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

weak and strong solutions for the incompressible navier stokes equations with damping in this paper we show that the cauchy problem of the navier stokes equations with damping u 1u 0 has global weak solutions for any 1 global strong solution for any 7 2 and that the strong solution is unique for any 7 2 5

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

nonlinear stability of large amplitude viscous shock wave for general viscous gas abstract in the present paper it is shown that the large amplitude viscous shock wave is nonlinearly stable for isentropic navier stokes equations in which the pressure could be general and includes law and the viscosity coefficient is a smooth function of density the strength of shock wave could be arbitrarily large the proof is given by introducing a new variable which can formulate the original system into a new one and the elementary energy method introduced in 21

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

invariant subspaces for polynomially bounded operators abstract let t be a polynomially bounded operator on a banach space x whose spectrum contains the unit circle then t has a nontrivial invariant subspace in particular if x is reflexive then t itself has a nontrivial invariant subspace this generalizes the well known result of brown chevreau and pearcy for hilbert space contractions

global existence of weak solutions to 3d compressible primitive equations with degenerate viscosity in this paper we investigate the compressible primitive equations cpes with density dependent viscosity for large initial data the cpe model can be derived from the 3d compressible and anisotropic navier stokes equations by hydrostatic approximation motivated by the work of vasseur and yu siam j math anal 48 1489 1511 2016 invent math 206 935 974 2016 in which the global existence of weak solutions to the compressible navier stokes equations with degenerate viscosity was obtained we construct approximate solutions and prove the global existence of weak solutions to the cpe in this paper in our proof we first present the vertical velocity as a function of density and horizontal velocity which plays a role in using the faedo galerkin method to obtain the global existence of the approximate solutions then we obtain the key estimates of lower bound of the density the bresch desjardins entropy on the approximate solutions finally we apply compactness arguments to obtain global existence of weak solutions by vanishing the parameters in our approximate system step by step

existence of global weak solutions for a 2d viscous shallow water equations and convergence to the quasi geostrophic model we consider a two dimensional viscous shallow water model with friction term existence of global weak solutions is obtained and convergence to the strong solution of the viscous quasi geostrophic equation with free surface term is proven in the well prepared case the ill prepared data case is also discussed

singular limits in bounded domains for quasilinear symmetric hyperbolic systems having a vorticity equation abstract a short time existence theorem is proven for the initial boundary value problem for a class of quasilinear symmetric hyperbolic systems containing a constant coefficient spatial operator multiplied by a large parameter solutions u are shown to remain bounded independently of for a time independent of provided that u 0 and u l 0 are bounded independently of and certain structural conditions are satisfied this result is applied to the quasigeostrophic approximation of the shallow water equations and the constant pressure approximation in combustion and solutions for these cases are shown to converge to solutions of limiting equations as

dual piecewise analytic bundle shift models of linear operators lettbe a banach space operator with empty point spectrum whose essential spectrum lies on a finite system of possibly intersecting curves under certain conditions ont dual analytic representations oftas a kind of bundle shift and of t as an adjoint bundle shift are constructed a specialization of this scheme is given which allows one to obtain in some cases concrete similarity models oftandt in certain analogues of smirnove2 spaces the latter construction is applied to toeplitz operators with smooth symbols

resolution of the nested hierarchy for rational sl n models we construct drinfel d twists for the rational sl n xxx model giving rise to a completely symmetric representation of the monodromy matrix we obtain a polarization free representation of the pseudoparticle creation operators figuring in the construction of the bethe vectors within the framework of the quantum inverse scattering method this representation enables us to resolve the hierarchy of the nested bethe ansatz for the sl n invariant rational heisenberg model our results generalize the findings of maillet and sanchez de santos for sl 2 models

