on center singularity for compressible spherically symmetric nematic liquid crystal flows abstract the paper is concerned with a simplified system proposed by ericksen 6 and leslie 20 modeling the flow of nematic liquid crystals in the first part we give a new serrin s continuation principle for strong solutions of general compressible liquid crystal flows based on new observations we establish a localized serrin s regularity criterion for the 3d compressible spherically symmetric flows it is proved that the classical solution loses its regularity in finite time if and only if either the concentration or vanishing of mass forms or the norm inflammation of gradient of orientation field occurs around the center

vanishing of vacuum states and blow up phenomena of the compressible navier stokes equations the navier stokes systems for compressible fluids with density dependent viscosities are considered in the present paper these equations in particular include the ones which are rigorously derived recently as the saint venant system for the motion of shallow water from the navier stokes system for incompressible flows with a moving free surface 14 these compressible systems are degenerate when vacuum state appears we study initial boundary value problems for such systems for both bounded spatial domains or periodic domains the dynamics of weak solutions and vacuum states are investigated rigorously

on the existence of local strong solutions to chemotaxis shallow water system with large data and vacuum abstract in this paper motivated by the chemotaxis navier stokes system arising from mathematical biology 43 a modified shallow water type chemotactic model is derived for large initial data allowing vacuum the local existence of strong solutions together with the blow up criterion is established

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

local existence of the strong solutions for the full navier stokes poisson equations abstract we study the full navier stokes poisson equations for viscous polytropic fluids we prove the existence and uniqueness of local strong solutions for all initial values satisfying some compatibility conditions it is important that we allow the initial density to be vacuum

local well posedness to the cauchy problem of the 2d compressible navier stokes smoluchowski equations with vacuum abstract this paper concerns the local strong solution to the compressible navier stokes smoluchowski equations with vacuum as far field density on the whole space r 2 without additional cho choe kim type compatibility conditions which provided the initial density and the initial particles density decay not too slow at infinity in particular we extend the results of ding et al 2016 13 and yang 2020 24 to the 2d case

global behavior of compressible navier stokes equations with a degenerate viscosity coefficient in this paper we study a free boundary problem for compressible navier stokes equations with density dependent viscosity precisely the viscosity coefficient is proportional to with open image in new window where is the density and 1 is the physical constant of polytropic gas under certain assumptions imposed on the initial data we obtain the global existence and uniqueness of the weak solution give the uniform bounds with respect to time of the solution and show that it converges to a stationary one as time tends to infinity moreover we estimate the stabilization rate in l norm weighted l2 norm and weighted h1 norm of the solution as time tends to infinity

navier stokes flow in the weighted hardy space with applications to time decay problem abstract the asymptotic expansions of the navier stokes flow in r n and the rates of decay are studied with aid of weighted hardy spaces fujigaki and miyakawa 12 miyakawa 28 proved the nth order asymptotic expansion of the navier stokes flow if initial data decays like 1 x n 1 and if nth moment of initial data is finite in the present paper it is clarified that the moment condition for initial data is essential in order to obtain higher order asymptotic expansion of the flow and to consider the rapid time decay problem the second author 39 established the weighted estimates of the strong solutions in the weighted hardy spaces with small initial data which belongs to l n and a weighed hardy space firstly the refinement of the previous work 39 is achieved with alternative proof then the existence time of the solution in the weighted hardy spaces is characterized without any hardy norm as a result in two dimensional case the smallness condition on initial data is completely removed as an application the rapid time decay of the flow is investigated with aid of asymptotic expansions and of the symmetry conditions introduced by brandolese 3

a blowup criterion for the compressible nematic liquid crystal flows in dimension two abstract in this paper we establish a blowup criterion for the two dimensional compressible nematic liquid crystal flows the criterion is given in terms of the density and the gradient of direction field where the later satisfies the serrin type blowup criterion for this result we do not need the initial density to be positive

