a blow up criterion for two dimensional compressible viscous heat conductive flows abstract we establish a blow up criterion in terms of the upper bound of the density and temperature for the strong solution to 2d compressible viscous heat conductive flows the initial vacuum is allowed

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

combinatorial point for fused loop models integrable loop models associated with higher representations spin l 2 of u q mathfrak sl 2 are investigated at the point q hbox e pm i pi ell 2 the ground state eigenvalue and eigenvectors are described introducing inhomogeneities into the models allows to derive a sum rule for the ground state entries

global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations abstract in this paper we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations in r 3 we prove the global existence of the smooth solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in h 3 framework moreover if additionally the initial data belong to l p with 1 p 6 5 the optimal convergence rates of the solutions in l q norm with 2 q 6 and its spatial derivatives in l 2 norm are obtained

on some integrals over the u n unitary group and their large n limit the integral over the u n unitary group i du exp tr au bu is re examined various approaches and extensions are first reviewed the second half of the paper deals with more recent developments relation with integrable toda lattice hierarchy diagrammatic expansion and combinatorics and what they teach us on the large n limit of log i

on the variety of projectors in this paper we are going to give an answer to the following two questions let k be a field and a4 the space of n x n square matrices with coefficients in k the first question we want to investigate is what are the equations of the subvariety a of ikf consisting of rank h matrices a such that a2 a by equations we mean of course a set of explicit generators of the prime ideal of x the second question is what are the equations of the varieties yh h n 2 which are the closures of the nilpotent conjugacy classes in h4 e consisting of matrices of rank h whose square is the matrix 0 in this case we shall also prove that these varieties y are normal and cohen macaulay in arbitrary characteristic thus generalizing a result of hesselink 3 to arbitrary characteristic note that in characteristic 0 all closures of nilpotent conjugacy classes have been proved to be normal and cohen macaulay by kraft and procesi 4 but our method is not suitable for generalizing their result the method used in this paper is the one of hodge algebras 11 namely we shall exhibit an explicit linear basis for the coordinate rings of the varieties x and y

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

robust hypothesis testing with a relative entropy tolerance this paper considers the design of a minimax test for two hypotheses where the actual probability densities of the observations are located in neighborhoods obtained by placing a bound on the relative entropy between actual and nominal densities the minimax problem admits a saddle point which is characterized the robust test applies a nonlinear transformation which flattens the nominal likelihood ratio in the vicinity of one results are illustrated by considering the transmission of binary data in the presence of additive noise

on the variety of a highest weight module 1 2 let u g be the enveloping algebra of g detailed information on the primitive spectrum prim u g of u g was summarized by conjecture 7 4 of 181 here a main question was to relate prim u g with the springer correspondence parts i ii of this conjecture were established in i ii and a partial solution to iii was given in 21 sufficient e g to treat g of type a meanwhile borho and brylinski 5 have established iii for induced ideals while barbasch and vogan 1 2 have established a version of iii in which gr i i e prim u g is replaced by a certain wavefront set known to be contained in y gr 1 this last work involves some case by case analysis and gives little indication as to why the springer correspondence should arise in the study of prim u g

discontinuous solutions of the navier stokes equations for multidimensional flows of heat conducting fluids we prove the global existence of weak solutions of the navier stokes equations for compressible heat conducting fluids in two and three space dimensions when the initial density is close to a constant in l 2 l the initial temperature is close to a constant in l 2 and the initial velocity is small in h s l 4 where s 0 when n 2 and when n 3 the l p norms must be weighted slightly when n 2 in particular the initial data may be discontinuous across a hypersurface of n a great deal of qualitative information about the solution is obtained for example we show that the velocity vorticity and temperature 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 central role in the entire analysis particularly in closing the required energy estimates and in understanding rates of regularization near the initial layer moreover f is precisely the quantity through which the hyperbolicity of the corresponding equations for inviscid fluids shows itself an effect which is crucial for obtaining time independent pointwise bounds for the density

