global well posedness of incompressible inhomogeneous fluid systems with bounded density or non lipschitz velocity in this paper we first prove the global existence of weak solutions to the d dimensional incompressible inhomogeneous navier stokes equations with initial data a 0 in l infty mathbb r d u 0 u 0 h u 0 d in dot b 1 frac d p p r mathbb r d which satisfy mu a 0 l infty u 0 h dot b 1 frac d p p r rm exp c r mu 2r u 0 d dot b 1 frac d p p r 2r leqq c 0 mu for some positive constants c 0 c r and 1 p d 1 r the regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non lipschitz velocity fields furthermore with additional regularity assumptions on the initial velocity or on the initial density we can also prove the uniqueness of such a solution we should mention that the classical maximal l p l q regularity theorem for the heat kernel plays an essential role in this context

on the decay and stability to global solutions of the 3 d inhomogeneous navier stokes equations in this chapter we consider the large time decay and stability to any given global smooth solution of the three dimensional inhomogeneous incompressible navier stokes equations

the local langlands conjecture for gsp 4 we prove the local langlands conjecture for gsp4 f where f is a nonarchimedean local eld of characteristic zero

solutions in mixed norm sobolev lorentz spaces to the initial value problem for the navier stokes equations abstract in this note for 0 m and index vectors q q 1 q 2 q d r r 1 r 2 r d where 1 q i 1 r i and 1 i d we introduce and study mixed norm sobolev lorentz spaces h l q r m which are more general than the classical sobolev spaces h q m then we investigate the existence and uniqueness of solutions to the navier stokes equations nse in the spaces l p 0 t h l q r m where p 2 t 0 and the initial datum is taken in the space i u 0 s r d d div u 0 0 e t u 0 l p 0 t h l q r m the results have a standard relation between existence time and data size large time with small datum or large datum with small time in the case of global solutions t and critical indexes 2 p i 1 d 1 q i m 1 the space i coincides with the homogeneous besov space b l q r m 2 p p in the case when m 0 q 1 q 2 q d r 1 r 2 r d our results recover those of fabes jones and riviere 10

a lagrangian approach for the incompressible navier stokes equations with variable density we investigate the cauchy problem for the inhomogeneous navier stokes equations in the whole n dimensional space under some smallness assumption on the data we show the existence of global in time unique solutions in a critical functional framework the initial density is required to belong to the multiplier space of input amssym dot b n p 1 p 1 bbb r n in particular piecewise constant initial densities are admissible data provided the jump at the interface is small enough and generate global unique solutions with piecewise constant densities using lagrangian coordinates is the key to our results as it enables us to solve the system by means of the basic contraction mapping theorem as a consequence conditions for uniqueness are the same as for existence 2012 wiley periodicals inc

global regularity to the cauchy problem of the 3d heat conducting incompressible navier stokes equations abstract this paper concerns the global regularity to the cauchy problem of the 3d heat conducting incompressible navier stokes equations with density temperature dependent viscosity and vacuum through the t weighted a priori estimates we establish the global existence and decay of strong solutions provided the initial energy is suitably small it should be noted that the absolute temperature can be large initially

random walk in a random medium in one dimension applying scaling and universality arguments the long time behavior of the probability distribution for a random walk in a one dimensional random medium satisfying sinai s constraint is obtained analytically the convergence to this asymptotic limit and the fluctuations of this distribution are evaluated by solving numerically the stochastic equations for this walk

global regularity for generalized anisotropic navier stokes equations we consider the global cauchy problem for the three dimensional generalized anisotropic navier stokes system with the viscosity term hu m2u where m is a fourier multiplier whose symbol m r r is non negative the case m 3 3 is essentially the navier stokes system while the case m 0 is the anisotropic navier stokes system we obtain the global regularity in the case that m 3 3 32 g 3 for all sufficiently large 3 where g r r is a nondecreasing function such that 1 dsg2 s slns then using the method of descent we can obtain the classical global regularity result of the two dimensional navier stokes system

unique solvability for the density dependent incompressible navier stokes korteweg system abstract this article concerns the strong solutions for the incompressible fluid models of korteweg type with a density dependent viscosity and capillary coefficient in a bounded domain r 3 it is proved that the initial boundary problem of the density dependent incompressible fluid of korteweg type admits a unique local strong solution which allows the existence of initial vacuum provided the initial data satisfies a compatibility condition