commutants and reflexivity of multiplication tuples on vector valued reproducing kernel hilbert spaces abstract motivated by the theory of weighted shifts on directed trees and its multivariable counterpart we address the question of identifying commutant and reflexivity of the multiplication d tuple m z on a reproducing kernel hilbert space h of e valued holomorphic functions on where e is a separable hilbert space and is a bounded domain in c d admitting bounded approximation by polynomials in case e is a finite dimensional cyclic subspace for m z under some natural conditions on the b e valued kernel associated with h the commutant of m z is shown to be the algebra h b e of bounded holomorphic b e valued functions on provided m z satisfies the matrix valued von neumann s inequality this generalizes a classical result of shields and wallen the case of dim e 1 and d 1 as an application we determine the commutant of a bergman shift on a leafless locally finite rooted directed tree t of finite branching index as the second main result of this paper we show that a multiplication d tuple m z on h satisfying the von neumann s inequality is reflexive this provides several new classes of examples as well as recovers special cases of various known results in one and several variables we also exhibit a family of tri diagonal b c 2 valued kernels for which the associated multiplication operators m z are non hyponormal reflexive operators with commutants equal to h b c 2 d

cauchy problem for viscous rotating shallow water equations abstract we consider the cauchy problem for a viscous compressible rotating shallow water system with a third order surface tension term involved derived recently in the modeling of motions for shallow water with free surface in a rotating sub domain marche 2007 19 the global existence of the solution in the space of besov type is shown for initial data close to a constant equilibrium state away from the vacuum unlike the previous analysis about the compressible fluid model without coriolis forces see for instance danchin 2000 10 haspot 2009 16 the rotating effect causes a coupling between two parts of hodge s decomposition of the velocity vector field and additional regularity is required in order to carry out the friedrichs regularization and compactness arguments

asymptotics of the 1d compressible navier stokes equations with density dependent viscosity abstract we are concerned with the time asymptotic behavior toward rarefaction waves for strong non vacuum solutions to the cauchy problem of the one dimensional compressible navier stokes equations with degenerate density dependent viscosity the case when the pressure p and the viscosity coefficient for some parameters r is considered for 0 max 1 if the initial data is assumed to be sufficiently regular without vacuum and mass concentrations we show that the cauchy problem of the one dimensional compressible navier stokes equations admits a unique global strong non vacuum solution which tends to the rarefaction waves as time goes to infinity here both the initial perturbation and the strength of the rarefaction waves can be arbitrarily large the proof is established via a delicate energy method and the key ingredient in our analysis is to derive the uniform in time positive lower and upper bounds on the specific volume

dynamical behaviors for 1d compressible navier stokes equations with density dependent viscosity abstract the dynamical behaviors of vacuum states for one dimensional compressible navier stokes equations with density dependent viscosity coefficient are considered it is first shown that a unique strong solution to the free boundary value problem exists globally in time the free boundary expands outwards at an algebraic rate in time and the density is strictly positive in any finite time but decays pointwise to zero time asymptotically then it is proved that there exists a unique global weak solution to the initial boundary value problem when the initial data contains discontinuously a piece of continuous vacuum and is regular away from the vacuum the solution is piecewise regular and contains a piece of continuous vacuum before the time t 0 which is compressed at an algebraic rate and vanishes at the time t meanwhile the weak solution becomes either a strong solution or a piecewise strong one and tends to the equilibrium state exponentially

on one dimensional compressible navier stokes equations with degenerate viscosity and constant state at far fields in this paper we are concerned with the cauchy problem for one dimensional compressible isentropic navier stokes equations with density dependent viscosity 0 and pressure p 1 we will establish the global existence and asymptotic behavior of weak solutions for any 0 and 1 under the assumption that the density function keeps a constant state at far fields in particular in the case that 0 12 we obtain the large time behavior of the strong solution obtained by mellet and vasseur when the solution has a lower bound no vacuum

