asymptotic behavior for the navier stokes equations with nonzero external forces abstract we estimate the asymptotic behavior for the stokes solutions with external forces first we found that if there are external forces then the energy decays slowly even if the forces decay quickly then we also obtain the asymptotic behavior in the temporal spatial direction for weak solutions of the navier stokes equations we also provide a simple example of external forces which shows that the stokes solution does not decay quickly

analytic continuation and boundary continuity of functions of several complex variables this note treats some questions about analytic continuation in several variables the first theorem in effect determines the envelops of holomorphy of certain domains in n the second main result is a continuity theorem if a bounded holomorphic function f on a convex domain in n has boundary values that are continuous on the complement in b of a set of the form int b where is a real hyperplane in n that misses then f is continuous on in addition we obtain what may be regarded as a local version of the theorem in our earlier paper concerning the one dimensional extension property our methods depend on hartogs theorem n 3 and directly on the bochnermartinelli formula n 2

multipliers paramultipliers and weak strong uniqueness for the navier stokes equations abstract in this article we describe spaces p such that if u is a weak in the sense of leray j leray sur le mouvement d un fluide visqueux remplissant l espace acta math 63 1934 193 248 solution of the navier stokes system for some initial data u 0 and if u belongs to p then u is unique in the class of weak solutions we say then that weak strong uniqueness holds it turns out that the proof of such results relies on the boundedness of a trilinear functional f l 2 h l 2 h p r where belong to 0 1 in order to find optimal conditions for the boundedness of f we are led to describing spaces of multipliers and of paramultipliers that is functions which map by classical pointwise product or by paraproduct a given sobolev spaces in another given sobolev space the study of these spaces enables us to give conditions for weak strong uniqueness which generalise all previously known results from the famous serrin criterion j serrin the initial value problem for the navier stokes equations in r e langer ed nonlinear problems univ of wisconsin press 1963 pp 69 98 to the recent conditions formulated by lemarie rieusset p g lemarie rieusset recent developments in the navier stokes problem chapman and hall 2003

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

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

l3 solutions of the liquid crystals system abstract in this article we study the simplified system of the original ericksen leslie equations for the flow of liquid crystals it is shown that the l 3 solutions of the cauchy problem for the three dimensional incompressible liquid crystal system are smooth

extension criterion via two components of vorticity on strong solutions to the 3 d navier stokes equations we shall show that only two components of vorticity play an essential role to determine possibility of extension of the time interval for the local strong solution to the navier stokes equations then we shall apply our extension theorem to regularity criterion on weak solutions due to serrin and beirao da veiga chae choe proved the same criterion as beirao da veiga only by means of the two components of vorticity we deal with the critical case which they excluded our criterion may be regarded as the generalization of the result of beal kato majda from l to bmo

random data cauchy problem for the navier stokes equations on t3 in this paper we study the random data initial value problem of the navier stokes equations in l2 t3 by using the randomization approach recently developed by n burq and n tzvetkov we prove that for almost all where is the sample space of a probability space a p for the randomized initial data f h l 2 t3 the navier stokes equations have a global solution u c 0 l 2 t3 or more precisely u et f h c 0 h 12 t3

on some class of sets with multi sheeted envelope of holomorphy let m be a smooth cr manifold of dimension 2n 2 contained in the unit sphere open image in new window we prove that if m is the finite union of boundaries of n 1 dimensional analytic varieties and m is globally minimal then the envelope of holomorphy of open image in new window is multi sheeted in some cases we prove that the number of sheets is exactly two it is also proved for 4 dimensional cr manifolds contained in open image in new window that the property of being globally minimal even being of finite type at each point is generic

existence and positivity of solutions of a fourth order nonlinear pde describing interface fluctuations we study the partial differential equation which arose originally as a scaling limit in the study of interface fluctuations in a certain spin system in that application x lies in r but here we study primarily the periodic case x e s we establish existence uniqueness and regularity of solutions locally in time for positive initial data in h s and prove the existence of several families of lyapunov functions for the evolution from the latter we establish a sharp connection between existence globally in time and positivity preservation if o t is a maximal half open interval of existence for a positive solution of the equation with t 1 16r2 then t 03 and lim m w r exists and is constant we discuss also some explicit solutions and propose a generalization to higher dimensions 0 1994 john wiley

components reduction regularity results for the navier stokes equations in general dimensions abstract we consider the cauchy problem of the navier stokes equations in arbitrary dimensions and establish several new components reduction regularity criteria

weighted estimate of the asymptotic profiles of the navier stokes flow in rn abstract we study the asymptotic profiles of the navier stokes flow and the weighted estimate of the spatial and temporal decay rate of the error term this paper extends the result of y fujigaki and t miyakawa in y fujigaki t miyakawa asymptotic profiles of nonstationary incompressible navier stokes flows in the whole space siam j math anal 33 3 2001 523 544 where they derived the asymptotic profiles of the navier stokes flow and estimated the temporal decay rate of the error term