global regularity for some classes of large solutions to the navier stokes equations in previous works by the first two authors classes of initial data to the three dimensional incompressible navier stokes equations were presented generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large the main feature of the initial data considered in one of those studies is that it varies slowly in one direction though in some sense it is well prepared its norm is large but does not depend on the slow parameter the aim of this article is to generalize that setting to an ill prepared situation the norm blows up as the small parameter goes to zero as in those works the proof uses the special structure of the nonlinear term of the equation

global well posedness for the 3d navier sokes equations with a large component of vorticity abstract in this paper we give a new approach to improve the leray s result concerning the cauchy problem to the 3d navier stokes equations in particular global well posedness with a large component of the initial vorticity is obtained our idea is considering the vorticity equations and using some suitable function spaces

global well posedness bkm blow up criteria and zero h limit for the 3d incompressible hall mhd equations abstract we consider the cauchy problem for the 3d incompressible hall mhd equations firstly we prove the global well posedness under some new conditions which improve the previous corresponding results in 3 4 24 then a beale kato majda bkm blow up criterion is established finally we show the explicit convergence rate of strong solutions from the hall mhd equations to the mhd equations as the hall coefficient h 0

finite size scaling analysis of biased diffusion on fractals abstract diffusion on a t fractal lattice under the influence of topological biasing fields is studied by finite size scaling methods this allows to avoid proliferation and singularities which would arise in a renormalization group approach on infinite system as a consequence of logarithmic diffusion within the scheme logarithmic diffusion is proved on the basis of an analysis of various temporal scales such as first passage time moments and survival probability characteristic time this confirms and puts on firmer basis previous renormalization group results a careful study of the asymptotic occupation probabilities of different kinds of lattice points allows to elucidate the mechanism of trapping into dangling ends which is responsible of the logarithmic time dependence of average displacement

large global solutions to the navier stokes equations slowly varying in one direction in to previous papers by the authors classes of initial data to the three dimensional incompressible navier stokes equations were presented generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large the aim of this article is to provide new examples of arbitrarily large initial data giving rise to global solutions in the whole space contrary to the previous examples the initial data has no particular oscillatory properties but varies slowly in one direction the proof uses the special structure of the nonlinear term of the equation

global stability of large solutions to the 3d navier stokes equations we prove the stability of mildly decaying global strong solutions to the navier stokes equations in three space dimensions combined with previous results on the global existence of large solutions with various symmetries this gives the first global existence theorem for large solutions with approximately symmetric initial data the stability of unforced 2d flow under 3d perturbations is also obtained

global solutions for the 3 d incompressible nonhomogeneous mhd equations abstract in this article we establish the global well posedness to the 3 d incompressible non homogeneous magnetohydrodynamics mhd equations for a class of large velocity fields and magnetic fields in critical besov spaces we improve recent results by chen et al 2015 and zhai et al 2015 by extending the validity range of the parameters related to the integrability from 1 q p 6 to the seemingly optimal range 1 p q 6 we also relax their assumption on exponential norm of the initial data by replacing it by a more natural polynomial condition

on the well posedness of 3 d inhomogeneous incompressible navier stokes equations with variable viscosity abstract in this paper we mainly study the well posedness for the 3 d inhomogeneous incompressible navier stokes equations with variable viscosity with some smallness assumption on the bmo norm of the initial density we first get the local well posedness of 1 1 in the critical besov spaces moreover if the viscosity coefficient is a constant we can extend this local solution to be a global one our theorem implies that we have successfully extended the integrability index p of the initial velocity which has been obtained by abidi gui and zhang in 3 burtea in 8 and zhai and yin in 32 to approach the ideal one i e 1 p 6 the main novelty of this work is to apply the crw theorem obtained by coifman rochberg weiss in 11 to get a new a priori estimate for an elliptic equation with variable coefficients the uniqueness of the solution also relies on a lagrangian approach as in 16 17 18

global strong solutions for equations related to the incompressible viscoelastic fluids with a class of large initial data abstract we investigate the global strong solutions for a system related to the incompressible viscoelastic fluids of oldroyd b type with a class of large initial data e 0 e 0 i j n n v 0 v 01 v 0 n we obtain the global existence and uniqueness of a solution provided that e 0 h v 0 h are sufficiently small in b 2 1 n 2 1 b 2 1 n 2 b 2 1 n 2 1 while e 0 n and v 0 n are large where e 0 h e 0 i j n 1 n v 0 h v 01 v 0 n 1 and e 0 n e 0 n 1 e 0 n n