global well posedness and long time dynamics for a higher order quasi geostrophic type equation abstract in this paper we study a higher order viscous quasi geostrophic type equation this equation was derived in 11 as the limit dynamics of a singularly perturbed navier stokes korteweg system with coriolis force when the mach rossby and weber numbers go to zero at the same rate the scope of the present paper is twofold first of all we investigate well posedness of such a model on the whole space r 2 we prove that it is well posed in h s for any s 3 globally in time interestingly enough we show that this equation owns two levels of energy estimates for which one gets existence and uniqueness of weak solutions with different regularities namely h 3 and h 4 regularities this fact can be viewed as a remainder of the so called bd entropy structure of the original system in the second part of the paper we investigate the long time behavior of these solutions we show that they converge to the solution of the corresponding linear parabolic type equation with same initial datum and external force our proof is based on dispersive estimates both for the solutions to the linear and non linear problems

effective velocity in compressible navier stokes equations with third order derivatives abstract a formulation of certain barotropic compressible navier stokes equations with third order derivatives as a viscous euler system is proposed by using an effective velocity variable the equations model for instance viscous korteweg or quantum navier stokes flows the formulation in the new variable allows for the derivation of an entropy identity which is known as the bd bresch desjardins entropy equation as a consequence of this estimate a new global in time existence result for the one dimensional quantum navier stokes equations with strictly positive particle densities is proved

on the uniqueness of strong solution to the incompressible navier stokes equations with damping abstract in this paper we show that the cauchy problem of the incompressible navier stokes equations with damping u 1 u 0 has global strong solution for any 3 and the strong solution is unique when 3 5 this improves earlier results

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

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

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

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

global existence of weak solutions to the compressible quantum navier stokes equations with degenerate viscosity we study the compressible quantum navier stokes qns equations with degenerate viscosity in the three dimensional periodic domains on the one hand we consider qns with additional damping terms motivated by the recent works of li and xin e print arxiv 1504 06826 and p antonelli and s spirito arch ration mech anal 225 1161 1199 2017 we construct a suitable approximate system which has smooth solutions satisfying the energy inequality and the bd entropy estimate using this system we obtain the global existence of weak solutions to the compressible qns equations with damping terms for large initial data moreover we obtain some new a priori estimates which can avoid using the assumption that the gradient of the velocity is a well defined function which was indeed used directly in the work of vasseur and yu siam j math anal 48 1489 1511 2016 invent math 206 935 974 2016 on the other hand in the absence of damping terms we also prove the global existence of weak solutions to the compressible qns equations without the lower bound assumption on the dispersive coefficient which improves the previous result of p antonelli and s spirito arch ration mech anal 225 1161 1199 2017

ekman layers of rotating fluids the case of general initial data in this paper we describe the weak solutions of the navier stokes equations with a large coriolis term as the rossby and the ekman numbers go to zero in a special domain with various boundary conditions this work extends the results in the case of well prepared initial data in 19 30 2000 john wiley sons inc

asymptotic behavior of a compressible two phase model with well formation interaction abstract in this work we consider a compressible gas liquid model with a well reservoir interaction term that is relevant for coupled wellbore reservoir flow systems involved in e g drilling operations main focus is on deriving estimates that are independent of time under suitable conditions on the well reservoir interaction term we obtain such estimates which allow prediction of the long time behavior of the gas and liquid masses moreover we also obtain a quantification of the convergence rates as a function of time and gain some insight into the role played by the rate characterizing how fast the well reservoir interaction must die out the model is investigated in a free boundary setting where the initial mass is a mixture of both phases i e no single phase zone exists

well posedness in critical spaces for the system of compressible navier stokes in larger spaces this paper is dedicated to the study of viscous compressible barotropic fluids in dimension n geq2 we address the question of well posedness for it large data having critical besov regularity our result improves the analysis of r danchin in cite dw of chen et al in cite c2 and of the author in cite h1 h2 inasmuch as we may take initial density in b nn p 1 with 1 leq p infty our result relies on a new a priori estimate for the velocity where we introduce a new unknown called textit effective velocity to weaken one the coupling between the density and the velocity in particular for the first time we obtain uniqueness without imposing hypothesis on the gradient of the density