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

global existence of classical solutions to the vlasov poisson boltzmann system with given magnetic field abstract this paper considers the vlasov poisson boltzmann system with given magnetic field the global existence of classical solutions was obtained when the initial data is a small perturbation around a global maxwellian the proof is based on the theory of compressible navier stokes poisson equations with forcing and the macro microdecomposition of the solution to the boltzmann equation with respect to the local maxwellian introduced in t p liu t yang s h yu energy method for the boltzmann equation physica d 188 3 4 2004 178 192 and elaborated in t yang h j zhao a new energy method for the boltzmann equation j math phys 47 2006 the result shows that the existence of solutions is independent of the magnetic field

blow up of smooth highly decreasing at infinity solutions to the compressible navier stokes equations abstract we prove that the smooth solutions to the cauchy problem for the navier stokes equations with conserved total mass finite total energy and finite momentum of inertia lose the initial smoothness within a finite time in the case of space of dimension 3 or greater even if the initial data are not compactly supported the cases of isentropic and incompressible fluids are also considered

on blowup of classical solutions to the compressible navier stokes equations in this paper we study the finite time blow up of smooth solutions to the compressible navier stokes system when the initial data contain vacuums we prove that any classical solutions of viscous compressible fluids without heat conduction will blow up in finite time as long as the initial data has an isolated mass group see definition 2 2 the results hold regardless of either the size of the initial data or the far fields being vacuum or not this improves the blowup results of xin comm pure appl math 51 229 240 1998 by removing the crucial assumptions that the initial density has compact support and the smooth solution has finite total energy furthermore the analysis here also yields that any classical solutions of viscous compressible fluids without heat conduction in bounded domains or periodic domains will blow up in finite time if the initial data have an isolated mass group satisfying some suitable conditions

remarks on blow up of smooth solutions to the compressible fluid with constant and degenerate viscosities abstract in this paper we will show the blow up of smooth solutions to the cauchy problem for the full compressible navier stokes equations and isentropic compressible navier stokes equations with constant and degenerate viscosities in arbitrary dimensions under some restrictions on the initial data in particular the results hold true for the full compressible euler equations and isentropic compressible euler equations and the blow up time can be computed in a more precise way it is not required that the initial data has compact support or contains vacuum in any finite regions moreover we will give a simplified and unified proof on the blow up results to the classical solutions of the full compressible navier stokes equations without heat conduction by xin 41 and with heat conduction by cho jin 5

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

global classical solution of the cauchy problem to 1d compressible navier stokes equations with large initial data in this paper we prove that the 1d cauchy problem of the compressible navier stokes equations admits a unique global classical solution u if the viscosity 1 with 0 the initial data can be arbitrarily large and may contain vacuum some new weighted estimates of the density and velocity are obtained when deriving higher order estimates of the solution

blow up of viscous heat conducting compressible flows abstract we show the blow up of strong solution of viscous heat conducting flow when the initial density is compactly supported this is an extension of z xin s result z xin blow up of smooth solutions to the compressible navier stokes equations with compact density comm pure appl math 51 1998 229 240 to the case of positive heat conduction coefficient but we do not need any information for the time decay of total pressure nor the lower bound of the entropy we control the lower bound of second moment by total energy and obtain the exact relationship between the size of support of initial density and the existence time we also provide a sufficient condition for the blow up in case that the initial density is positive but has a decay at infinity

asymptotic behavior of solutions to the compressible navier stokes equation around a parallel flow the initial boundary value problem for the compressible navier stokes equation is considered in an infinite layer of mathbf r 2 it is proved that if the reynolds and mach numbers are sufficiently small then strong solutions to the compressible navier stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations the large time behavior of the solution is described by a solution of a one dimensional viscous burgers equation the proof is given by a combination of spectral analysis of the linearized operator and a variant of the matsumura nishida energy method

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

existence and uniqueness of steady solutions to the magnetohydrodynamic equations of compressible flows abstract we establish the existence and uniqueness of a strong solution to the steady magnetohydrodynamic equations for the compressible barotropic fluids in a bounded smooth domain with a perfectly conducting boundary under the assumption that the external force field is small

blow up criterion for the compressible magnetohydrodynamic equations with vacuum abstract in this paper the 3 d compressible magnetohydrodynamic mhd equations with initial vacuum or infinite electric conductivity is considered we prove that the l norms of the deformation tensor d u and the absolute temperature control the possible blow up see 18 23 of strong solutions especially for the non resistive mhd system when the magnetic diffusion vanishes this conclusion means that if a solution of the compressible mhd equations is initially regular and loses its regularity at some later time then the formation of singularity must be caused by losing the bound of d u or as the critical time approaches the viscosity coefficients are only restricted by the physical conditions our criterion see 1 17 is similar to 17 for 3 d incompressible euler equations and to 12 for 3 d compressible isentropic navier stokes equations