on spherically symmetric solutions of the compressible isentropic navier stokes equations we prove the global existence of weak solutions to the cauchy problem for the compressible isentropic navier stokes equations in n n 2 3 when the cauchy data are spherically symmetric the proof is based on the exploitation of the one dimensional feature of symmetric solutions and use of a new multidimensional property induced by the viscous flux the present paper extends lions existence theorem 15 to the case 1 n for spherically symmetric initial data where is the specific heat ratio in the pressure n 3 2 for n 2 and n 9 5 for n 3 dedicated to professor rolf leis on the occasion of his 70th birthday

blowup criterion for the three dimensional equations of compressible viscous micropolar fluids with vacuum abstract in this paper we prove a blowup criterion of strong solutions to the cauchy problem for the three dimensional equations of compressible viscous micropolar fluids it is shown that if the density and the velocity satisfy l 0 t l 1 2 u l s 0 t l r with 2 s 3 r 1 and 3 r then the strong solution will exist globally on r 3 0 t in particular the initial density may vanish on open sets that is the initial vacuum is allowed

global existence of strong solutions of navier stokes equations with non newtonian potential for one dimensional isentropic compressible fluids abstract the aim of this paper is to discuss the global existence and uniqueness of strong solution for a class of the isentropic compressible navier stokes equations with non newtonian in one dimensional bounded intervals we prove two global existence results on strong solutions of the isentropic compressible navier stokes equations the first result shows only the existence and the second one shows the existence and uniqueness result based on the first result but the uniqueness requires some compatibility condition

strong solutions of the navier stokes equations for isentropic compressible fluids we study strong solutions of the isentropic compressible navier stokes equations in a domain r3 we first prove the local existence of unique strong solutions provided that the initial data 0 and u0 satisfy a natural compatibility condition the important point in this paper is that we allow the initial vacuum the initial density may vanish in an open subset of we then prove a new uniqueness result and stability result our results are valid for unbounded domains as well as bounded ones

on the strong solutions of one dimensional navier stokes poisson equations for compressible non newtonian fluids we study the strong solutions of 1d navier stokes poisson equations for compressible non newtonian fluids in bounded intervals the model is raised from the viscous isentropic gas flow under considering an external force and the non newtonian gravitational force term by using the iterative method we prove the local existence and uniqueness of strong solutions based on some compatibility condition the main condition is that the initial density vacuum is allowed

regularity of 1d compressible isentropic navier stokes equations with density dependent viscosity abstract in this paper we consider one dimensional compressible isentropic navier stokes equations with the viscosity depending on density and with free boundary the viscosity coefficient is proportional to with 0 1 where is the density the existence and uniqueness of global weak solutions in h 1 0 1 have been established in s jiang z xin p zhang global weak solutions to 1d compressible isentropic navier stokes equations with density dependent viscosity methods appl anal 12 2005 239 252 we will establish the regularity of global solution under certain assumptions imposed on the initial data by deriving some new a priori estimates

space time decay estimates for the incompressible viscous resistive mhd and hall mhd equations abstract in this paper we address the space time decay properties for strong solutions to the incompressible viscous resistive hall mhd equations we obtained the same space time decay rates as those of the heat equation based on the temporal decay results in 10 we find that one can obtain weighted estimates of the magnetic field b by direct weighted energy estimate and then by regarding the magnetic convection term as a forcing term in the velocity equations we can obtain the weighted estimates for the vorticity which yields the corresponding estimates for the velocity field the higher order derivative estimates will be obtained by using a parabolic interpolation inequality proved in 22 it should be emphasized that the magnetic field has stronger decay properties than the velocity field in the sense that there is no restriction on the exponent of the weight the same arguments also yield the sharp space time decay rates for strong solutions to the usual mhd equations

existence results for viscous polytropic fluids with vacuum abstract we consider the full navier stokes equations for viscous polytropic fluids with nonnegative thermal conductivity we prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition the initial density need not be positive and may vanish in an open set moreover our results hold for both bounded and unbounded domains

