singular behavior of vacuum states for compressible fluids in this survey paper we will present the recent work on the study of the compressible fluids with vacuum states by illustrating its interesting and singular behavior through some systems of fluid dynamics that is euler equations euler poisson equations and navier stokes equations the main concern is the well posedness of the problem when vacuum presents and the singular behavior of the solution near the interface separating the vacuum and the gas furthermore the relation of the solutions for the gas dynamics with vacuum to those of the boltzmann equation will also be discussed in fact the results obtained so far for vacuum states are far from being complete and satisfactory therefore this paper can only be served as an introduction to this interesting field which has many open and challenging mathematical problems moreover the problems considered here are limited to the author s interest and knowledge in this area

fluid dynamic limits of kinetic equations i formal derivations the connection between kinetic theory and the macroscopic equations of fluid dynamics is described in particular our results concerning the incompressible navier stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of hilbert or chapman enskog the moment formalism shows that the limit leading to the incompressible navier stokes equations like that leading to the compressible euler equations is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible navier stokes equations some indications of the validity of these limits are given more specifically the connection between the diperna lions renormalized solution of the classical boltzmann equation and the leray solution of the navier stokes equations is discussed

compressible navier stokes equations with degenerate viscosity coefficient and vacuum in this paper we consider the compressible navier stokes equations for isentropic flow of finite total mass when the initial density is either of compact or infinite support the viscosity coefficient is assumed to be a power function of the density so that the cauchy problem is well posed new global existence results are established when the density function connects to the vacuum states continuously for this some new a priori estimates are obtained to take care of the degeneracy of the viscosity coefficient at vacuum we will also give a non global existence theorem of regular solutions when the initial data are of compact support in eulerian coordinates which implies singularity forms at the interface separating the gas and vacuum

the aw rascle traffic model with locally constrained flow abstract we consider solutions of the aw rascle model for traffic flow fulfilling a constraint on the flux at x 0 two different kinds of solutions are proposed at x 0 the first one conserves both the number of vehicles and the generalized momentum while the second one conserves only the number of cars we study the invariant domains for these solutions and we compare the two riemann solvers in terms of total variation of relevant quantities finally we construct ad hoc finite volume numerical schemes to compute these solutions

on the vacuum state for the equations of isentropic gas dynamics condition c which is dictated by physics is a source of essential difficulties in the mathematical treatment of this problem there have been extensive studies on the smooth solutions for equations e 2 61 these studies are concerned with states p u in the region strictly away from the vacuum that is p l u p0 for some fixed positive constant p a necessary condition for the existence of a global bounded continuous solution is

strong convergence to global solutions for multidimensional flows of compressible viscous fluids with polytropic equations of state and discontinuous initial data we extend to general polytropic pressures p k 1 the existence theory of 8 for isothermal 1 flows of navier stokes fluids in two and three space dimensions with fairly general initial data specifically we require that the initial density be close to a constant in l2 and l and that the initial velocity be small in l2 and bounded in l2n in two dimensions the l2 norms must be weighted slightly solutions are obtained as limits of approximate solutions corresponding to mollified initial data the key point is that the approximate densities are shown to converge strongly so that nonlinear pressures can be accommodated even in the absence of any uniform regularity information for the approximate densities

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

energy method for boltzmann equation abstract a basic simple energy method for the boltzmann equation is presented here it is based on a new macro micro decomposition of the boltzmann equation as well as the h theorem this allows us to make use of the ideas from hyperbolic conservation laws and viscous conservation laws to yield the direct energy method as an illustration we apply the method for the study of the time asymptotic nonlinear stability of the global maxwellian states previous energy method starting with grad and finishing with ukai involves the spectral analysis and regularity of collision operator through sophisticated weighted norms

uniqueness and nonuniqueness for positive radial solutions of δu f u r 0 on considere le probleme de l unicite pour les solutions positives du probleme de dirichlet non lineaire u f u x 0 dans u 0 est une boule ou un anneau de r n et f 0 est superlineaire en u et satisfaisait f 0 x 0

existence of bv solutions for a non conservative constrained aw rascle zhang model for vehicular traffic abstract the main aim of this paper is to study the aw rascle zhang arz model with non conservative local point constraint on the density flux introduced in 10 its motivation being for instance the modeling of traffic across a toll gate we prove the existence of weak solutions under assumptions that result to be more general than those required in 11 more precisely we do not require that the waves of the first characteristic family have strictly negative speeds of propagation the result is achieved by showing the convergence of a sequence of approximate solutions constructed via the wave front tracking algorithm the case of solutions attaining values at the vacuum is considered we also present an explicit numerical example to describe some qualitative features of the solutions