mild solutions to the time fractional navier stokes equations in r n abstract this paper addresses the existence and uniqueness of mild solutions to the navier stokes equations with time fractional differential operator of order 0 1 several interesting properties about the solution are also highlighted like regularity and decay rate in lebesgue spaces which will depend on the fractional exponent moreover it is shown that the l p exponent range which the solution belongs to is different from the range for the solution of the classical problem with 1

some global results on extension of cr objects in complex manifolds we prove some results concerning the holomorphic extendability of cr objects defined on real hypersurfaces of a complex manifold after a preliminary generalization of the classic theorem on the extendability from the boundary of a relatively compact domain we discuss the extendability from a part of the boundary of such a domain the one side extendability from a piece of hypersurface and the extendability from the boundary of an unbounded domain let m be a noncompact connected complex analytic manifold with countable topology of complex dimension n 2 the present paper deals with holomorphic extension of cr objects defined on real hypersurfaces of m what in fact we do for the most part is to discuss some generalizations in different directions of the well known global extension theorem which asserts the existence and uniqueness of a holomorphic extension on an open domain d c m for a cr function defined on the boundary bd of d under the condition h1 m 0 cf 9 the cr objects considered here are cr sections cr distribution sections and cr hyperfunction sections of a holomorphic complex vector bundle on m our first result theorem 1 1 relates to the holomorphic extension of these kinds of cr objects from the boundary bd of a domain d c m which need not be relatively compact but the closure of which is supposed to belong to some paracompactifying family in m not the whole family of closed sets such a generalization of the global extension theorem does not seem to be already found in the literature actually although it does not require any essentially new ideas is worth being discussed as it allows one to derive other results of some independent interest on extension of cr objects later in 2 we discuss the question of holomorphic extendability on all of a domain d c m for cr objects which need not be defined on the whole boundary bd but just on the complement bd k of a closed set k c bd this question has already been investigated in more particular settings cf 11 13 14 16 17 22 23 by using integral representations techniques here we received by the editors september 10 1987 and in revised form march 3 1988 and november 15 1988 1980 mathematics subject classification 1985 revision primary 32d99 1990 american mathematical society 0002 9947 90 1 00 25 per page

existence and uniqueness of time periodic physically reasonable navier stokes flow past a body let be a three dimensional exterior domain of class c2 0 1 assume that a navier stokes liquid is moving in under the action of a body force f that is time periodic of period t and that the velocity of the liquid is zero at spatial infinity in this paper we show that if f satisfies suitable conditions and its norm in appropriate function spaces is sufficiently small there is at least one time periodic strong solution furthermore the velocity field v of such a solution decays to zero for large x as x 1 and its spatial gradient decays as x 2 both uniformly in time in addition the pressure p decays like x 2 and its gradient like x 3 for almost all t 0 t in the special case where f is time independent these solutions are also time independent and coincide with finn s physically reasonable solutions 4 moreover we show that our time periodic solutions are unique in a very large class namely the class of time periodic weak solutions satisfying the energy inequality and with corresponding pressure fields verifying mild summability conditions in 0 t

the initial boundary value problem for thin plate flow of incompressible non newtonian viscous fluids abstract in this paper the particular model that describes the thin plate flow of incompressible non newtonian viscous fluids is investigated this model which arises from polymer processing concerns the initial boundary problem of nonstationary thin plate flow of non newtonian viscous incompressible fluids by using monotone operator theory and schauder s fixed point theorem an existence and uniqueness theorem for a generalized solution is obtained and the regularity for this solution is tested

diffusive expansion for solutions of the boltzmann equation in the whole space this paper is concerned with the diffusive expansion for solutions of the rescaled boltzmann equation in the whole space 0 1 tf v xf 1 q f f x rn v rn t 0 with prescribed initial data 0 2 f t x v t 0 f0 x v x rn v rn our main purpose is to justify the global validity of the diffusive expansion 0 3 f t x v f1 t x v 2f2 t x v n 1fn 1 t x v nfn t x v for a solution f t x v of the rescaled boltzmann equation 0 1 in the whole space rn for all t 0 with initial data f0 x v satisfying the initial expansion 0 4 f0 x v f1 0 x v 2f2 0 x v n 1fn 1 0 x v nfn 0 x v x v r2n here v 2 n2exp v 22 is a normalized global maxwellian under the assumption that the fluid components of the coefficients fm 0 x v 1 m n of the initial expansion f0 x v have divergence free velocity fields um0 x as well as temperature fields m0 x if we assume further that the velocity temperature fields u10 x 10 x of f1 0 x v have small amplitude in hs rn s 2 n n 2 we can determine these coefficients fm t x v 1 m n in the diffusive expansion 0 3 uniquely by an iteration method and energy method the hydrodynamic component of these coefficients fm t x v 1 m n satisfies the incompressible condition the boussinesq relations and or the navier stokes fourier system respectively while the microscopic component of these coefficients is determined by a recursive formula compared with the corresponding problem inside a periodic box studied in y guo 2006 18 the main difficulty here is due to the fact that poincare s inequality is not valid in the whole space rn and this difficulty is overcome by using the lp lq estimate on the riesz potential moreover by exploiting the energy method we can also deduce certain the space time energy estimates on these coefficients fm t x v 1 m n once the coefficients fm t x v 1 m n in the diffusive expansion 0 3 are uniquely determined and some delicate estimates have been obtained the uniform estimates with respect to on the remainders fn t x v are then established via a unified nonlinear energy method and such an estimate guarantees the validity of the diffusive expansion 0 3 in the large provided that 0 5 n 2n 2 notice that for m 2 um t x is no longer a divergence free vector and it is worth to pointing out that for m 3 it was in deducing certain estimates on pm t x by the lp lq estimate on the riesz potential that we need to require that n 2n 2

