weak solutions for an inviscid two phase flow model in physical vacuum abstract we investigate a free boundary problem for the inviscid two phase flow model with moving physical vacuum boundary which is a degenerate and characteristic hyperbolic system of conservation laws based on a suitable transformation a special degenerate parabolic approximation and delicate uniform energy estimates we prove the existence and uniqueness of the local weak solutions this can be regarded as a generalization of the results in coutand and shkoller 2011 5 from single phase gas model to two phase gas liquid model

global existence and optimal convergence rates for the strong solutions in h 2 to the 3d viscous liquid gas two phase flow model abstract this paper is concerned with the cauchy problem of the three dimensional viscous liquid gas two phase flow model we prove the global existence of a strong solution when h 2 norm of the initial perturbation around a constant state is sufficiently small and its l 1 norm is bounded moreover the optimal convergence rates are also obtained for such a solution

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

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

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

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

a blow up criterion for a 2d viscous liquid gas two phase flow model abstract in this paper we prove a blow up criterion in terms of the upper bound of the liquid mass for the strong solution to the two dimensional 2d viscous liquid gas two phase flow model in a smooth bounded domain the result also applies to three dimensional 3d case

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

existence and uniqueness of global weak solution to a two phase flow model with vacuum in this paper we study a two phase liquid gas model with constant viscosity coefficient when both the initial liquid and gas masses connect to vacuum continuously just as in evje and karlsen commun pure appl anal 8 1867 1894 2009 and evje et al nonlinear anal 70 3864 3886 2009 the gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid we use a new technique to get the upper and lower bounds of gas and liquid masses n and m then we get the global existence of weak solution by the line method also we obtain the uniqueness of the weak solution

decay of the navier stokes poisson equations we establish the time decay rates of the solution to the cauchy problem for the compressible navier stokes poisson system via a refined pure energy method in particular the optimal decay rates of the higher order spatial derivatives of the solution are obtained the h s 0 s 3 2 negative sobolev norms are shown to be preserved along time evolution and enhance the decay rates as a corollary we also obtain the usual lp l2 1 p 2 type of the optimal decay rates compared to the compressible navier stokes system and the compressible irrotational euler poisson system our results imply that both the dispersion effect of the electric field and the viscous dissipation contribute to enhance the decay rate of the density our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis

decay estimates of solutions for the equations of motion of compressible viscous and heat conductive gases in an exterior domain in ℝ3 we consider the equations of motion of compressible viscous and heat conductive gases in an exterior domain in 3 we give the l q l p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem

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

well posedness in smooth function spaces for moving boundary 1 d compressible euler equations in physical vacuum we prove well posedness for the 3 d compressible euler equations with moving physical vacuum boundary with an equation of state given by the so called gamma gas law for gamma 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a new higher order hardy type inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions out methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

a priori estimates for the free boundary 3d compressible euler equations in physical vacuum we prove a priori estimates for the three dimensional compressible euler equations with moving physical vacuum boundary with an equation of state given by p c for 1 the vacuum condition necessitates the vanishing of the pressure and hence density on the dynamic boundary which creates a degenerate and characteristic hyperbolic free boundary system to which standard methods of symmetrizable hyperbolic equations cannot be applied

a blow up criterion of strong solutions to a viscous liquid gas two phase flow model with vacuum in 3d abstract in this paper we get a blow up criterion for the strong solutions to the 3d viscous liquid gas two phase flow model with vacuum more precisely we prove that the bound of l t 1 l x norm of the deformation tensor of velocity gradient controls the possible breakdown of the strong solutions

global stability for a class of predator prey systems this paper deals with the question of global stability of the positive locally asymptotically stable equilibrium in a class of predator prey systems the dulac s criterion is applied and liapunov functions are constructed to establish the global stability

global existence of weak solutions for a viscous two phase model abstract the purpose of this paper is to explore a viscous two phase liquid gas model relevant for well and pipe flow our approach relies on applying suitable modifications of techniques previously used for studying the single phase isothermal navier stokes equations a main issue is the introduction of a novel two phase variant of the potential energy function needed for obtaining fundamental a priori estimates we derive an existence result for weak solutions in a setting where transition to single phase flow is guaranteed not to occur when the initial state is a true mixture of both phases some numerical examples are also included in order to demonstrate characteristic behavior of solutions in particular we illustrate how two phase flow is genuinely different compared to single phase flow concerning the behavior of an initial mass discontinuity