on large time behavior of solutions to the compressible navier stokes equations in the half space in r3 the navier stokes equation for compressible viscous fluid is considered on the half space in r3 under the zero dirichlet boundary condition for the momentum with initial data near an arbitrarily given equilibrium of positive constant density and zero momentum time decay properties in l2 norms for solutions of the linearized problem are investigated to obtain the rate of convergence in l2 norms of solutions to the equilibrium when initial data are sufficiently close to the equilibrium in some lower bounds are derived for solutions to the linearized problem one of which indicates a nonlinear phenomenon not appearing in the case of the cauchy problem on the whole space

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

zero dissipation limit of the 1d linearized navier stokes equations for a compressible fluid abstract in this paper we study the asymptotic limiting behavior of the solutions to the initial boundary value problem for linearized one dimensional compressible navier stokes equations we consider the characteristic boundary conditions that is we assume that an eigenvalue of the associated inviscid euler system vanishes uniformly on the boundary the aim of this paper is to understand the evolution of the boundary layer to construct the asymptotic ansatz which is uniformly valid up to the boundary and to obtain rigorously the uniform convergence to the solution of the euler equations without the weakness assumption on the boundary layer

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

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

maximin performance of binary input channels with uncertain noise distributions we consider uncertainty classes of noise distributions defined by a bound on the divergence with respect to a nominal noise distribution the noise that maximizes the minimum error probability for binary input channels is found the effect of the reduction in uncertainty brought about by knowledge of the signal to noise ratio is also studied the particular class of gaussian nominal distributions provides an analysis tool for near gaussian channels the asymptotic behavior of the least favorable noise distribution and the resulting error probability are studied in a variety of scenarios namely asymptotically small divergence with and without power constraint asymptotically large divergence with and without power constraint and asymptotically large signal to noise ratio

existence of global weak solutions for 3d degenerate compressible navier stokes equations in this paper we prove the existence of global weak solutions for 3d compressible navier stokes equations with degenerate viscosity the method is based on the bresch and desjardins commun math phys 238 211 223 2003 entropy conservation the main contribution of this paper is to derive the mellet and vasseur commun partial differ equ 32 431 452 2007 type inequality for weak solutions even if it is not verified by the first level of approximation this provides existence of global solutions in time for the compressible barotropic navier stokes equations the result holds for any gamma 1 in two dimensional space and for 1 gamma 3 in three dimensional space in both case with large initial data possibly vanishing on the vacuum this solves an open problem proposed by lions mathematical topics in fluid mechanics vol 2 compressible models 1998

large time behavior of the semigroup on lp spaces associated with the linearized compressible navier stokes equation in a cylindrical domain abstract large time behavior of solutions to the linearized compressible navier stokes equation around the motionless state in a cylindrical domain is investigated the l p decay estimates of the associated semigroup are established for all 1 p it is also shown that the time asymptotic leading part of the semigroup is given by a one dimensional heat semigroup

grobner geometry of vertex decompositions and of flagged tableaux we relate a classic algebro geometric degeneration technique dating at least to hodge 1941 to the notion of vertex decompositions of simplicial com plexes the good case is when the degeneration is reduced and we call this a geometric vertex decomposition our main example in this paper is the family of vexillary matrix schubert vari eties whose ideals are also known as one sided ladder determinantal ideals us ing a diagonal term order to specify the grobner degeneration we show that these have geometric vertex decompositions into simpler varieties of the same type from this together with the combinatorics of the pipe dreams of fomin kirillov 1996 we derive a new formula for the numerators of their multigraded hilbert series the double grothendieck polynomials in terms of flagged set valued tableaux this unifies work of wachs 1985 on flagged tableaux and buch 2002 on set valued tableaux giving geometric meaning to both this work focuses on diagonal term orders giving results complementary to those of knutson miller 2004 where it was shown that the generating minors form a grobner basis for any antidiagonal term order and any matrix schubert va riety we show here that under a diagonal term order the only matrix schubert varieties for which these minors form grobner bases are the vexillary ones reach ing an end toward which the ladder determinantal literature had been building contents