global well posedness of classical solutions with large oscillations and vacuum to the three dimensional isentropic compressible navier stokes equations we establish the global existence and uniqueness of classical solutions to the cauchy problem for the isentropic compressible navier stokes equations in three spatial dimensions with smooth initial data that are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or nonvacuum the initial density is allowed to vanish and the spatial measure of the set of vacuum can be arbitrarily large in particular the initial density can even have compact support these results generalize previous results on classical solutions for initial densities being strictly away from vacuum and are the first for global classical solutions that may have large oscillations and can contain vacuum states 2012 wiley periodicals inc

blowup criterion for viscous baratropic flows with vacuum states we prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth strong solutions for the 3 dimensional 3d barotropic compressible navier stokes equations more precisely if a solution of the 3d barotropic compressible navier stokes equations is initially regular and loses its regularity at some later time then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches our result is the same as ponce s criterion for 3 dimensional incompressible euler equations ponce in commun math phys 98 349 353 1985 in addition initial vacuum states are allowed in our cases

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

hyperbolic compressible navier stokes equations abstract we consider the non isentropic compressible navier stokes equations with hyperbolic heat conduction and a law for the stress tensor which is modified correspondingly by maxwell s law these two relaxations turning the whole system into a hyperbolic one are not only treated simultaneously but are also considered in a version having galilean invariance for this more complicated relaxed system the global well posedness is proved for small data moreover for vanishing relaxation parameters the solutions are shown to converge to solutions of the classical system

existence of global solutions for 2 dimensional viscous compressible flows abstract we prove the existence of global weak solutions for 2 dimensional unsteady viscous compressible flows of an ideal isothermal gas whatever the size of the data may be this is achieved by a suitable coupling of the method of hopf 17 with standard techniques in orlicz spaces

electromagnetic wave diffraction by semi infinite strip grating a nonlinear operator equation in terms of the operator of reflection by a plane periodic semi infinite grating of thin metal strips is obtained for the h polarized electromagnetic wave incidence this equation is solved numerically for an incident plane wave the algorithm created allows the solution of a nonlinear operator equation in the fourier amplitude of the reflected field with the method of successive iterations the initial approach chosen is substantiated and the solution convergence is numerically verified the field reflected by a semi infinite grating is calculated and analyzed for different numerical values of the exciting wave angle of incidence the strip width and the incident field wavelength

pointwise estimates of solution for the navier stokes poisson equations in multi dimensions abstract the cauchy problem of the navier stokes poisson equations in multi dimensions n 3 is considered we obtain the pointwise estimates of the solution when it is a perturbation of the constant state then we have the optimal l p r n p 1 convergence rate of the solution

on the existence of local classical solutions to the navier stokes equations with degenerate viscosities in this paper we consider the three dimensional cauchy problem of compressible navier stokes equations with degenerate viscosities depending on density with the initial data containing vacuum at far fields we prove the local existence of regular solutions with some compatibility conditions the regular solution is classical due to the smoothing effect of the velocity u in this paper we consider the three dimensional cauchy problem of compressible navier stokes equations with degenerate viscosities depending on density with the initial data containing vacuum at far fields we prove the local existence of regular solutions with some compatibility conditions the regular solution is classical due to the smoothing effect of the velocity u

existence and uniqueness of solutions for a class of non newtonian fluids with singularity and vacuum abstract the aims of this paper are to discuss existence and uniqueness of local solutions for a class of non newtonian fluids with singularity and vacuum in one dimensional bounded intervals there are two important points in this paper one is that we allow the initial vacuum another one is that the viscosity term of momentum equation is with singularity and fully nonlinearity