global strong solutions to radial symmetric compressible navier stokes equations with free boundary in this paper we consider the two dimensional barotropic compressible navier stokes equations with stress free boundary condition imposed on the free surface as the viscosity coefficients satisfies 2 1 we establish the existence of global strong solution for arbitrarily large spherical symmetric initial data even if the density vanishes across the free boundary in particular we show that the density is strictly positive and bounded from the above and below in any finite time if the initial density is strictly positive and the free boundary propagates along the particle path and expand outwards at an algebraic rate

two level mixed finite element methods for the navier stokes equations with damping abstract in this paper the mixed finite element methods are considered for the navier stokes equations with damping optimal error estimates of the h 1 norm and l 2 norm for the velocity and the l 2 norm for the pressure are obtained moreover two kinds of two level methods based on the oseen correction and newton correction are presented finally two numerical examples are implemented to confirm the analysis

lagrange structure and dynamics for solutions to the spherically symmetric compressible navier stokes equations the compressible navier stokes system cns with density dependent viscosity coefficients is considered in multi dimension the prototype of the system is the viscous saint venat model for the motion of shallow water a spherically symmetric weak solution to the free boundary value problem for cns with stress free boundary condition and arbitrarily large data is shown to exist globally in time with the free boundary separating fluids and vacuum and propagating at finite speed as particle path which is continuous away from the symmetry center detailed regularity and lagrangian structure of this solution have been obtained in particular it is shown that the particle path is uniquely defined starting from any non vacuum region away from the symmetry center along which vacuum states shall not form in any finite time and the initial regularities of the solution is preserved starting from any non vacuum point at a later on time a particle path is also uniquely defined backward in time which either reaches at some initial non vacuum point or stops at a small middle time and connects continuously with vacuum in addition the free boundary is shown to expand outward at an algebraic rate in time and the fluid density decays and tends to zero almost everywhere away from the symmetry center as the time grows up this finally leads to the formation of vacuum state almost everywhere as the time goes to infinity

on the dynamics of navier stokes equations for a shallow water model in this paper we study a free boundary problem of one dimensional compressible navier stokes equations with a density dependent viscosity which include in particular a shallow water model under some suitable assumptions on the initial data we obtain the global existence uniqueness and the large time behavior of weak solutions in particular it is shown that a stationary wave pattern connecting a gas to the vacuum continuously is asymptotically stable for small initial general perturbations

global existence for 1d compressible isentropic navier stokes equations with large initial data we prove the global existence of weak solutions of the cauchy problem for the navier stokes equations of compressible isentropic flow of a polytropic gas in one space dimension the initial velocity and density are assumed to be in l2 and l2 n bv respectively modulo additive constants in particular no smallness assumptions are made about the intial data in addition we prove a result concerning the asymptotic decay of discontinuities in the solution when the adiabatic constant exceeds 3 2

supersymmetric vertex models with domain wall boundary conditions by means of the drinfeld twists we derive the determinant representations of the partition functions for the gl 1 1 and gl 2 1 supersymmetric vertex models with domain wall boundary conditions in the homogeneous limit these determinants degenerate to simple functions

phragmen lindelof alternative results for the shallow water equations for transient compressible viscous flow abstract in this paper the spatial behavior of the viscous shallow water equations in a two dimensional channel domain is studied a differential inequality for suitable energy associated with the solutions of the viscous shallow water equations in a semi infinite channel is derived from which we show the phragmen lindelof alternative results the main tool used is the energy method our results can be viewed as a version of saint venant s principle to the transient compressible viscous flow

drinfeld twists and algebraic bethe ansatz of the supersymmetric model associated with uq gl m n we construct the drinfeld twists or factorizing f matrices of the supersymmetric model associated with quantum superalgebra u q gl m vertical bar n and obtain the completely symmetric representations of the creation operators of the model in the f basis provided by the f matrix as an application of our general results we present the explicit expressions of the bethe vectors in the f basis for the u q gl 2 vertical bar 1 model the quantum t j model