global well posedness of the cauchy problem of two dimensional compressible navier stokes equations in weighted spaces abstract in this paper we study the global well posedness of classical solution to 2d cauchy problem of the compressible navier stokes equations with large initial data and vacuum it is proved that if the shear viscosity is a positive constant and the bulk viscosity is the power function of the density that is with 3 then the 2d cauchy problem of the compressible navier stokes equations on the whole space r 2 admits a unique global classical solution u which may contain vacuums in an open set of r 2 note that the initial data can be arbitrarily large to contain vacuum states various weighted estimates of the density and velocity are obtained in this paper and these self contained estimates reflect the fact that the weighted density and weighted velocity propagate along with the flow

worst case additive noise for binary input channels and zero threshold detection under constraints of power and divergence additive noise channels with binary inputs and zero threshold detection are considered we study worst case noise under the criterion of maximum error probability with constraints on both power and divergence with respect to a given symmetric nominal noise distribution particular attention is focused on the cases of a gaussian nominal distributions and b asymptotic increase in worst case error probability when the divergence tolerance tends to zero

quantization based methods additive attacks performance analysis the main goal of this study consists in the development of the worst case additive attack wcaa for ary quantization based data hiding methods using as design criteria the error probability and the maximum achievable rate of reliable communications our analysis focuses on the practical scheme known as distortion compensated dither modulation dc dm from the mathematical point of view the problem of the worst case attack wca design using probability of error as a cost function is formulated as the maximization of the average probability of error subject to the introduced distortion for a given decoding rule when mutual information is selected as a cost function a solution of the minimization problem should provide such an attacking noise probability density function pdf that will maximally decrease the rate of reliable communications for an arbitrary decoder structure the obtained results demonstrate that within the class of additive attacks the developed attack leads to a stronger performance decrease for the considered class of embedding techniques than the additive white gaussian or uniform noise attacks

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

from the boltzmann equation to the incompressible navier stokes equations on the torus a quantitative error estimate we investigate the boltzmann equation depending on the knudsen number in the navier stokes perturbative setting on the torus using hypocoer civity we derive a new proof of existence and exponential decay for solutions close to a global equilibrium with explicit regularity bounds and rates of convergence these results are uniform in the knudsen number and thus allow us to obtain a strong derivation of the incompressible navier stokes equations as the knudsen number tends to 0 moreover our method is also used to deal with other kinetic models finally we show that the study of the hydrodynamical limit is rather different on the torus than the one already proved in the whole space as it requires averaging in time unless the initial layer conditions are satisfied

blow up of smooth solutions to the compressible magnetohydrodynamic flows abstract in this paper we prove the blow up phenomena of smooth solutions to the cauchy problem for the full compressible magnetohydrodynamic equations and isentropic compressible magnetohydrodynamic equations with constant and degenerate viscosities under some restrictions on the initial data in particular our results do not require that the initial data have compact support or contain vacuum in any finite region

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

the cauchy problem for the compressible navier stokes equations in the l p framework the global in time existence of regular solutions to the compressible navier stokes equations in the whole space is obtained the solutions are close to nontrivial equilibrium solutions moreover the result is sharp in the lp framework the velocity of the fluid belongs to wr2 1 and the initial density to w1r with r 3

existence of global strong solutions in critical spaces for barotropic viscous fluids this paper is dedicated to the study of viscous compressible barotropic fluids in dimension n 2 we address the question of the global existence of strong solutions for initial data close to a constant state having critical besov regularity first this article shows the recent results of charve and danchin arch ration mech anal 198 1 233 271 2010 and chen et al commun pure appl math 63 1173 1224 2010 with a new proof our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity which allows us to cancel out the coupling between the density and the velocity as in haspot well posedness in critical spaces for barotropic viscous fluids 2009 second we improve the results of charve and danchin 2010 and chen et al 2010 by adding as in charve and danchin 2010 some regularity on the initial data in low frequencies in this case we obtain global strong solutions for a class of large initial data which rely on the results of hoff arch rational mech anal 139 303 354 1997 hoff commun pure appl math 55 11 1365 1407 2002 and hoff j math fluid mech 7 3 315 338 2005 and those of charve and danchin 2010 and chen et al 2010 we conclude by generalizing these results for general viscosity coefficients