a proof of langland s conjecture on plancherel measures complementary series of p adic groups analysis of p adic reductive groups our first result theorem 7 9 proves a conjecture of langlands on normalization of intertwining operators by means of local langlands root numbers and l functions at least when the group is quasi split and the inducing representation is generic assuming two natural conjectures in harmonic analysis of p adic groups we also prove the validity of the conjecture in general theorem 9 5 as our second result we obtain all the complementary series and special representations of quasi split p adic groups coming from rank one parabolic subgroups and generic supercuspidal represen

global solutions to the 3d incompressible nematic liquid crystal system abstract in this paper we consider the global well posedness of the cauchy problem of the 3d incompressible nematic liquid crystal system with initial data in the critical besov space b 2 1 1 2 r 3 b 2 1 3 2 r 3 in particular we prove that there exists a positive constant c 0 such that the nematic liquid crystal system has a unique global solution with initial data u 0 d 0 u 0 h u 0 3 d 0 which satisfies 1 1 d 0 d 0 b 2 1 3 2 1 u 0 h b 2 1 1 2 exp c 0 2 u 0 3 b 2 1 1 2 1 2 c 0 for some c 0 sufficiently small where d 0 is a constant vector with d 0 1 here and are two positive viscosity constants

stability of the local gamma factor arising from the doubling method let g be either a symplectic group or an orthogonal group over a local non archimedean field f of characteristic zero we consider the local gamma factor associated to an irreducible admissible representation of g by the doubling method of piatetski shapiro and rallis ps r g ps r l r denote this gamma factor by s where is a character of f and is a fixed non trivial character of f in this paper we prove

scaling of the first passage time and the survival probability on exact and quasi exact self similar structures the authors calculate the asymptotic behaviour of the moments of the first passage time and survival probability for random walks on an exactly self similar tree and on a quasi self similar comb by applying an exact decimation approach to the master equations for the hierarchical comb a transition from ordinary to anomalous diffusion occurs at r 2 where r is the ratio of teeth length in successive iterations of the structure in the anomalous regime r 2 the positive integer moments of the first passage time tq scale as ltau q with tau q 1 2q 1 ln r ln 2 where l is the linear distance from input to output the asymptotic behaviour of the survival probability is studied using both scaling theory and by a direct solution of the master equations they find that the characteristic time t in the asymptotic exponential decay of the survival probability exp t t scales as t approximately ltau with tau ln r2 ln 2 i e tau is distinct from tau 1 however substantial corrections to this asymptotic form for tau exist and these are needed to account for the recent simulation data of havlin and matan 1988

on the global wellposedness to the 3 d incompressible anisotropic navier stokes equations corresponding to the wellposedness result 2 for the classical 3 d navier stokes equations ns with initial data in the scaling invariant besov space mathcal b 1 frac3p p infty here we consider a similar problem for the 3 d anisotropic navier stokes equations ans where the vertical viscosity is zero in order to do so we first introduce the besov sobolev type spaces mathcal b frac12 frac12 4 and mathcal b frac12 frac12 4 t then with initial data in the scaling invariant space mathcal b frac12 frac12 4 we prove the global wellposedness for ans provided the norm of initial data is small enough compared to the horizontal viscosity in particular this result implies the global wellposedness of ans with high oscillatory initial data 1 2

global well posedness of compressible navier stokes equations for some classes of large initial data we prove the global well posedness of three dimensional compressible navier stokes equations for some classes of large initial data which may have large oscillation for the density and large energy for the velocity the proof uses the special structure of the system especially the effective viscous flux

fujita kato theorem for the 3 d inhomogeneous navier stokes equations abstract in this paper we prove the global existence and uniqueness of the solution with discontinuous density for the 3 d inhomogeneous navier stokes equations if the initial data 0 u 0 l r 3 h s r 3 with s 1 2 satisfies 0 c 0 0 c 0 u 0 h 1 2 e for some small e 0 depending only on c 0 c 0 furthermore we introduce the dual method to show that if u 0 l p r 3 for p 6 5 2 the velocity satisfies the decay estimate k u t l 2 c 1 t k 2 p for t 1 k 0 1 with p 3 2 1 p 1 2