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

global existence of weak solutions for the multicomponent reaction flows abstract a multidimensional model is introduced for the dynamic reversible reaction of compressible multicomponent gaseous mixture the purpose of this work is to develop a complete existence result in the case where the pressure depends only on the mixture composition and the barotropic correction without any additional assumption the existence of globally weak solutions of the navier stokes for multicomponent reaction flows is established by using weak convergence methods in the spirit of mellet vasseur 14 and li xin 12

superclose and superconvergence of finite element discretizations for the stokes equations with damping a class of conforming mixed finite element methods fems are applied to the stokes equations with damping with the help of the prior estimates for the exact and approximated solutions and the appropriate choice of the parameters a n and r appeared in the problem the superclose and superconvergence results for the velocity in h 1 norm and the pressure in l 2 norm are derived to our best knowledge this is the first superconvergence analysis of conforming mixed fems for this kind nonlinear problem

global weak solutions to 3d compressible navier stokes poisson equations with density dependent viscosity abstract in this paper we study the global existence of weak solutions to the compressible navier stokes poisson n s p equations with density dependent viscosity and non monotone pressure in a three dimensional torus our approach is based on the faedo galerkin method and the compactness arguments motivated by vasseur yu 29 and 30 we construct the approximate solutions and the key estimates ling in the elementary energy estimates b d entropy and mellet vasseur type inequality for the weak solutions here we need the conditions that the adiabatic constant satisfies 4 3 3 for 1 or 1 3 for 1 where is a sign constant of poisson equation which determines the physical meaning of the n s p system

a shimorin type analytic model on an annulus for left invertible operators and applications abstract a new analytic model for left invertible operators which extends both shimorin s analytic model for left invertible and analytic operators and gellar s model for bilateral weighted shift is introduced and investigated we show that a left invertible operator t which satisfies certain conditions can be modeled as a multiplication operator m z on a reproducing kernel hilbert space of vector valued analytic functions on an annulus or a disc a similar result for composition operators in l 2 spaces is established

polarization free generators for the belavin model employing a change of basis the so called factorizing drinfel d twist we construct polarization free and completely symmetric creation operators for a face type model equivalent to the belavin model a resolution of the nested structure of the bethe vectors is achieved

finite energy method for compressible fluids the navier stokes korteweg model this is the first of a series of papers devoted to the initial value problem for the one dimensional euler system of compressible fluids and augmented versions containing higher order terms in the present paper we are interested in dispersive shock waves and analyze the zero viscosity capillarity limit associated with the navier stokes korteweg system specifically we establish the existence of finite energy solutions as well as their convergence toward entropy solutions to the euler system our method of proof combines energy and effective energy estimates a nonlinear sobolev inequality high integrability properties for the mass density and for the velocity and compactness properties based on entropies 2015 wiley periodicals inc

determinant representation of correlation functions for the uq gl 1 1 free fermion model with the help of the factorizing f matrix the scalar products of the uq gl 1 1 free fermion model are represented by determinants by means of these results we obtain the determinant representations of correlation functions of the model

on the compactness of the reduced gravity two and a half layer equations abstract we consider the reduced gravity two and a half model in oceanic fluid dynamics and prove the stability of weak solutions in periodic domain t 2 the proof is based on the uniform a priori estimates and the method of weak convergence and the limit is carried out with the help of a new higher regularity estimate of the velocity which is derived by constructing a special test function

asymptotic behavior of weak solutions to the damped navier stokes equations abstract whether or not weak solutions of classical navier stokes equations decay to zero in l 2 as time tends to infinity was posed by leray in his pioneering paper 12 13 and has been well solved see 15 for instance this paper examines the three dimensional navier stokes equations with damping term u 1 u and proves that the weak solutions decay to zero in l 2 as time tends to infinity for any 1 by developing local in space estimate for weak solutions

determinant representations of correlation functions for the supersymmetric t j model working in the f basis provided by the factorizing f matrix the scalar products of bethe states for the supersymmetric t j model are represented by determinants by means of these results we obtain determinant representations of correlation functions for the model