the decay rate of solution for the bipolar navier stokes poisson system in this paper the cauchy problem for the bipolar navier stokes poisson system is studied we obtain the global existence and the hs decay rate of classical solutions for this problem our approach is mainly based on the analysis of the green s function of the linearized system and some elaborate energy estimates it should be mentioned that with the help of long wave short wave decomposition the decay rate of the higher order derivatives of the solutions is obtained furthermore based on the hs decay rate of the solutions we also give the lp estimate of the solution

bifurcation analysis of the degond lucquin desreux morrow model for gas discharge abstract the main purpose of this paper is to investigate mathematically gas discharge townsend discovered and mechanisms which are essential for ionization of gas and then derived a threshold of voltage at which gas discharge can happen in this derivation he used some simplification such as discretization of time therefore it is an interesting problem to analyze the threshold by using the degond lucquin desreux morrow model and also to compare the results of analysis with townsend s theory note that gas discharge never happens in townsend s theory if mechanism is not taken into account in this paper we study an initial boundary value problem to the model with mechanism but no mechanism this problem has a trivial stationary solution of which the electron and ion densities are zero it is shown that there exists a threshold of voltage at which the trivial solution becomes unstable from stable then we conclude that gas discharge can happen for a voltage greater than this threshold even if mechanism is not taken into account it is also of interest to know the asymptotic behavior of solutions to this initial boundary value problem for the case that the trivial solution is unstable to this end we establish bifurcation of non trivial stationary solutions by applying crandall and rabinowitz s theorem and show the linear stability and instability of those non trivial solutions

entropy bounded solutions to the one dimensional compressible navier stokes equations with zero heat conduction and far field vacuum abstract the entropy is one of the fundamental states of a fluid and in the viscous case the equation that it satisfies is highly singular in the region close to the vacuum in spite of its importance in the gas dynamics the mathematical analyses on the behavior of the entropy near the vacuum region were rarely carried out in particular in the presence of vacuum either at the far field or at some isolated interior points it was unknown whether the entropy remains its boundedness the results obtained in this paper indicate that the ideal gases retain their uniform boundedness of the entropy locally or globally in time if the vacuum occurs at the far field only and the density decays slowly enough at the far field precisely we consider the cauchy problem to the one dimensional full compressible navier stokes equations without heat conduction and establish the local and global existence and uniqueness of entropy bounded solutions in the presence of vacuum at the far field only it is also shown that different from the case that with compactly supported initial density the compressible navier stokes equations with slowly decaying initial density can propagate the regularities in inhomogeneous sobolev spaces

time periodic solutions of compressible navier stokes equations abstract in this paper we study the navier stokes equations with a time periodic external force in r n we show that a time periodic solution exists when the space dimension n 5 under some smallness assumption the main idea is to combine the energy method and the spectral analysis for the optimal decay estimates on the linearized solution operator with the optimal decay estimates we prove the existence and uniqueness of time periodic solution in some suitable function space by the contraction mapping theorem in addition we also study the time asymptotic stability of the time periodic solution

low mach number limit of non isentropic magnetohydrodynamic equations in a bounded domain abstract this paper verifies the low mach number limit of the non isentropic magnetohydrodynamic equations with zero thermal coefficient in a two dimensional bounded domain with navier s slip boundary condition for velocity and perfectly conducting boundary condition for magnetic field

the initial value problem for the compressible navier stokes equations without heat conductivity abstract in this paper we are concerned with the global existence and convergence rates of strong solutions for the compressible navier stokes equations without heat conductivity in r 3 the global existence and uniqueness of strong solutions are established by the delicate energy method under the condition that the initial data are close to the constant equilibrium state in h 2 framework furthermore if additionally the initial data belong to l 1 the optimal convergence rates of the solutions in l 2 norm and convergence rates of their spatial derivatives in l 2 norm are obtained

helically symmetric solutions to the 3 d navier stokes equations for compressible isentropic fluids abstract we prove the existence of global weak solutions to the navier stokes equations for compressible isentropic fluids for any 1 when the cauchy data are helically symmetric where the constant is the specific heat ratio moreover a new integrability estimate of the density in any neighborhood of the symmetry axis the singularity axis is obtained