stability to the global large solutions of 3 d navier stokes equations in this chapter we consider the stability of global large solutions to the 3 d classical isotropic navier stokes equations in the anisotropic sobolev spaces

global large solutions to 3 d inhomogeneous navier stokes system with one slow variable under a nonlinear smallness condition on the isotropic critical besov norm to the fluctuation of the initial density and the critical anisotropic besov norm of the horizontal components of the initial velocity which have to be exponentially small compared with the critical anisotropic besov norm to the third component of the initial velocity we investigate the global wellposedness of 3 d inhomogeneous incompressible navier stokes equations 1 2 in the critical besov spaces the novelty of this results is that the isotropic space structure to the inhomogeneity of the initial density function is consistent with the propagation of anisotropic regularity for the velocity field in the second part we apply the same idea to prove the global wellposedness of 1 2 with some large data which are slowly varying in one direction

between homogeneous and inhomogeneous navier stokes systems the issue of stability abstract we construct large velocity vector solutions to the three dimensional inhomogeneous navier stokes system the result is proved via the stability of two dimensional solutions with constant density under the assumption that initial density is point wisely close to a constant key elements of our approach are estimates in the maximal regularity regime and the lagrangian coordinates considerations are done in the whole r 3

global well posedness for the 3 d incompressible inhomogeneous mhd system in the critical besov spaces abstract in this paper we study the cauchy problem for the incompressible inhomogeneous magnetohydrodynamic mhd system in r 3 we obtain the global existence of a unique solution for this system without any small conditions imposed on the third component of the initial velocity field and magnetic field in the critical besov spaces our methods mainly rely upon the fourier frequency localization and bony s paraproduct decomposition

well posedness for the navier stokes equations where u is the velocity and p is the pressure it is well known that the navierstokes equations are locally well posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at for instance it is easy to see see 9 for much more general results that if s n 2 then for any h initial data there exists a unique c 0 t h r local solution with a pressure p c 0 t h r in the sequel we consider solutions for less regular initial data this has to be understood in the sense that the map from the initial data to the solution extends continuously to rougher function spaces the question we are interested in is the global well posedness for small data and local well posedness for large data with respect to a certain space of

on the wellposedness of three dimensional inhomogeneous navier stokes equations in the critical spaces we prove the local wellposedness of three dimensional incompressible inhomogeneous navier stokes equations with initial data in the critical besov spaces without assumptions of small density variation furthermore if the initial velocity field is small enough in the critical besov space dot b 1 2 2 1 mathbb r 3 this system has a unique global solution

global small solutions to an mhd type system the three dimensional case in this paper we consider the global well posedness of a three dimensional incompressible mhd type system with smooth initial data that is close to some nontrivial steady state it is a coupled system between the navier stokes equations and a free transport equation with a universal nonlinear coupling structure the main difficulty of the proof lies in exploring the dissipative mechanism of the system due to the fact that there is a free transport equation of in the coupled equations and only the horizontal derivatives of is dissipative with respect to time to achieve this we first employ anisotropic littlewood paley analysis to establish the key l1 lip 3 estimate to the third component of the velocity field then we prove the global well posedness to this system by the energy method which depends crucially on the divergence free condition of the velocity field 2014 wiley periodicals inc

multidimensional random walks in random environments with subclassical limiting behavior in this paper we will describe and analyze a class of multidimensional random walks in random environments which contain the one dimensional nearest neighbor situation as a special case and have the pleasant feature that quite a lot can be said about them our results make rigorous a heuristic argument of marinari et al 1983 and show that in anyd we can have a xn is recurrent and b xn logn 2

monte carlo study of biased diffusion at the percolation threshold computer simulations confirm dhar s prediction ibid vol 17 p l257 1984 that the asymptotic exponent for the rms distance against time relation is zero right at the percolation threshold if a particle diffuses under the influence of a constant force in a random medium for intermediate times and weak force the exponent for the mean displacement as a function of time in the direction of the force confirms the prediction of ohtsuki and keyes phys rev lett vol 52 p 1177 1984

