently small we also show that the smallness assumption imposed oil the strength of the diffusion waves is a necessary condition to guarantee the nonlinear stability result and compared with the corresponding results obtained by l hsiao and t p liu 1992 comm math phys 143 599 605 the smallness conditions we imposed on the initial disturbance are much more weaker c 2001 academic press

global existence and large time behavior for the system of compressible adiabatic flow through porous media in r 3 abstract the system of compressible adiabatic flow through porous media is considered in r 3 in the present paper the global existence and uniqueness of classical solutions are obtained when the initial data is near its equilibrium we also show that the pressure of the system converges to its equilibrium state at the same l 2 rate 1 t 3 4 as the navier stokes equations without heat conductivity but the velocity of the system decays at the l 2 rate 1 t 5 4 which is faster than the l 2 rate 1 t 3 4 for the navier stokes equations without heat conductivity duan and ma 2008 3

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 pointwise estimates of solutions for euler equations with damping in multi dimensions we study the time asymptotic behavior of solutions for the isentropic euler equations with damping in multi dimensions the global existence and pointwise estimates of the solutions are obtained furthermore we obtain the optimal lp 1 p convergence rate of the solution when it is a perturbation of a constant state our approach is based on a detailed analysis of the green function of the linearized system and some energy estimates

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

compressible euler equations with vacuum abstract in this paper we first show that the regular solutions of compressible euler equations in r 3 with damping will not be global if the initial density function has compact support this implies that c 2 cannot be smooth across the boundary separating the gas and the vacuum after a finite time where c is the speed of sound then we study the local existence of solutions for isentropic gas flow in r when c 0 1 is smooth across using the energy method and the characteristics method

darcy s law as long time limit of adiabatic porous media flow abstract it is conjectured that darcy s law governs the motion of compressible porous media flow in large time this has been justified for one dimensional isentropic flows in this work we show the conjecture is true for one dimensional adiabatic flows with generic small smooth initial data

optimal convergence rates to nonlinear diffusion waves for the compressible euler poisson system with damping abstract in this work we study the 1 d isentropic bipolar hydrodynamic model this model takes the form of compressible euler poisson system with nonlinear damping added to the momentum equations under some smallness conditions the solutions to the cauchy problem of the system globally exist and convergence to the nonlinear diffusion waves which are the corresponding solutions of nonlinear parabolic equations given by the darcy s law with a specified initial data the optimal convergence rates are obtained by green function method when the initial perturbation is in l 1 space

green s functions of wave equations in mathbb r n_ times mathbb r _ we study the d alembert equation with a boundary we introduce the notions of rayleigh surface wave operators delayed advanced mirror images wave recombinations and wave cancellations this allows us to obtain the complete and simple formula of the green s functions for the wave equation with the presence of various boundary conditions we are able to determine whether a rayleigh surface wave is active or virtual and study the lacunas of the wave equation in three dimensional with the presence of a boundary in the case of a virtual rayleigh surface wave

asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping we consider the asymptotic behavior of the solution of quasilinear hyperbolic equation with linear dampingvtt a vx x vt 0 x t r 0 subsequent to k nishihara j differential equations131 1996 171 188 in that article the system with dampingvt ux 0 ut p v x u p v 0 was treated and the convergence rates to the diffusion wave by l hsiao and t p liu comm math phys 143 1992 599 605 and l hsiao and t p liu chinese ann math ser b14 1993 465 480 were improved for initial data v0 u0 x satisfying v0 u0 v0 0 and v0 x v0 dx 0 this system is reduced to the quasilinear hyperbolic equation with linear dampingvtt p v0 vx p v0 x vt 0 and the decay rates v t l o t 1 2 vx t l o t 1 vt t l o t 3 2 were obtained provided that v vt t 0 h3 h2 is small and inl1 l1 from these decay rates vis expected to behave as a solution to the parabolic equation the aims of this paper are to seek the asymptotic profile x t satisfying t l o t 1 2 x t l o t 1 t t l o t 3 2 and to show that the solution of satisfies v t l o t 1 v x t l o t 3 2 v t t l o t 2 the proof is based on the elementary energy method and the green function method for the parabolic equation

on the classical solutions to the hydrodynamic model for semiconductors abstract this paper mainly deals with the multidimensional hydrodynamic model for semiconductors inspired by the previous papers y shizuta s kawashima systems of equations of hyperbolic parabolic type with applications to the discrete boltzmann equation hokkaido math j 14 1985 249 275 s kawashima w a yong dissipative structure and entropy for hyperbolic systems of balance laws arch ration mech anal 174 2004 345 364 w a yong entropy and global existence for hyperbolic balance laws arch ration mech anal 172 2004 247 266 we develop some new frequency localization estimates to establish the global existence and exponential stability of small classical solutions in a class of critical besov spaces which are different from estimates in our recent paper d y fang j xu t zhang global exponential stability of classical solutions to the hydrodynamic model for semiconductors math models methods appl sci 17 2007 1507 1530 furthermore this new method can also be applied to the multidimensional euler equations with damping the analytic tool used is the littlewood paley decomposition