scalar conservation laws with moving constraints arising in traffic flow modeling an existence result abstract we consider a strongly coupled pde ode system that describes the influence of a slow and large vehicle on road traffic the model consists of a scalar conservation law accounting for the main traffic evolution while the trajectory of the slower vehicle is given by an ode depending on the downstream traffic density the moving constraint is expressed by an inequality on the flux which models the bottleneck created in the road by the presence of the slower vehicle we prove the existence of solutions to the cauchy problem for initial data of bounded variation

compressible navier stokes equations with degenerate viscosity coefficient and vacuum ii abstract this is a continuation of the paper comm math phys 230 2002 329 on the study of the compressible navier stokes equations for isentropic flow when the initial density connects to vacuum continuously the degeneracy appears in the initial data and has effect on the viscosity coefficient because the coefficient is assumed to be a power function of the density this assumption comes from physical consideration and it also gives the well posedness of the cauchy problem a new global existence result is established by some new a priori estimates so that the interval for the power of the density in the viscosity coefficient is enlarged to 0 1 3

global validity of the boltzmann equation for two and three dimensional rare gas in vacuum erratum and improved result we point out an error in our earlier papers 1 and 2 and present a more direct and natural proof which although based on the same physical ideas of the previous ones saves and actually improves the validity results for the boltzmann equation given in 1 and 2

fluid dynamic limits of kinetic equations ii convergence proofs for the boltzmann equation using relative entropy estimates about an absolute maxwellian it is shown that any properly scaled sequence of diperna lions renormalized solutions of some classical boltzmann equations has fluctuations that converge to an infinitesimal maxwellian with fluid variables that satisfy the incompressibility and boussinesq relations moreover if the initial fluctuations entropically converge to an infinitesimal maxwellian then the limiting fluid variables satisfy a version of the leray energy inequality if the sequence satisfies a local momentum conservation assumption the momentum densities globaly converge to a solution of the stokes equation a similar discrete time version of this result holds for the navier stokes limit with an additional mild weak compactness assumption the continuous time navier stokes limit is also discussed 01993 john wiley sons inc

receding horizon boundary control of nonlinear conservation laws with shock avoidance this paper deals with the boundary control for scalar non linear hyperbolic systems of conservation laws one of the issues arising from these systems is the occurrence of singularities called shocks in the domain we will show that to avoid the shock a set of constraints on both the control and the state at the boundary has to be fulfilled then a proof of the exponential stability of the system is established provided that there is no shock and that the state at the boundary is exponentially stable these conditions are shown to be achieved by the receding horizon optimal control approach a simulation is finally carried out with the freeway traffic model to demonstrate the effectiveness of the here proposed control

finite volume schemes for locally constrained conservation laws this paper is devoted to the numerical analysis of the road traffic model proposed by colombo and goatin j differ equ 234 2 654 675 2007 the model involves a standard conservation law supplemented by a local unilateral constraint on the flux at the point x 0 modelling a road light a toll gate etc we first show that the problem can be interpreted in terms of the theory of conservation laws with discontinuous flux function as developed by adimurthi et al j hyperbolic differ equ 2 4 783 837 2005 and burger et al siam j numer anal 47 3 1684 1712 2009 we reformulate accordingly the notion of entropy solution introduced by colombo and goatin j differ equ 234 2 654 675 2007 and extend the well posedness results to the l framework then starting from a general monotone finite volume scheme for the non constrained conservation law we produce a simple scheme for the constrained problem and show its convergence the proof uses a new notion of entropy process solution numerical examples modelling a green wave are presented

existence and non existence of global smooth solutions for p system with relaxation abstract in this paper we consider the cauchy problem for p system with relaxation under the assumption that the relaxation time e is sufficiently small we prove the existence of the global smooth solution to the cauchy problem with c1 initial data provided the c0 norm of the derivative of the initial data is of the order of e here is a small positive constant on the other hand when the initial density has compact support but is not identically zero we prove the global regular solution for the cauchy problem does not exist