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

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

global stability of weak solutions for a multilayer saint venant model with interactions between the layers in this paper we investigate the existence of global weak solutions for the multilayer model introduced by audusse et al which is related to incompressible free surface flows we prove the global stability of weak solutions over the torus we observe that this model admits the so called bd entropy and a gain of integrability on the velocity in the spirit of the work of mellet and vasseur it allows us to obtain enough compactness estimates in order to show the stability of global weak solutions

two grid mfeas for the incompressible stokes type variational inequality with damping abstract in this paper we study two grid mixed finite element algorithms mfeas for the stationary incompressible stokes equations with damping and nonlinear slip boundary conditions we set up the weak formulation and obtain the well posedness then employing the mixed finite element method some optimal error estimates between the exact solution and the approximation solution are derived two kinds of two grid mfeas based on the stokes correction and newton type correction are established some error estimates for two grid mfeas are obtained several numerical results are presented to confirm our theoretical findings

asymptotic behavior of compressible navier stokes equations with density dependent viscosity and vacuum in this paper we study the one dimensional navier stokes equations connecting to vacuum state with a jump in density when the viscosity depends on the density precisely when the viscosity coefficient is proportional to with 0 where is the density we give the asymptotic behavior and the decay rate of the density function x t furthermore the behavior of the density function x t near the interfaces separating the gas from vacuum and the expanding rate of the interfaces are also studied the analysis is based on some new mathematical techniques and some new useful estimates this fills a final gap on studying navier stokes equations with the viscosity coefficient dependent on the density

highly rotating viscous compressible fluids in presence of capillarity effects we study here a singular limit problem for a navier stokes korteweg system with coriolis force in the domain mathbb r 2 times 0 1 and for general ill prepared initial data taking the mach and the rossby numbers proportional to a small parameter varepsilon rightarrow 0 we perform the incompressible and high rotation limits simultaneously moreover we consider both the constant and vanishing capillarity regimes in this last case the limit problem is identified as a 2 d incompressible navier stokes equation in the variables orthogonal to the rotation axis if the capillarity is constant instead the limit equation slightly changes keeping however a similar structure due to the presence of an additional surface tension term in the vanishing capillarity regime various rates at which the capillarity coefficient goes to 0 are considered in general this produces an anisotropic scaling in the system the proof of the results is based on suitable applications of the rage theorem combined with microlocal symmetrization arguments

pressureless euler with nonlocal interactions as a singular limit of degenerate navier stokes system abstract we show that weak solutions of degenerate navier stokes equations converge to the strong solutions of the pressureless euler system with linear drag term newtonian repulsion and quadratic confinement the proof is based on the relative entropy method using the artificial velocity formulation for the one dimensional navier stokes system

mac finite difference scheme for stokes equations with damping on non uniform grids abstract in this paper we consider the numerical methods for stationary stokes equations with damping the mark and cell mac method has been applied to discretize the problem on non uniform grids we establish the lbb condition and the stability for the mac scheme then we obtain the second order super convergence in l2 norm for both velocity and pressure on non uniform grids we also obtain the second order convergence for some terms of h1 norm of the velocity and the other terms of h1 norm are second order convergence on uniform grids numerical experiments using the mac scheme show agreement of the numerical results with theoretical analysis

global existence of strong solution for viscous shallow water system with large initial data on the irrotational part abstract we are interested in studying the cauchy problem for the viscous shallow water system in dimension n 2 we show the existence of global strong solutions with large initial data on the irrotational part of the velocity for the scaling of the equations more precisely our smallness assumption on the initial data is supercritical for the scaling of the equations it allows us to give a first kind of answer to the problem of the existence of global strong solution with large initial energy data in dimension n 2 to do this we introduce the notion of quasi solutions which consists in solving the pressureless viscous shallow water system we can obtain such solutions at least for irrotational data which are subject to regularizing effects both on the velocity and on the density this smoothing effect is purely nonlinear and is crucial in order to build solution of the viscous shallow water system as perturbations of the quasi solutions indeed the pressure term can be considered as a remainder term which becomes small in high frequencies for the scaling of the equations to finish we prove the existence of global strong solution with large initial data when n 2 provided that the mach number is sufficiently large