kinetic formulation of the isentropic gas dynamics and p systems we consider the 2 x 2 hyperbolic system of isentropic gas dynamics in both eulerian or lagrangian variables also called the p system we show that they can be reformulated as a kinetic equation using an additional kinetic variable such a formulation was first obtained by the authors in the case of multidimensio nal scalar conservation laws a new phenomenon occurs here namely that the advection velocity is now a combination of the macroscopic and kinetic velocities various applications are given we recover the invariant regions deduce new l estimates using moments lemma and prove l w stability for 7 3

regular hilberg processes an example of processes with a vanishing entropy rate a regular hilberg process is a stationary process that satisfies both a hyperlogarithmic growth of maximal repetition and a power law growth of topological entropy which are a kind of dual conditions the hyperlogarithmic growth of maximal repetition has been experimentally observed for texts in natural language whereas the power law growth of topological entropy implies a vanishing shannon entropy rate and thus probably does not hold for natural language in this paper we provide a constructive example of regular hilberg processes which we call random hierarchical association rha processes our construction does not apply the standard cutting and stacking method for the constructed rha processes we demonstrate that the expected length of any uniquely decodable code is the orders of magnitude larger than the shannon block entropy of the ergodic component of the rha process our proposition supplements the classical result by shields concerning nonexistence of universal redundancy rates

asymptotic behavior of solutions to bipolar euler poisson equations with time dependent damping abstract in this paper we study the one dimensional euler poisson equations of bipolar hydrodynamic model for semiconductor device with time dependent damping effect j 1 t for 1 1 where the damping effect is time gradually degenerate for 0 and time gradually enhancing for 0 such a damping effect makes the hydrodynamic system possess the nonlinear diffusion phenomena time asymptotic weakly or strongly based on technical observation and by using the time weighted energy method where the weights are artfully chosen we prove that the system admits a unique global smooth solution which time asymptotically converges to the corresponding diffusion wave when the initial perturbation around the diffusion wave is small enough the convergence rates are specified in the algebraic forms o t 3 4 1 and o t 1 according to different values of in 1 1 7 and 1 7 1 respectively where 1 7 is the critical point and the convergence rate at the critical point is o t 6 7 ln t all these convergence rates obtained in different cases are optimal in the sense when the initial perturbations are l 2 integrable particularly when 1 7 the convergence rate is the fastest namely the asymptotic profile of the original system at the critical point is the best

pointwise convergence to shock waves for viscous conservation laws we are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws it is shown that besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and burgers equations a perturbation also gives rise to algebraically decaying terms which measure the coupling of waves of different characteristic families our technique is a combination of time asymptotic expansion construction of approximate green functions and analysis of nonlinear wave interactions the pointwise estimates yield optimal lp convergence of the perturbation to the shock and diffusion waves 1 p the new approach of obtaining pointwise estimates based on the green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks these are being explored elsewhere 1997 john wiley sons inc

asymptotic behavior for scalar viscous conservation laws with boundary effect abstract we consider the asymptotic stability of viscous shock wave for scalar viscous conservation lawsut f u x uxxon the half space 0 with boundary valuesu x u u x 0 u our problem is divided into three cases depending on the sign of shock speedsof the shock u u whens 0 the asymptotic state ofubecomes d t whered t depends implicitly on the initial datau x 0 and is related to the boundary layer of the solution at the boundaryx 0 the stability of this state fors 0 is also treated

half space problem of the system of compressible adiabatic flow through porous media in multi d abstract in this paper we consider the half space problem for the system of compressible adiabatic flow through porous media in 2 d and 3 d combining energy method green s function method and bootstrap method one establishes the decaying energy estimates which lead to global existence and optimal l 2 decaying rate of the solution finally for more large time information especially in spatial variables one applies green s function solution representation obtained by duhamel s principle and a priori estimate from energy method to obtain the space time pointwise structure of the solution

existence and asymptotic behavior of c1 solutions to the multi dimensional compressible euler equations with damping abstract in this paper the existence and asymptotic behavior of c 1 solutions to the multi dimensional compressible euler equations with damping on the framework of besov space are considered comparing with the well posedness results of sideris thomases wang t sideris b thomases d h wang long time behavior of solutions to the three dimensional compressible euler with damping comm partial differential equations 28 2003 953 978 we weaken the regularity assumptions on the initial data the global existence lies on a crucial a priori estimate which is obtained by the spectral localization method the main analytic tools are the littlewood paley decomposition and bony s paraproduct formula

nonlinear diffusion waves for hyperbolic p system with nonlinear damping abstract this paper is concerned with the p system of hyperbolic conservation laws with nonlinear damping when the constant states are small the solutions of the cauchy problem for the damped p system globally exist and converge to their corresponding nonlinear diffusion waves which are the solutions of the corresponding nonlinear parabolic equation given by the darcy s law the optimal convergence rates are also obtained in order to overcome the difficulty caused by the nonlinear damping a couple of correction functions have been technically constructed the approach adopted is the elementary energy method together with the technique of approximating green function on the other hand when the constant states are large the solutions of the cauchy problem for the p system will blow up at a finite time