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

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

well posedness for the motion of an incompressible liquid with free surface boundary we study the motion of an incompressible perfect liquid body in vacuum this can be thought of as a model for the motion of the ocean or a star the free surface moves with the velocity of the liquid and the pressure vanishes on the free surface this leads to a free boundary problem for euler s equations where the regularity of the boundary enters to highest order we prove local existence in sobolev spaces assuming a physical condition related to the fact that the pressure of a fluid has to be positive

hopf bifurcation analysis of a predator prey system with holling type iv functional response and time delay this paper is concerned with a predator prey system with holling type iv functional response and time delay our aim is to investigate how the time delay affects the dynamics of the predator prey system by choosing the delay as a bifurcation parameter the local asymptotic stability of the positive equilibrium and existence of local hopf bifurcations are analyzed based on the normal form and the center manifold theory the formulaes for determining the properties of hopf bifurcation of the predator prey system are derived finally to support these theoretical results some numerical simulations are given to illustrate the results

optimal lp lq convergence rates for the compressible navier stokes equations with potential force in this paper we are concerned with the optimal lp lq convergence rates for the compressible navier stokes equations with a potential external force in the whole space under the smallness assumption on both the initial perturbation and the external force in some sobolev spaces the optimal convergence rates of the solution in lq norm with 2 q 6 and its first order derivative in l2 norm are obtained when the initial perturbation is bounded in lp with 1 p 6 5 the proof is based on the energy estimates on the solution to the nonlinear problem and some lp lq estimates on the semigroup generated by the corresponding linearized operator

prime factors of dynamical sequences let f t a q t have degree d f 2 for a given rational number x0 define xn 1 f xn for each nf 0 if this sequence is not eventually periodic and if f does not lie in one of two explicitly determined a ne conjugacy classes of rational functions then xn 1 xn has a primitive prime factor in its numerator for all su ciently large n the same result holds for the exceptional maps provided that one looks for primitive prime fac tors in the denominator of xn 1 xn hence the result for each rational function f of degree at least 2 implies a new proof that there are infinitely many primes the question of primi tive prime factors of xn dxn is also discussed for d uniformly bounded

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

global strong solution for a three dimensional viscous liquid gas two phase flow model with vacuum this paper is concerned with the cauchy problem of the 3d viscous liquid gas two phase flow model where the initial vacuum is allowed the existence of the global strong solution is proved when the energy of the initial data is small enough

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

delay induced oscillation in predator prey system with beddington deangelis functional response abstract the beddington deangelis predator prey system with distributed delay is studied in this paper at first the positive equilibrium and its local stability are investigated then with the mean delay as a bifurcation parameter the system is found to undergo a hopf bifurcation the bifurcating periodic solutions are analyzed by means of the normal form and center manifold theorems finally numerical simulations are also given to illustrate the results

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

local well posedness of dynamics of viscous gaseous stars we establish the local in time well posedness of strong solutions to the vacuum free boundary problem of the compressible navier stokes poisson system in the spherically symmetric and isentropic motion our result captures the physical vacuum boundary behavior of the lane emden star configurations for all adiabatic exponents gamma frac 6 5

asymptotic stability of the compressible gas liquid model with well formation interaction and gravity abstract in this paper we are concerned with the initial boundary value problem of the compressible gas liquid model with well formation interaction and gravity the asymptotic behavior of solutions to steady states is established also the time decay rates of perturbed solutions in the sense of l norm are obtained under some suitable assumptions on the initial date if 1 associated with pressure law of gas and 0 2 0 0 3 where characterizes the viscosity coefficient and describes the mass decay rate at the boundary a main purpose of this work is to clarify the role played by the well reservoir interaction term the analysis demonstrates that it is essential to take into account information about sign as well as size of the interaction term in order to obtain time independent estimates when it operates in combination with gravity