construction of solutions for compressible isentropic navier stokes equations in one space dimension with nonsmooth initial data we prove the global existence of weak solutions for the cauchy problem for the navier stokes equations for one dimensional isentropic flow when the initial velocity is in l 2 and the initial density is in l 2 bv solutions are obtained as limits of approximations obtained by building heuristic jump conditions into a semi discrete difference scheme this allows for a rather simple analysis in which pointwise control is achieved through piecewise h 1 and total variation estimates

asymptotic limit for rotational quantum compressible navier stokes equations with multiple scales abstract in this paper we consider the degenerate quantum compressible navier stokes equations giving rise to a variety of mathematical problems in many areas we study the asymptotic limit for the rotational compressible navier stokes equations with quantum term and the ill prepared initial data

dilations and commutant lifting for jointly isometric operators a geometric approach abstract a tuple of commuting contractions t t 1 t 2 t n is called a joint isometry if t j t j i we give a geometric proof that joint isometries have a regular unitary dilation and that its commutant lifts we also show that t is subnormal and that its minimal normal extension is also jointly isometric

the linearized 2d inviscid shallow water equations in a rectangle boundary conditions and well posedness we consider the linearized 2d inviscid shallow water equations in a rectangle a set of boundary conditions is proposed which make these equations well posed several different cases occur depending on the relative values of the reference velocities u 0 v 0 and reference height phi 0 sub or super critical flow at each part of the boundary

existence of global weak solution for a reduced gravity two and a half layer model we investigate the existence of global weak solution to a reduced gravity two and a half layer model in one dimensional bounded spatial domain or periodic domain also we show that any possible vacuum state has to vanish within finite time then the weak solution becomes a unique strong one

a mixed fem for navier stokes type variational inequality with nonlinear damping term abstract in this work we consider a mixed finite element method fem for incompressible navier stokes equations with nonlinear damping term and friction boundary conditions after establishing the variational formulation we show the well posedness of this problem subsequently we focus our attention on the mixed fem and analyze the galerkin approximation to this system then some optimal error estimates are deduced in the end some iterative algorithms are presented and numerical results are given to verify the theoretical analysis

lp estimates for quantities advected by a compressible flow abstract we consider the evolution of a quantity advected by a compressible flow and subject to diffusion when this quantity is scalar it can be for instance the temperature of the flow or the concentration of some pollutants because of the diffusion term one expects the equations to have a regularizing effect however in their euler form the equations describe the evolution of the quantity multiplied by the density of the flow the parabolic structure is thus degenerate near vacuum when the density vanishes in this paper we show that we can nevertheless derive uniform l p bounds that do not depend on the density in particular the bounds do not degenerate near vacuum furthermore the result holds even when the density is only a measure we investigate both the scalar and the system case in the former case we obtain l bounds in the latter case the quantity being investigated could be the velocity field in compressible navier stokes type of equations and we derive uniform l p bounds for some p depending on the ratio between the two viscosity coefficients the main additional difficulty in that case being to deal with the second viscosity term involving the divergence of the velocity such estimates are to our knowledge new and interesting since they are uniform with respect to the density the proof relies mostly on a method introduced by de giorgi to obtain regularity results for elliptic equations with discontinuous diffusion coefficients

well posedness for the viscous rotating shallow water equations with friction terms we consider the cauchy problem for viscous rotating shallow water equations with friction terms the global existence of the solution in some hybrid spaces is shown for the initial data close to a constant equilibrium state away from the vacuum

global existence of weak solution for quantum navier stokes poisson equations in this paper we consider the compressible quantum navier stokes poisson equations with a linear density dependent viscosity by the use of a singular pressure close to vacuum we prove the global in time existence of weak solutions in a three dimensional torus for large data in the sense of distribution