decay of the non isentropic navier stokes poisson equations abstract we establish the time decay rates of the solution to the cauchy problem for the non isentropic 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 as a corollary we also obtain the usual l p l 2 1 p 2 type of the optimal decay rates the h s 0 s 3 2 negative sobolev norms are shown to be preserved along time evolution and enhance the decay rates we use a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis

blow up criterion for the 3d compressible non isentropic navier stokes equations without thermal conductivity abstract in this paper we prove a blow up criterion in terms of the density and the pressure p for the strong solutions with vacuum to cauchy problem of the 3d compressible non isentropic navier stokes equations without thermal conductivity more precisely we show that the strong solution exists globally if the norm p l 0 t r 3 is bounded from above

time decay rates for the compressible viscoelastic flows abstract in this paper we investigate the time decay rates of the global smooth solutions to the compressible viscoelastic flows in r 3 the global existence and uniqueness of smooth solutions are obtained under the assumption that the h 3 norm of the initial data is small but the higher order derivatives can be arbitrarily large moreover if the initial data is bounded in h s 0 s 3 2 or b 2 s 0 s 3 2 we establish the optimal decay rates of the smooth solutions by a regularity interpolation trick and delicate energy methods

formation of singularities in three dimensional compressible fluids presented are several results on the formation of singularities in solutions to the three dimensional euler equations for a polytropic ideal fluid under various assumptions on the initial data in particular it is shown that a localized fluid which is initially compressed and outgoing on average will develop singularities regardless of the size of the initial disturbance

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

global existence for the dirichlet problem for the viscous shallow water equations abstract a global existence and uniqueness theorem of strong solutions for the initial boundary value problem with dirichlet boundary conditions is established for small forcing and small initial data an exponentialc0decay rate is also established and the solution is shown to be classical fort 0 the positivity of the fluid heighthfort 0 is also proved

optimal decay estimates on the linearized boltzmann equation with time dependent force and their applications although the decay in time estimates of the semi group generated by the linearized boltzmann operator without forcing have been well established there is no corresponding result for the case with general external force this paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted sobolev spaces for the linearized boltzmann equation with a time dependent external force no time decay assumption is made on the force the proof is based on both the energy method through the macro micro decomposition and the lp lq estimates from the spectral analysis the decay estimates thus obtained are applied to the study on the global existence of the cauchy problem to the nonlinear boltzmann equation with time dependent external force and source precisely for space dimension n 3 the global existence and decay rates of solutions to the cauchy problem are obtained under the condition that the force and source decay in time with some rates this time decay restriction can be removed for space dimension n 5 moreover the existence and asymptotic stability of the time periodic solution are given for space dimension n 5 when the force and source are time periodic with the same period

global existence and asymptotic behavior for the 3d compressible navier stokes equations without heat conductivity in a bounded domain abstract in this paper we investigate the global existence and uniqueness of strong solutions to the initial boundary value problem for the 3d compressible navier stokes equations without heat conductivity in a bounded domain with slip boundary the global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in h 2 furthermore the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods

finite time blow up of compressible navier stokes equations on half space or outside a fixed ball abstract in this paper we consider the initial boundary value problem to the compressible navier stokes equations for ideal gases without heat conduction in the half space or outside a fixed ball in r n with n 1 we prove that any classical solutions u in the class c 1 0 t h m m n 2 2 with bounded from below initial entropy and compactly supported initial density which allows to touch the physical boundary must blow up in finite time as long as the initial mass is positive this paper extends the classical result by xin 1998 19 in which the cauchy problem is considered to the case that with physical boundary

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

global existence and optimal convergence rates of solutions for 3d compressible magneto micropolar fluid equations abstract the cauchy problem for the three dimensional compressible magneto micropolar fluid equations is considered existence of global in time smooth solutions is established under the condition that the initial data are small perturbations of some given constant state moreover we obtain the time decay rates of the higher order spatial derivatives of the solution by combining the l p l q estimates for the linearized equations and the fourier splitting method if the initial perturbation is small in h 3 norm and bounded in l 1 norm