well posedness in smooth function spaces for the moving boundary three dimensional compressible euler equations in physical vacuum we prove well posedness for the three dimensional compressible euler equations with moving physical vacuum boundary with an equation of state given by p c for 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a higher order version of hardy s inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

well posedness for compressible euler equations with physical vacuum singularity an important problem in the theory of compressible gas flows is to understand the singular behavior of vacuum states the main difficulty lies in the fact that the system becomes degenerate at the vacuum boundary where the characteristic speeds u c coincide and have unbounded spatial derivative since c behaves like x 1 2 close to the boundary in this paper we overcome this difficulty by presenting a new formulation and new energy spaces we establish the local in time well posedness of one dimensional compressible euler equations for isentropic flows with the physical vacuum singularity in some spaces adapted to the singularity 2009 wiley periodicals inc

global existence in critical spaces for flows of compressible viscous and heat conductive gases we are concerned with global existence and uniqueness of strong solutions for a general model of viscous and heat conductive gases the initial data are supposed to be close to a stable equilibrium with constant density and temperature using uniform estimates for the linearized system with a convection term we get global well posedness in a functional setting invariant with respect to the scaling of the associated equations in space dimension n 3 we also show a smoothing effect on the velocity and the temperature and a decay on the difference between the density and the constant reference state these results extend a previous paper devoted to the barotropic case see 5

hopf bifurcation analysis for a ratio dependent predator prey system with two delays and stage structure for the predator the ratio dependent theory is favored by researchers since it is more suitable for describing the relationship between predator and its prey in this paper a ratio dependent predator prey system with holling type ii functional response two time delays and stage structure for the predator is investigated firstly by choosing the two time delays as the bifurcation parameter the sufficient conditions for the local stability and the existence of hopf bifurcation with respect to both delays are established furthermore based on the normal form method and center manifold theorem explicit formulas are derived to determine the direction of hopf bifurcation and stability of the bifurcating periodic solution finally numerical simulations are given to verify the theoretical analysis

stability and hopf bifurcation for a delayed prey predator system with diffusion effects abstract this paper is concerned with a delayed lotka volterra prey predator system with diffusion effects and neumann boundary conditions the main purpose is to investigate the stability of spatially homogeneous positive equilibrium and give the explicit formulae determining the direction and stability of hopf bifurcation by linearizing the system at positive equilibrium and analyzing the associated characteristic equation the stability of positive equilibrium and the existence of hopf bifurcation are demonstrated by means of the normal form theory and the center manifold reduction for partial functional differential equations pfdes the direction and stability of periodic solutions occurring through hopf bifurcation are determined finally in order to verify our theoretical results some numerical simulations are also included

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

global weak solutions for a viscous liquid gas model with transition to single phase gas flow and vacuum abstract this work deals with a viscous two phase liquid gas model relevant to the flow in wells and pipelines the liquid is treated as an incompressible fluid whereas the gas is assumed to be polytropic the model is rewritten in terms of lagrangian coordinates and is studied in a free boundary setting where the liquid and gas masses are of compact support initially and continuous at the boundary consequently the initial masses involve a transition to single phase gas flow and vacuum at the boundary an appropriate balance between pressure and viscous forces is identified which allows obtaining pointwise upper and lower estimates of masses these estimates rely on the assumption of a certain relation between the rate of degeneracy of the viscosity coefficient and the rate that determines how fast the initial masses are vanishing at the boundary by combining these estimates with basic energy type of estimates higher order regularity estimates are obtained the existence of global weak solutions is then proved by showing compactness for a class of semi discrete approximations

global weak solutions for a compressible gas liquid model with well formation interaction abstract the objective of this work is to explore a compressible gas liquid model designed for modeling of well flow processes we build into the model well reservoir interaction by allowing flow of gas between well and formation surrounding reservoir inflow of gas and subsequent expansion of gas as it ascends towards the top of the well a so called gas kick represents a major concern for various well operations in the context of petroleum engineering we obtain a global existence result under suitable assumptions on the regularity of initial data and the rate function that controls the flow of gas between well and formation uniqueness is also obtained by imposing more regularity on the initial data the key estimates are to obtain appropriate lower and upper bounds on the gas and liquid masses for that purpose we introduce a transformed version of the original model that is highly convenient for analysis of the original model in particular in the analysis of the transformed model additional terms representing well formation interaction can be treated by natural extensions of arguments that previously have been employed for the single phase navier stokes model the analysis ensures that transition to single phase regions do not appear when the initial state is a true gas liquid mixture