a theory of l 1 dissipative solvers for scalar conservation laws with discontinuous flux we propose a general framework for the study of l1 contractive semigroups of solutions to conservation laws with discontinuous flux u t mathfrak f x u x 0 qquad mathfrak f x u left begin array ll f l u x 0 end array right quad quad quad rm cl where the fluxes fl fr are mainly assumed to be continuous developing the ideas of a number of preceding works baiti and jenssen in j differ equ 140 1 161 185 1997 towers in siam j numer anal 38 2 681 698 2000 towers in siam j numer anal 39 4 1197 1218 2001 towers et al in skr k nor vidensk selsk 3 1 49 2003 adimurthi et al in j math kyoto university 43 1 27 70 2003 adimurthi et al in j hyperbolic differ equ 2 4 783 837 2005 audusse and perthame in proc roy soc edinburgh a 135 2 253 265 2005 garavello et al in netw heterog media 2 159 179 2007 burger et al in siam j numer anal 47 1684 1712 2009 we claim that the whole admissibility issue is reduced to the selection of a family of elementary solutions which are piecewise constant weak solutions of the form c x c l11 left x 0 right we refer to such a family as a germ it is well known that cl admits many different l1 contractive semigroups some of which reflect different physical applications we revisit a number of the existing admissibility or entropy conditions and identify the germs that underly these conditions we devote specific attention to the vanishing viscosity germ which is a way of expressing the condition of diehl j hyperbolic differ equ 6 1 127 159 2009 for any given germ we formulate germ based admissibility conditions in the form of a trace condition on the flux discontinuity line x 0 in the spirit of vol pert math ussr sbornik 2 2 225 267 1967 and in the form of a family of global entropy inequalities following kruzhkov math ussr sbornik 10 2 217 243 1970 and carrillo arch ration mech anal 147 4 269 361 1999 we characterize those germs that lead to the l1 contraction property for the associated admissible solutions our approach offers a streamlined and unifying perspective on many of the known entropy conditions making it possible to recover earlier uniqueness results under weaker conditions than before and to provide new results for other less studied problems several strategies for proving the existence of admissible solutions are discussed and existence results are given for fluxes satisfying some additional conditions these are based on convergence results either for the vanishing viscosity method with standard viscosity or with specific viscosities adapted to the choice of a germ or for specific germ adapted finite volume schemes

traffic flow modelling with junctions abstract motivated by the modelling of a roundabout we are led to study the traffic on a road with points of entry and exit in this note we would like to describe the modellisation of a junction and solve the riemann problem for such a model more precisely between each point of discontinuity we use a multi class extension of the lwr model to describe the evolution of the density of the vehicles the multi class approach being used in order to distinguish the vehicles after their origin and destination then we treat the points of entry and exit thanks to special boundary conditions that give bounds on the flows of the different types of vehicles in the case of the one t road we obtain a result of existence and uniqueness this first step allows us to obtain a similar result for the n t road we describe these results and also some properties of the obtained solutions in order to see how long this model is valid

solutions of euler poisson equations for gaseous stars in this paper we study the euler poisson equations governing gas motion under self gravitational force we are interested in the evolution of the gaseous stars for which the density function has compact support we establish existence theory for the stationary solutions and describe the behavior of the solutions near the vacuum boundary the boundary behavior thus obtained agrees with the physical boundary condition proposed and studied in l ly for both euler equations with damping and the euler poisson equations existence non existence uniqueness and instability of the stationary solutions with vacuum are also discussed in terms of the adiabatic exponent and the entropy function and the phenomena of the blowup that is the collapsing of the star to a single point with finite mass as well as the drifting of part of the star to infinity in space are also studied and shown to agree with the conjecture from the physical considerations

hybrid finite element fast spectral domain multilayer boundary integral modeling of doubly periodic structures hybrid finite element fe boundary integral bi analysis of infinite periodic arrays is extended to include planar multilayered green s functions in this manner a portion of the volumetric dielectric region is modeled via the fe method whereas uniform multilayered regions are characterized using a multilayered green s function as such thick uniform substrates can be modeled without loss of efficiency and accuracy the multilayered green s function is analytically computed in the spectral domain and the resulting bi matrix vector products are evaluated via the fast spectral domain algorithm fsda without explicit generation of the bi matrix furthermore the number of floquet modes in the bi expansion is kept very few by appropriately placing the bi surfaces within the computational unit cell as a result the computational cost of the method is very little and the model can easily be adapted to various requirements examples of frequency selective surface fss arrays are analyzed with this method to demonstrate the accuracy and capability of the approach

error estimate for godunov approximation of locally constrained conservation laws we consider a model of traffic flow with unilateral constraint on the flux introduced by colombo and goatin j differential equations 234 2007 pp 654 675 for which the convergence of numerical approximation using monotone finite volume schemes has been performed by andreianov goatin and seguin numer math 115 2010 pp 609 645 we derive for this problem a new rm bv estimate and make use of it to provide an error estimate for the godunov approximation of the problem of order h 1 3 that is improved into the optimal order h 1 2 under a reasonable assumption numerical experiments are then provided to illustrate the optimality of the result