viscosity effect on the degenerate lake equations abstract this paper concerns the effect of viscosity on the degenerate lake equations anelastic limit when the bottom topography vanishes on the shore we establish the existence and uniqueness of a global weak solutions for various choices of viscosity term in weighted sobolev spaces where the weight is assumed to be a power type weight our solution is constructed by adapting carefully the known works on the incompressible navier stokes equations in unweighted context finally we study in one case the vanishing viscosity limit when the solution of the inviscid lake equations is regular enough generalizing the results by martino et al 2001 to the degenerate case

wold type decompositions and wandering subspaces for operators close to isometries for operators t satisfying certain inequalities we obtain a weak analog of the classical wold kolmogorov decomposition theorem representing t as a direct sum of a unitary operator and a shift operator acting in some hilbert space of analytic functions the concept of a dual operator is introduced which reflects relationships between shift operators acting in two hilbert spaces of analytic functions dual to each other with respect to the cauchy pairing among different aspects of this duality we consider relationships between hereditary inequalities and properties of reproducing kernels

analysis of an inviscid zero mach number system in endpoint besov spaces for finite energy initial data abstract the present paper is the continuation of work 18 devoted to the study of an inviscid zero mach number system in the framework of endpoint besov spaces of type b r s r d r 1 d 2 which can be embedded in the lipschitz class c 0 1 in particular the largest case b 1 1 and the case of holder spaces c 1 are taken into account the local in time well posedness of this system is proved under an additional finite energy hypothesis on the initial data the key to get this result is new a priori estimates for parabolic equations with variable coefficients in endpoint spaces b r s r d which are of independent interest in the special case of space dimension d 2 we are able to give a lower bound for the lifespan such that the solutions tend to be globally defined when the initial inhomogeneity is small there we will show refined a priori estimates in endpoint besov spaces for transport equations

generalized multipliers for left invertible analytic operators and their applications to commutant and reflexivity abstract we introduce generalized multipliers for left invertible analytic operators we show that they form a banach algebra and characterize the commutant of such operators in its terms in the special case we describe the commutant of balanced weighted shift only in terms of its weights in addition we prove two independent criteria for reflexivity of weighted shifts on directed trees

on the flow of chemically reacting gaseous mixture abstract we consider the cauchy problem for the system of equations governing flow of isothermal reactive mixture of compressible gases our main contribution is to prove sequential stability of weak solutions when the state equation essentially depends on the species concentration and the viscosity coefficients vanish on vacuum moreover under additional assumption on the cold component of the pressure in the regions of small density we prove the existence of weak solutions for arbitrary large initial data

an asymptotic limit of a navier stokes system with capillary effects a combined incompressible and vanishing capillarity limit in the barotropic compressible navier stokes equations for smooth solutions is proved the equations are considered on the two dimensional torus with well prepared initial data the momentum equation contains a rotational term originating from a coriolis force a general korteweg type tensor modeling capillary effects and a density dependent viscosity the limiting model is the viscous quasi geostrophic equation for the rotated velocity potential the proof of the singular limit is based on the modulated energy method with a careful choice of the correction terms

global weak solutions and asymptotic behavior to 1d compressible navier stokes equations with density dependent viscosity and vacuum abstract this paper is concerned with existence of global weak solutions to a class of compressible navier stokes equations with density dependent viscosity and vacuum when the viscosity coefficient is proportional to with 1 2 max 3 3 2 a global existence result is obtained which improves the previous results in fang and zhang 2004 4 vong et al 2003 27 yang and zhu 2002 30 here is the density moreover we prove that the domain where fluid is located on expands outwards into vacuum at an algebraic rate as the time grows up due to the dispersion effect of total pressure it is worth pointing out that our result covers the interesting case of the saint venant model for shallow water i e 1 2

stabilized low order finite elements for stokes equations with damping abstract in this paper we consider the stationary stokes equations with damping low order mixed finite element spaces are used to approximate the velocity and the pressure and a local pressure projection stabilization method is used for the pairs to overcome the lack of the inf sup condition the stability of this method is proved and the optimal order error estimates are derived by some nonlinear analysis techniques at last two numerical examples are implemented to test the stability and effectiveness of the proposed method