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

well posedness for the motion of a compressible liquid with free surface boundary we study the motion of a compressible perfect liquid body in vacuum this can be thought of as a model for the motion of the ocean or a star the free surface moves with the velocity of the liquid and the pressure vanishes on the free surface this leads to a free boundary problem for euler s equations where the regularity of the boundary enters to highest order we prove local existence in sobolev spaces assuming a physical condition related to the fact that the pressure of a fluid has to be positive

time periodic solution to a two phase model with magnetic field in a periodic domain abstract in the paper we consider a two phase model with magnetic field and a time periodic external force in a periodic domain in r n the existence of the time periodic solutions to the model is obtained by the topological degree theory and parabolic regularization method

a global existence result for the compressible navier stokes equations in the critical l p framework the present paper is dedicated to the global well posedness issue for the barotropic compressible navier stokes system in the whole space mathbb r d with d 2 we aim at extending the work by danchin inventiones mathematicae 141 3 579 614 2000 to a critical framework which is not related to the energy space for small perturbations of a stable equilibrium state in the sense of suitable lp type besov norms we establish the global existence as a consequence like for incompressible flows one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true in passing we obtain new estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows

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

free boundary value problem for a viscous two phase model with mass dependent viscosity abstract in this paper we study a free boundary value problem for two phase liquid gas model with mass dependent viscosity coefficient when both the initial liquid and gas masses connect to vacuum with a discontinuity this is an extension of the paper s evje k h karlsen global weak solutions for a viscous liquid gas model with singular pressure law http www irisresearch no docsent emp nsf wvansatte sev just as in s evje k h karlsen global weak solutions for a viscous liquid gas model with singular pressure law http www irisresearch no docsent emp nsf wvansatte sev the gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid we give the proof of the global existence and uniqueness of weak solutions when 0 1 which have improved the previous result of evje and karlsen and get the asymptotic behavior result also we obtain the regularity of the solutions by energy 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

global well posedness for systems of hyperbolic parabolic composite type with center manifold abstract concerning the global existence of classical solution to systems of hyperbolic parabolic composite type a well known general theory was established by kawashima in 4 where the dissipation condition kawashima shizuta condition to the linearized system plays a fundamental role recently systems with much weaker dissipations have attracted a lot of attentions see 1 2 10 11 among others the typical feature of this kind of system is that the corresponding linearized system has one eigenvalue with the real part equals to zero this violates the kawashima shizuta stability conditions in this paper we develop a general global well posedness theory for this kind of system moreover as the applications of the general theory several examples are given

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

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

blow up criterion for 3d viscous liquid gas two phase flow model abstract this work deals with the blow up criterion for the strong solution to the three dimensional viscous liquid gas two phase flow model in terms of the l 1 0 t l norm of the gradient of the velocity with two types of boundary conditions the dirichlet boundary condition and the navier slip boundary condition there is no extra restriction on viscosity coefficients the result also applies to the whole space case

large time behavior for the system of a viscous liquid gas two phase flow model in r3 abstract the cauchy problem of a three dimensional compressible viscous liquid gas two phase flow model is considered in the present paper the global existence and uniqueness of solutions are established when the initial data is close to its equilibrium in the framework of sobolev space h 3 r 3 moreover the optimal l 2 l 2 convergence rates are also obtained for the solution

some estimates of solutions for the equations of motion of compressible viscous fluid in the three dimensional exterior domain abstract we consider the equations of motion of compressible viscous fluid in an exterior domain in r 3 we give the l q l p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem in particular we shall give l 1 estimates which implies the diffusion wave of compressible navier stokes flow