the interactions between ergodic theory and information theory information theorists frequently use the ergodic theorem likewise entropy concepts are often used in information theory recently the two subjects have become partially intertwined as deeper results from each discipline find use in the other a brief history of this interaction is presented in this paper together with a more detailed look at three areas of connection namely recurrence theory blowing up bounds and direct sample path methods

on the vocabulary of grammar based codes and the logical consistency of texts this paper presents a new interpretation for zipf mandelbrot s law in natural language which rests on two areas of information theory firstly we construct a new class of grammar based codes and secondly we investigate properties of strongly nonergodic stationary processes the motivation for the joint discussion is to prove a proposition with a simple informal statement if a text of length n describes n independent facts in a repetitive way then the text contains at least n log n different words under suitable conditions on n in the formal statement two modeling postulates are adopted firstly the words are understood as nonterminal symbols of the shortest grammar based encoding of the text secondly the text is assumed to be emitted by a finite energy strongly nonergodic source whereas the facts are binary iid variables predictable in a shift invariant way

global classical solutions for partially dissipative hyperbolic system of balance laws the basic existence theory of kato and majda enables us to obtain local in time classical solutions to generally quasilinear hyperbolic systems in the framework of sobolev spaces in x with higher regularity however it remains a challenging open problem whether classical solutions still preserve well posedness in the case of critical regularity this paper is concerned with partially dissipative hyperbolic system of balance laws under the entropy dissipative assumption we establish the local well posedness and blow up criterion of classical solutions in the framework of besov spaces with critical regularity with the aid of the standard iteration argument and friedrichs regularization method then we explore the theory of function spaces and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous chemin lerner spaces mixed space time besov spaces this fact allows us to capture the dissipation rates generated from the partial dissipative source term and further obtain the global well posedness and stability by assuming at all times the shizuta kawashima algebraic condition as a direct application the corresponding well posedness and stability of classical solutions to the compressible euler equations with damping are also obtained

decay of entropy solutions of nonlinear conservation laws we are concerned with the asymptotic behavior of entropy solutions of nonlinear conservation laws the main objective of this paper is to present an analytical approach and to explore its applications to studying the large time behavior of periodic entropy solutions of hyperbolic conservation laws the asymptotic decay of periodic solutions of nonlinear hyperbolic conservation laws is an important nonlinear phenomenon it is observed that the genuine nonlinearity of equations forces the waves of each characteristic family to interact vigorously and to cancel each other the insightful analysis of glimm lax gl for scalar equations and 2 2 systems has indicated that the resultant mutual cancellation of interacting shock and rarefaction waves of the same family induces the decay of periodic solutions such a result was first shown by lax l1 in 1957 for one dimensional convex scalar conservation laws dafermos d1 applying his uniform processes proved the decay result for the case that the set of inflection points of the flux does not have an accumulation point the glimm lax theory gl indicates that for 2 2 strictly hyperbolic and genuinely nonlinear systems any periodic glimm solution decays like o 1 t this result was proved by using the approximate characteristic method in the glimm difference solutions provided that the oscillation of the corresponding initial data is small recently using the method of generalized characteristics dafermos d3 showed that any periodic solution with local bounded variation and small oscillation for 2 2 systems decays asymptotically with the same detailed structure pas found by lax l1 for the

global and blow up solutions for compressible euler equations with time dependent damping abstract this paper deals with the cauchy problem for the compressible euler equations with time dependent damping where the time vanishing damping in the form of 1 t makes some fantastic variety of the dynamic system for 0 1 and 0 or 1 but 2 the solutions are proved to exist globally in time when the derivatives of the initial data are small but the initial data themselves can be arbitrarily large this is the so called challenging case of global solutions with large initial data while when the initial riemann invariants are monotonic and their derivatives with absolute value are large at least at one point then the solutions are still bounded but their derivatives will blow up at finite time somewhat like the singularity formed by shock waves for 1 and 0 or 1 but 0 1 the derivatives of solutions will blow up even for all initial data including the interesting case of blow up solutions with small initial data here the initial riemann invariants are monotonic in fact such a blow up phenomenon is determined by the mechanism of the dynamic system itself in order to prove the global existence of solutions with large initial data we introduce a new energy functional related to the riemann invariants which crucially enables us to build up the maximum principle for the corresponding riemann invariants and the uniform boundedness for the local solutions finally numerical simulations in different cases are carried out which further confirm our theoretical results

local existence with physical vacuum boundary condition to euler equations with damping abstract in this paper we consider the local existence of solutions to euler equations with linear damping under the assumption of physical vacuum boundary condition by using the transformation introduced in lin and yang methods appl anal 7 3 2000 495 to capture the singularity of the boundary we prove a local existence theorem on a perturbation of a planar wave solution by using littlewood paley theory and justifies the transformation introduced in liu and yang 2000 in a rigorous setting

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