equidistribution theorems for holomorphic siegel modular forms for gsp_4 hecke fields and n level density this paper is a continuation of kim et al j inst math jussieu 2018 we supplement four results on a family of holomorphic siegel cusp forms for gsp 4 mathbb q first we improve the result on hecke fields namely we prove that the degree of hecke fields is unbounded on the subspace of genuine forms which do not come from functorial lift of smaller subgroups of gsp 4 second we prove simultaneous vertical sato tate theorem namely we prove simultaneous equidistribution of hecke eigenvalues at finitely many primes third we compute the n level density of degree 4 spinor l functions and thus we can distinguish the symmetry type depending on the root numbers this is conditional on certain conjecture on root numbers fourth we consider equidistribution of paramodular forms in this case we can prove the conjecture on root numbers main tools are the equidistribution theorem in our previous work and shin templier s compos math 150 12 2003 2053 2014 work

large global solutions to 3 d inhomogeneous navier stokes equations slowly varying in one variable abstract motivated by chemin and gallagher 2010 8 we consider the global wellposedness to the 3 d incompressible inhomogeneous navier stokes equations with large initial velocity slowly varying in one space variable in particular we proved that when the initial density is close enough to a positive constant then given divergence free initial velocity field of the type v 0 h w 0 h w 0 3 x h x 3 as that in chemin and gallagher 2010 8 for the classical navier stokes system we shall prove the global wellposedness of ins for sufficiently small the main difficulty here lies in the fact that we will have to obtain the l 1 r lip r 3 estimate for convection velocity in the transport equation of ins toward this and due to the strong anisotropic properties of the approximate solutions we will have to work in the framework of anisotropic type besov spaces here

global well posedness for the 3d incompressible inhomogeneous navier stokes equations and mhd equations abstract the present paper is dedicated to the global well posedness for the 3d inhomogeneous incompressible navier stokes equations in critical besov spaces without smallness assumption on the variation of the density we aim at extending the work by abidi gui and zhang 2012 2 and 2013 3 to a lower regularity index about the initial velocity the key to that improvement is a new a priori estimate for an elliptic equation with nonconstant coefficients in besov spaces which have the same degree as l 2 in r 3 finally we also generalize our well posedness result to the inhomogeneous incompressible mhd equations

global solutions to the 3 d incompressible inhomogeneous navier stokes system in this paper we consider the global well posedness of the 3 d incompressible inhomogeneous navier stokes equations with initial data in the critical besov spaces a0 bq 13q r3 u0 u0h u03 bp 1 1 3p r3 for p q satisfying 1 q p 6 and 1q 1p 13 more precisely we prove that there exist two positive constants c0 c0 such that if a0 bq 13q u0h bp 1 1 3p exp c0 u03 bp 1 1 3p2 2 c0 then 1 3 has a unique global solution a l r bq 13q r3 u l r bp 1 1 3p r3 l1 r bp 11 3p r3 in particular this result implies the global well posedness result in abidi and paicu 2007 2 for the inhomogeneous navier stokes system with small initial data

erratum to global wellposed problem for the 3 d incompressible anisotropic navier stokes equations in an anisotropic space in this paper we consider a global wellposed problem for the 3 d incompressible anisotropic navier stokes equations ans we prove the global wellposedness for ans provided the initial horizontal data are sufficient small in the scaling invariant besov sobolev type space b 0 frac 1 2 in particular the result implies the global wellposedness of ans with large initial vertical velocity

global solutions to the 3 d incompressible anisotropic navier stokes system in the critical spaces in this paper we consider the global wellposedness of the 3 d incompressible anisotropic navier stokes equations with initial data in the critical besov sobolev type spaces mathcal b and mathcal b frac12 frac12 4 see definitions 1 1 and 1 2 below in particular we proved that there exists a positive constant c such that ans has a unique global solution with initial data u 0 u 0 h u 0 3 which satisfies u 0 h mathcal b exp bigl frac c nu 4 u 0 3 mathcal b 4 bigr leq c 0 nu or u 0 h mathcal b frac12 frac12 4 exp bigl frac c nu 4 u 0 3 mathcal b frac12 frac12 4 4 bigr leq c 0 nu for some c 0 sufficiently small to overcome the difficulty that gronwall s inequality can not be applied in the framework of chemin lerner type spaces widetilde l p t mathcal b we introduced here sort of weighted chemin lerner type spaces widetilde l 2 t f mathcal b for some apropriate l 1 function f t