global existence of strong solutions of the navier stokes equations for isentropic compressible fluids with density dependent viscosity abstract this paper is concerned with global strong solutions of the isentropic compressible navier stokes equations with density dependent viscosity coefficient in one dimensional bounded intervals precisely the viscosity coefficient and the pressure p is proportional to with 1 the important point in this paper is that the initial density may vanish in an open subset we also show that the strong solution obtained above is unique provided that the initial data satisfies additional regularity and a compatible condition compared with former results obtained by hyunseok kim in h kim global existence of strong solutions of the navier stokes equations for one dimensional isentropic compressible fluids available at http com2mac postech ac kr papers 2001 01 38 pdf we deal with density dependent viscosity coefficient

asymptotic expansion for solutions of the navier stokes equations with potential forces abstract we derive an asymptotic expansion for smooth solutions of the navier stokes equations in weighted spaces this result removes previous restrictions on the number of terms of the asymptotics as well as on the range of the polynomial weights we also write the expansion in terms of expressions involving non linear quantities

global solutions of the navier stokes equations for multidimensional compressible flow with discontinuous initial data abstract we prove the global existence of weak solutions of the navier stokes equations for compressible isothermal flow in two and three space dimensions when the initial density is close to a constant in l 2 and l and the initial velocity is small in l 2 and bounded in l 2 n in two dimensions the l 2 norms must be weighted slightly a great deal of qualitative information about the solution is obtained for example we show that the velocity and vorticity are relatively smooth in positive time as is the effective viscous flux f which is the divergence of the velocity minus a certain multiple of the pressure we find that f plays a crucial role in the entire analysis particularly in closing the required energy estimates understanding rates of regularization near the initial layer and most important obtaining time independent pointwise bounds for the density

on temporal asymptotics for the pth power viscous reactive gas abstract in this paper we investigate the long time behaviour of solutions to the system governing a heat conductive viscous reactive pth power gas confined between two parallel plates for initial boundary value problems with the end points held at a prescribed temperature or insulated we prove the global existence of physically relevant solutions and establish their rate of convergence to equilibria for generic initial data the estimates for different boundary conditions are presented in a unified manner

local existence of strong solution for a class of compressible non newtonian fluids with non newtonian potential the aim of this paper is to discuss the existence and uniqueness of local solutions for a class of isentropic compressible non newtonian fluids with non newtonian potential in one dimensional bounded intervals the first difficult point in this paper is that we allow the initial vacuum another one is that the viscosity term and newtonian potential term are fully nonlinear that is the viscosity term adopts the o a ladyzhenskaya model p 2 the potential term employs the q laplace form 1

global classical large solutions to 1d compressible navier stokes equations with density dependent viscosity and vacuum abstract in this paper we investigate an initial boundary value problem for 1d compressible isentropic navier stokes equations with large initial data density dependent viscosity external force and vacuum making full use of the local estimates of the solutions in cho and kim 2006 3 and the one dimensional properties of the equations and the sobolev inequalities we get a unique global classical solution u where c 1 0 t h 1 0 1 and u h 1 0 t h 2 0 1 for any t 0 as it is pointed out in xin 1998 31 that the smooth solution u c 1 0 t h 3 r 1 t is large enough of the cauchy problem must blow up in finite time when the initial density is of nontrivial compact support it seems that the regularities of the solutions we obtained can be improved which motivates us to obtain some new estimates with the help of a new test function 2 u t t such as lemmas 3 2 3 6 this leads to further regularities of u where c 1 0 t h 3 0 1 u h 1 0 t h 3 0 1 it is still open whether the regularity of u could be improved to c 1 0 t h 3 0 1 with the appearance of vacuum since it is not obvious that the solutions in c 1 0 t h 3 0 1 to the initial boundary value problem must blow up in finite time

global behavior of spherically symmetric navier stokes equations with density dependent viscosity abstract in this paper we study a free boundary problem for compressible spherically symmetric navier stokes equations without a solid core under certain assumptions imposed on the initial data we obtain the global existence and uniqueness of the weak solution give some uniform bounds with respect to time of the solution and show that it converges to a stationary one as time tends to infinity moreover we obtain the stabilization rate estimates of exponential type in l norm and weighted h 1 norm of the solution by constructing some lyapunov functionals the results show that such system is stable under the small perturbations and could be applied to the astrophysics