a scheme related to the brauer loop model abstract we introduce the brauer loop scheme e m m n c m m 0 where is a certain degeneration of the ordinary matrix product its components of top dimension n 2 2 correspond to involutions s n having one or no fixed points in the case n even this scheme contains the upper upper scheme from a knutson some schemes related to the commuting variety j algebraic geom in press math ag 0306275 as a union of n 2 of its components one of those is a degeneration of the commuting variety of pairs of commuting matrices the brauer loop model is an integrable stochastic process studied in j de gier b nienhuis brauer loops and the commuting variety j stat mech 2005 p01006 math ag 0410392 based on earlier related work in m j martins b nienhuis r rietman an intersecting loop model as a solvable super spin chain phys rev lett 81 1998 504 507 cond mat 9709051 and some of the entries of its perron frobenius eigenvector were observed conjecturally to equal the degrees of the components of the upper upper scheme our proof of this equality follows the program outlined in p di francesco p zinn justin inhomogeneous model of crossing loops and multidegrees of some algebraic varieties math ph 0412031 in that paper the entries of the perron frobenius eigenvector were generalized from numbers to polynomials which allowed them to be calculated inductively using divided difference operators we relate these polynomials to the multi degrees of the components of the brauer loop scheme defined using an evident torus action on e as a consequence we obtain a formula for the degree of the commuting variety previously calculated up to 4 4 matrices

global existence of small solutions to a class of nonlinear evolution equations on etudie le comportement global des solutions d une classe d equations d evolution non lineaires on considere le probleme aux valeurs initiales pour des equations quasilineaires qui sont des perturbations d equations lineaires dissipatives

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

initial boundary value problems for the compressible viscoelastic fluid abstract this paper is concerned with the initial boundary value problems of the system modeling the compressible viscoelastic fluid of oldroyd type the global in time solution is proved to exist uniquely near the equilibrium state in h 2

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

universal attractor in h4 for the nonlinear one dimensional compressible navier stokes equations abstract this paper is concerned with the regularity exponential stability of solutions and existence of universal attractor in h 4 for a nonlinear one dimensional compressible navier stokes equations describing a motion of heat conductive viscous real gas in a bounded domain 0 1 some new ideas and more delicated estimates are introduced to prove these results

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

on the existence and stability of time periodic solution to the compressible navier stokes equation on the whole space the existence of a time periodic solution of the compressible navier stokes equation on the whole space is proved for a sufficiently small time periodic external force when the space dimension is greater than or equal to 3 the proof is based on the spectral properties of the time t map associated with the linearized problem around the motionless state with constant density in some weighted l and sobolev spaces the time periodic solution is shown to be asymptotically stable under sufficiently small initial perturbations and the l norm of the perturbation decays as time goes to infinity

formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum abstract we study the cauchy problem for multi dimensional compressible radiation hydrodynamics equations with vacuum first we present some sufficient conditions on the blow up of smooth solutions in multi dimensional space then we obtain the invariance of the support of density for the smooth solutions with compactly supported initial mass density by the property of the system under the vacuum state based on the above mentioned results we prove that we cannot get a global classical solution no matter how small the initial data are as long as the initial mass density is of compact support finally we will see that some of the results that we obtained are still valid for the isentropic flows with degenerate viscosity coefficients as well as for one dimensional case

b orbits in solutions to the equation x2 0 in triangular matrices abstract consider in k the set of solutions to the equation x2 0 in n n strictly upper triangular matrices over field k in this paper we construct the bijection from the set of adjoint orbits in in k onto the set of involutions in symmetric group sn for a finite field k we compute the number of points in the orbits

on the solvability of a boundary value problem for model equations of linearly viscous materials this paper deals with model equations for linearly viscous materials viscosity is allowed to vary with temperature the global solvability of an initial boundary value problem for nonlinear equations is proved by continuation of a local solution with the help of a priori estimates the main attention is paid to the global estimates

inhomogenous model of crossing loops and multidegrees of some algebraic varieties we consider a quantum integrable inhomogeneous model based on the brauer algebra b 1 and discuss the properties of its ground state eigenvector in particular we derive various sum rules and show how some of its entries are related to multidegrees of algebraic varieties

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

incompressible limits of the navier stokes equations for all time abstract we study the asymptotic behaviors of the regular solutions to the compressible navier stokes equations for well prepared initial data for all time as the mach number tends to zero by deriving a differential inequality with certain decay property the estimates obtained in this paper are uniform both in time and mach number