on the field dependence of random walks in the presence of random fields numerical simulations and scaling arguments are used to study the field dependence of a random walk in a one dimensional system with a bias field on each site the bias is taken randomly with equal probability to be e or e the probability density p x t is found to scale asymptotically as left a e beta 2 ln 2 t right exp left left x a e beta 2 ln 2 t right alpha right witha e ln 1 e 1 e 4 25 and 1 25 the mean square displacement scales as langle x 2 rangle sim a e beta f ta beta e where f u ln4u asymptotically

global small solutions of 2 d incompressible mhd system abstract in this paper we consider the global wellposedness of 2 d incompressible magneto hydrodynamical system with smooth initial data which is close to some non trivial steady state it is a coupled system between the navier stokes equations and a free transport equation with a universal nonlinear coupling structure the main difficulty of the proof lies in exploring the dissipative mechanism of the system to achieve this and to avoid the difficulty of propagating anisotropic regularity for the free transport equation we first reformulate our system 1 1 in the lagrangian coordinates 2 19 then we employ anisotropic littlewood paley analysis to establish the key a priori l 1 r lip r 2 estimate for the lagrangian velocity field y t with this estimate we can prove the global wellposedness of 2 19 with smooth and small initial data by using the energy method we emphasize that the algebraic structure of 2 19 is crucial for the proofs to work the global wellposedness of the original system 1 1 then follows by a suitable change of variables

galois representations for holomorphic siegel modular forms we prove local global compatibility up to a quadratic twist of galois representations associated to holomorphic hilbert siegel modular forms in many cases induced from borel or klingen parabolic and as a corollary we obtain a conjecture of skinner and urban for siegel modular forms when the local representation is an irreducible principal series we get local global compatibility without a twist we achieve this by proving a version of rigidity strong multiplicity one for gsp 4 using on the one hand the doubling method to compute the standard l function and on the other hand the explicit classification of the irreducible local representations of gsp 4 over p adic fields then we use the existence of a globally generic hilbert siegel modular form weakly equivalent to the original and we refer to sorensen mathematica 15 623 670 2010 for local global compatibility in that case

global well posedness of 3 d inhomogeneous navier stokes system with initial velocity being a small perturbation of 2 d solenoidal vector field abstract motivated by 21 we consider the global wellposedness to the 3 d incompressible inhomogeneous navier stokes equations with large horizontal velocity in particular we proved that when the initial density is close enough to a positive constant then given divergence free initial velocity field of the type v 0 h 0 x h w 0 h w 0 3 x h x 3 we shall prove the global wellposedness of 1 1 the main difficulty here lies in the fact that we will have to obtain the l 1 r lip r 3 estimate for convection velocity in the transport equation of 1 1 toward this and due to the strong anisotropic properties of the approximate solutions we will have to work in the framework of anisotropic littlewood paley theory here

e factor of representations of classical groups abstract we let g be the isometry group of a nondegenerate symmetric or skew symmetric form h on a space v over some local or global field k i e g u v h we let lg be the l group of go the identity component of g and r be the natural representation of this group we develop here a theory of l functions and e factors attached to this r in particular we develop a theory of local and global zeta integrals for such g we use very special intertwining operators therein we require an analysis of the proper normalization of such operators with such techniques we determine the correct e factors associated to local representations the issue of the correct local l factor is discussed

regularity results for two dimensional flows of multiphase viscous fluids global regularity results for weak solutions of the navier stokes equations for two dimensional multiphase incompressible fluids are proved under suitable conditions on the viscosity without assuming positive lower bounds on the initial density as an application we deduce regularity properties for the integral curves of the corresponding velocity field finally we prove regularity results in the small for strong solutions

global well posedness of 3 d anisotropic navier stokes system with large vertical viscous coefficient abstract in this paper we first prove the global well posedness of 3 d anisotropic navier stokes system provided that the vertical viscous coefficient of the system is sufficiently large compared to some critical norm of the initial data then we shall construct a family of initial data u 0 which vary fast enough in the vertical variable and which can be arbitrarily large in the space b m o 1 yet u 0 still generates a unique global solution to the classical 3 d navier stokes system provided that is sufficiently large

global well posedness for the incompressible navier stokes equations with density dependent viscosity coefficient abstract this paper is concerned with an initial boundary value problem of the incompressible navier stokes equations with density dependent viscosity in a smooth bounded domain r 3 the global well posedness of strong solutions with large oscillations is established in both non vacuum and vacuum cases provided the initial velocity is suitably small in certain sense it is worth pointing out that there isn t any smallness condition on the density and its gradient