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

on the weighted decay for solutions of the navier stokes system abstract we consider the question of to what extent the temporal decay u t l 2 o t 0 for solutions u of the navier stokes equation influences the decay of weighted norms we prove that x a u t l 2 o t 0 a 2 holds under the condition 0 a n 2 1 this extends the range of weights obtained in i kukavica j j torres weighted l p decay for solutions of the navier stokes equations commun partial differential equations 32 2007 819 831 a wider range of exponents is also obtained for the decay of l p norms

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

global existence for compressible navier stokes poisson equations in three and higher dimensions the compressible navier stokes poisson system is concerned in the present paper and the global existence and uniqueness of the strong solution is shown in the framework of hybrid besov spaces in three and higher dimensions

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 strong solution for a class of compressible full non newtonian models abstract the aim of this paper is to discuss the existence and uniqueness of a local solution for a class of isentropic compressible full non newtonian models in one dimensional bounded intervals the first important point in this paper is that we allow the initial vacuum another one is that the viscosity term and the newtonian potential term are fully nonlinear

global solutions to a model for exothermically reacting compressible flows with large discontinuous initial data we prove the global existence of solutions of the navier stokes equations describing the dynamic combustion of a compressible exothermically reacting fluid and we study the large time behavior of solutions giving necessary and sufficient conditions for complete combustion in certain cases the adiabatic constants and specific heats of the burned product and unburned reactant fluids may differ and the initial data may be large and discontinuous

bipolar isothermal non newtonian compressible fluids the existence and uniqueness of a solution to isothermal non newtonian bipolar fluid is proved we consider a special case where the stress tensor is expressed in the form of potentials depending oneijand eij xk the second part is devoted to the problem of the stability of the rest state it is proved that the perturbations of velocity and density tend to zero in appropriate norms along a chosen time sequence moreover we obtain the asymptotic stability of the rest state under the assumption of the regularity of the potential forces

global existence of classical solutions for isentropic compressible navier stokes equations with small initial density abstract in this paper we establish the global existence of classical solutions to the cauchy problem for the 3 d isentropic compressible navier stokes equations with smooth initial data that are small density but possibly large energy which could be either vacuum or non vacuum

the cauchy problem for the pressureless euler isentropic navier stokes equations we present a new hydrodynamic model consisting of the pressureless euler equations and the isentropic compressible navier stokes equations where the coupling of two systems is through the drag force this coupled system can be derived in the hydrodynamic limit from the particle fluid equations that are frequently used to study the medical sprays aerosols and sedimentation problems for the proposed system we first construct the local in time classical solutions in an appropriate l2l2 sobolev space we also establish the a priori large time behavior estimate by constructing a lyapunov functional measuring the fluctuation of momentum and mass from the averaged quantities and using this together with the bootstrapping argument we obtain the global classical solution the large time behavior estimate asserts that the velocity functions of the pressureless euler and the compressible navier stokes equations are aligned exponentially fast as time tends to infinity

a remark on time analyticity for the kuramoto sivashinsky equation many applications of time analyticity of solutions of dissipative evolution equations raise the question of estimating the uniform time analyticity radius of solutions for instance the rate of decay of power spectrum which is the quantity often measured in experiments can be estimated in terms of the uniform time analyticity radius 1 also in 8 10 11 uniform time analyticity plays a role in establishing algebraic approximations of attractors moreover a rigorous explicit estimate of the time analyticity domain is important for constructing these approximations since better analyticity estimates lead to better estimates on accuracy in this note we address time analyticity of solutions of the kuramoto sivashinsky equation ut uxxxx uxx uux 0 with the initial data u0 lp r since the time analyticity is not a local property it is expected that the analyticity domain will depend on the initial data in theorem 2 1 we provide this estimate as a function of the initial data and p in the case p 2 this has been obtained in 6 9 15 by energy estimates on rays in the complex plane here we estimate the fundamental solutions corresponding to each ray in order to optimize the shape of domains we employ the fact that the fundamental solution of the main linear part is contracting along every ray if p 2 and then use the riesz interpolation theorem to cover other values of p cf lemma 3 3