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

hyperbolic phase transitions in traffic flow this paper provides a mathematical model of the phenomenon of phase transitions in traffic flow the model consists of a scalar conservation law coupled with a 2 2 system of conservation laws the coupling is achieved via a free boundary where the phase transition takes place for this model the riemann problem is stated and globally solved the cauchy problem is proved to admit a solution defined globally in time without any assumption about the smallness of the initial data or the number of phase boundaries qualitative properties of real traffic flow are shown to agree with properties of the solutions of the model

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

a vacuum problem for the one dimensional compressible navier stokes equations with density dependent viscosity this paper is concerned with the free boundary problem for the one dimensional compressible navier stokes equations with density dependent viscosity a local in time existence result is established when the initial density is of compact support and connects to the vacuum continuously c 2002 elsevier science usa

the failure on continuous dependence on initial data for the navier stokes equations of compressible flow it is shown that physical solutions of the navier stokes equations for one dimensional compressible flow need not depend continuously on their initial data at least when vacuum states are allowed specifically two fluid regions initially separated by a third region of very low density 6 are considered it is shown that as delta to 0 the unique solutions corresponding to delta 0 do not in fact converge to a physical solution but rather to a nonphysical weak solution in which the two fluids cannot collide independent of their initial velocities and whose separate momenta need not be conserved a particular consequence is that solutions of the cavity problem delta 0 are not unique

blowup of smooth solutions to the compressible navier stokes equation with compact density we present a sufficient condition on the blowup of smooth solutions to the compressible navier stokes equations in arbitrary space dimensions with initial density of compact support as an immediate application it is shown that any smooth solutions to the compressible navier stokes equations for polytropic fluids in the absence of heat conduction will blow up in finite time as long as the initial densities have compact support and an upper bound which depends only on the initial data on the blowup time follows from our elementary analysis immediately another implication is that there is no global small decay in time or even bounded in the case that all the viscosity coefficients are positive smooth solutions to the compressible navier stokes equations for polytropic fluids no matter how small the initial data are as long as the initial density is of compact support this is in contrast to the classical theory of global existence of small solutions to the same system with initial data being a small perturbation of a constant state that is not a vacuum the blowup of smooth solutions to the compressible euler system with initial density and velocity of compact support is a simple consequence of our argument 1998 john wiley sons inc

a well posed conservation law with a variable unilateral constraint this paper considers the cauchy problem for a conservation law with a variable unilateral constraint its motivation being for instance the modeling of a toll gate along a highway this problem is solved by means of nonclassical shocks and its well posedness is proved then the solutions so obtained are shown to coincide with the limits of the classical solutions to suitable conservation laws with discontinuous flux function that approximate the constrained problem

rotating fluids with self gravitation in bounded domains in this paper we study the steady solutions of euler poisson equations in bounded domains with prescribed angular velocity this models a rotating newtonian star consisting of a compressible perfect fluid with given equation of state p es when the domain is a ball and the angular velocity is constant we obtain both existence and non existence theorems depending on the adiabatic gas constant in addition we obtain some interesting properties of the solutions e g monotonicity of the radius of the star with both angular velocity and central density we also prove that the radius of a rotating spherically symmetric star with given constant angular velocity and constant entropy is uniformly bounded independent of the central density this is physically striking and in sharp contrast to the case of the non rotating star for general domains and variable angular velocities both an existence result for the isentropic equations of state and non existence result for the non isentropic equation of state are also obtained

the boltzmann equation global existence for a rare gas in an infinite vacuum solutions of the boltzmann equation are proved to exist globally in time under conditions that include the case of a finite volume of gas in an infinite vacuum when the mean free path of the gas is large enough it is also proved as might be expected in this case that the density of the gas at each point in space goes to zero as time goes to infinity

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

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

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