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

global well posedness and decay rates of strong solutions to a non conservative compressible two fluid model in this paper we consider a compressible two fluid model with constant viscosity coefficients and unequal pressure functions p neq p as mentioned in the seminal work by bresch desjardins et al arch rational mech anal 196 599 629 2010 for the compressible two fluid model where p p common pressure is used and capillarity effects are accounted for in terms of a third order derivative of density the case of constant viscosity coefficients cannot be handled in their settings besides their analysis relies on a special choice for the density dependent viscosity refer also to another reference commun math phys 309 737 755 2012 by bresch huang and li for a study of the same model in one dimension but without capillarity effects in this work we obtain the global solution and its optimal decay rate in time with constant viscosity coefficients and some smallness assumptions in particular capillary pressure is taken into account in the sense that delta p p p f neq 0 where the difference function f is assumed to be a strictly decreasing function near the equilibrium relative to the fluid corresponding to p this assumption plays an key role in the analysis and appears to have an essential stabilization effect on the model in question

classification of bifurcation diagrams for a multiparameter diffusive logistic problem with holling type iv functional response abstract we study exact multiplicity and bifurcation diagrams of positive solutions for a multiparameter diffusive logistic problem with holling type iv functional response u x f u 0 1 x 1 u 1 u 1 0 where the growth rate function f u r u 1 u q u 1 u 2 q r are positive dimensionless parameters and 0 is a bifurcation parameter assume that either r 1 q and q r lies above the curve 1 q r q a 1 3 a 2 2 a r a 1 3 a 2 1 a 2 2 0 a 1 3 or r 2 q for some constants 1 0 618 and 2 0 601 then on the u plane we give a classification of four qualitatively different bifurcation diagrams an s shaped curve a broken s shaped curve a shaped curve and a monotone increasing curve

the interaction between quasilinear elastodynamics and the navier stokes equations the interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations coupled to one another along the moving material interface through the continuity of the velocity and traction vectors we prove the existence and uniqueness locally in time of strong solutions in sobolev spaces for quasilinear elastodynamics coupled to the incompressible navier stokes equations unlike our approach in 5 for the case of linear elastodynamics we cannot employ a fixed point argument on the nonlinear system itself and are instead forced to regularize it by a particular parabolic artificial viscosity term we proceed to show that with this specific regularization we obtain a time interval of existence which is independent of the artificial viscosity together with a priori estimates we identify the global solution in both phases as well as the interface motion as a weak limit in strong norms of our sequence of regularized problems

well posedness of the free surface incompressible euler equations with or without surface tension we provide a new method for treating free boundary problems in perfect fluids and prove local in time well posedness in sobolev spaces for the free surface incompressible 3d euler equations with or without surface tension for arbitrary initial data and without any irrotationality assumption on the fluid this is a free boundary problem for the motion of an incompressible perfect liquid in vacuum wherein the motion of the fluid interacts with the motion of the free surface at highest order

global existence in critical spaces for compressible navier stokes equations we investigate global strong solutions for isentropic compressible fluids with initial data close to a stable equilibrium we obtain the existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations more precisely the initial velocity has the same critical regularity index as for the incompressible homogeneous navier stokes equations and one more derivative is needed for the density we point out a smoothing effect on the velocity and a l 1 decay on the difference between the density and the constant reference state the proof lies on uniform estimates for a mixed hyperbolic parabolic linear system with a convection term

persistence and global stability in a delayed predator prey system with michaelis menten type functional response a delayed three species predator prey food chain model with michaelis menten type functional response is investigated it is proved that the system is uniformly persistent under some appropriate conditions by means of constructing suitable lyapunov functional sufficient conditions are derived for the global asymptotic stability of the positive equilibrium of the system

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

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

on the wave structure of two phase flow models we explore the relationship between two common two phase flow models usually denoted as the two fluid and drift flux models they differ in their mathematical description of momentum transfer between the phases in this paper we provide a framework in which these two model formulations are unified the drift flux model employs a mixture momentum equation and treats interphasic momentum exchange indirectly through the slip relation which gives the relative velocity between the phases as a function of the flow parameters this closure law is in general highly complex which makes it difficult to analyze the model algebraically to facilitate the analysis we express the quasi linear formulation of the drift flux model as a function of three parameters the derivatives of the slip with respect to the vector of unknown variables the wave structure of this model is investigated using a perturbation technique then we rewrite the drift flux model with a general slip relation such that it is expressed in term