semi infinite arrays of isotropic point scatterers a unified approach we solve the two dimensional problem of acoustic scattering by a semi infinite pe riodic array of identical isotropic point scatterers i e objects whose size is negligible compared to the incident wavelength and which are assumed to scatter incident waves uniformly in all directions this model is appropriate for scatterers on which dirichlet boundary conditions are applied in the limit as the ratio of wavelength to body size tends to infinity the problem is also relevant to the scattering of an e polarized electromagnetic wave by an array of highly conducting wires the actual geometry of each scatterer is characterized by a single parameter in the equations related to the single body scattering problem and determined from a harmonic boundary value problem using a mixture of analytical and numerical techniques we confirm that a number of phenomena reported for specific geometries are in fact present in the general case such as the presence of shadow boundaries in the far field and the vanishing of the circular wave scattered by the end of the array in certain specific directions we show that the semi infinite array problem is equivalent to that of inverting an infinite toeplitz matrix which in turn can be formulated as a discrete wiener hopf problem numerical results are presented which compare the amplitude of the wave diffracted by the end of the array for scatterers having different shapes

space time decay of navier stokes flows invariant under rotations we show that the solutions to the non stationary navier stokes equations in open image in new window d 2 3 which are left invariant under the action of discrete subgroups of the orthogonal group o d decay much faster as open image in new window than in generic case and we compute for each subgroup the precise decay rates in space time of the velocity field

global well posedness of strong solutions with large oscillations and vacuum to the compressible navier stokes poisson equations subject to large and non flat doping profile abstract this paper concerns an initial value problem of compressible navier stokes poisson equations subject to large and non flat doping profile in whole space r 3 the global well posedness of strong solutions with large oscillations and vacuum is established provided the initial data are of small energy and the steady state is strictly away from vacuum a weak strong uniqueness result is also obtained

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

uniform boundedness of the solutions for a one dimensional isentropic model system of compressible viscous gas this paper studies an initial boundary value problem for a one dimensional isentropic model system of compressible viscous gas with large external forces represented by v t u x 0 u t av x u x v x f 0 vdx t with v x 0 u x 0 v 0 x u 0 x u 0 t u 1 t 0 especially the uniform boundedness of the solution in time is investigated it is proved that for arbitrary large initial data and external forces the problem uniquely has an uniformly bounded global in time solution with also uniformly positive mass density provided the adiabatic constant 1 is suitably close to 1 the proof is based on l 2 energy estimates and a technique used in 9

wave diffraction by semi infinite venetian blind type grating the h polarized electromagnetic wave diffraction by semi infinite periodic venetian blind type grating is considered the grating consists of strips placed in the parallel planes the problem is reduced to a nonlinear operator equation with respect to the unknown reflection operator of a structure for its solution a complex approach which consists in using the successive over relaxation method and the gauss seidel method is suggested the near and far fields scattered by the grating are calculated the dependences of the reflection coefficient and diffraction patterns are analyzed for different incidence angles width of strips and distance between them

strong solutions of the compressible nematic liquid crystal flow abstract we study strong solutions of the simplified ericksen leslie system modeling compressible nematic liquid crystal flows in a domain r 3 we first prove the local existence of a unique strong solution provided that the initial data 0 u 0 d 0 are sufficiently regular and satisfy a natural compatibility condition the initial density function 0 may vanish on an open subset i e an initial vacuum may exist we then prove a criterion for possible breakdown of such a local strong solution at finite time in terms of blow up of the quantities l t l x and d l t 3 l x

global regularity of the three dimensional equations for nonhomogeneous incompressible fluids abstract this paper is concerned with the global well posedness of strong and classical solutions for the three dimensional nonhomogeneous incompressible navier stokes equations subject to vacuum and external forces let 0 m 0 and f be the initial density initial momentum and potential external force respectively we first show that there exists a global strong solution u on r 3 0 t for any 0 t provided that the viscosity coefficient 0 is sufficiently large or 0 l or m 0 2 0 l 1 0 l 2 f l 2 or m 0 0 l 2 f h 1 is small enough although the density may vanish in some open sets it is only assumed that u 0 m 0 0 is well defined and satisfies 0 1 2 u 0 u 0 l 2 a uniqueness result is also proved next if the given data are more regular and satisfy an additional compatibility condition used in choe and kim 2003 for the existence of strong solution then the strong solution is indeed a classical one

global behavior of spherically symmetric navier stokes poisson system with degenerate viscosity coefficients in this paper we study a free boundary problem for compressible spherically symmetric navier stokes poisson equations with degenerate viscosity coefficients and without a solid core under certain assumptions that are imposed on the initial data we obtain the global existence and uniqueness of the weak solution and give some uniform bounds with respect to time of the solution moreover we obtain some stabilization rate estimates of the solution the results show that such a system is stable under small perturbations and could be applied to the astrophysics

global classical solutions for 3d compressible navier stokes equations with vacuum and a density dependent viscosity coefficient nnsfc 10801111 11271306 10971171 fundamental research funds for the central universities 2010121006 2012121005 natural science foundation of fujian province of china 2010j05011

non existence of classical solutions with finite energy to the cauchy problem of the compressible navier stokes equations the well posedness of classical solutions with finite energy to the compressible navier stokes equations cns subject to arbitrarily large and smooth initial data is a challenging problem in the case when the fluid density is away from vacuum strictly positive this problem was first solved for the cns in either one dimension for general smooth initial data or multi dimension for smooth initial data near some equilibrium state that is small perturbation antontsev et al in boundary value problems in mechanics of nonhomogeneous fluids north holland publishing co amsterdam 1990 kazhikhov in sibirsk mat zh 23 60 64 1982 kazhikhov et al in prikl mat meh 41 282 291 1977 matsumura and nishida in proc jpn acad ser a math sci 55 337 342 1979 j math kyoto univ 20 67 104 1980 commun math phys 89 445 464 1983 in the case that the flow density may contain a vacuum the density can be zero at some space time point it seems to be a rather subtle problem to deal with the well posedness problem for cns the local well posedness of classical solutions containing a vacuum was shown in homogeneous sobolev space without the information of velocity in l2 norm for general regular initial data with some compatibility conditions being satisfied initially cho et al in j math pures appl 9 83 243 275 2004 cho and kim in j differ equ 228 377 411 2006 manuscr math 120 91 129 2006 choe and kim in j differ equ 190 504 523 2003 and the global existence of a classical solution in the same space is established under the additional assumption of small total initial energy but possible large oscillations huang et al in commun pure appl math 65 549 585 2012 however it was shown that any classical solutions to the compressible navier stokes equations in finite energy inhomogeneous sobolev space cannot exist globally in time since it may blow up in finite time provided that the density is compactly supported xin in commun pure appl math 51 229 240 1998 in this paper we investigate the well posedess of classical solutions to the cauchy problem of navier stokes equations and prove that the classical solution with finite energy does not exist in the inhomogeneous sobolev space for any short time under some natural assumptions on initial data near the vacuum this implies in particular that the homogeneous sobolev space is as crucial as studying the well posedness for the cauchy problem of compressible navier stokes equations in the presence of a vacuum at far fields even locally in time

on the ideal compressible magnetohydrodynamic equations abstract in this paper we consider the short time strong solutions to the ideal compressible magnetohydrodynamic equations with initial vacuum where the velocity field satisfies the navier slip condition inspired by kato and lax s idea we use the contraction mapping argument to prove the local existence moreover under the navier slip condition we establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of l norm of the deformation tensor d u