on the navier stokes equations in the half space with initial and boundary rough data in morrey spaces abstract we consider the initial boundary value problem for the navier stokes equations in the half space with data in morrey spaces existence of small global solutions is proved in spaces with the right homogeneity to allow self similar solutions moreover we analyze the long time behavior of the solutions and obtain a class of asymptotically self similar ones

on a larger class of stable solutions to the navier stokes equations in exterior domains where w w x w1 x wn x and x denote the unknown velocity vector and the unknown pressure of the fluid at point x respectively while f f x f i j x i j 1 n is the given tensor with div f n j 1 f 1 j xj nj 1 f n j xj denoting the external force in our previous paper 23 we established the theory of existence uniqueness and regularity for solutions to n s0 in the class w ln

on certain integrals associated to cr functions we construct explicit 0 primitives of cauchy fantappie type kernels on analytic subvarieties of domains in cn outside the zero set of a holomorphic map defined on the variety then we use these primitives to derive by means of a process of passing to a residue integral formulas for cr functions defined on subsets of the boundary of a variety certain geometric restrictions on the varieties are imposed these primitives apply in the particular case of the bochner martinelli kernel in domains in cn and we use these primitives in this case to prove a criterion for holomorphic extendability of cr functions defined on certain subsets of the boundary of a domain in cn introduction let d g c cn be bounded domains with smooth boundary and d c g let h g c be a holomorphic map and set z h c e g h o in 2 we constructed kernels ah 4 z and h so that if f is a holomorphic function on z h and z e z h n d then 1 f z f 4c a 4 z a fh c jez h n od the differential forms a h and a h were explicitly defined in terms of h a hefer decomposition hi of h i e holomorphic functions hij c z in gx g with n hi 4 hi z e hij c z j zj i 1 p c z e g x g j 1 and in terms of a function y od x d cn so that n e yj c z cj zj 1 for c z e od x d j l certain geometric restrictions transversality conditions were imposed which guarantee that z h n 3d is a smooth manifold in this paper we pursue further these constructions by proving an integral formula satisfied by cr functions which are defined on certain parts of the boundary of a variety first we construct another kernel i h z which is a received by the editors february 15 1988 1980 mathematics subject classification 1985 revision primary 32a25 32a27 32d1 5 1989 american mathematical society 0002 9947 89 1 00 25 per page

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

asymptotic stability of large solutions with large perturbation to the navier stokes equations abstract consider weak solutions w of the navier stokes equations in serrin s class w l 0 l q for 2 3 q 1 with 3 q where is a general unbounded domain in r 3 we shall show that although the initial and external disturbances from w are large every perturbed flow v with the energy inequality converges asymptotically to w as v t w t l 2 0 v t w t l 2 o t 1 2 as t

long time decay to the lei lin solution of 3d navier stokes equations abstract in this paper we prove if u c 0 x 1 r 3 is a global solution of 3d navier stokes equations then u t x 1 decays to zero as time goes to infinity fourier analysis and standard techniques are used

finite energy of generalized suitable weak solutions to the navier stokes equations and liouville type theorems in two dimensional domains abstract introducing a new notion of generalized suitable weak solutions we first prove validity of the energy inequality for such a class of weak solutions to the navier stokes equations in the whole space r n although we need certain growth condition on the pressure we may treat the class even with infinite energy quantity except for the initial velocity we next handle the equation for vorticity in 2d unbounded domains under a certain condition on the asymptotic behavior at infinity we prove that the vorticity and its gradient of solutions are both globally square integrable as their applications loiuville type theorems are obtained

generalized multiscale finite element methods gmsfem in this paper we propose a general approach called generalized multiscale finite element method gmsfem for performing multiscale simulations for problems without scale separation over a complex input space as in multiscale finite element methods msfems the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space in the proposed approach we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage the enrichment in the online stage is performed based on a spectral decomposition of the offline space in the online stage for any input parameter a multiscale space is constructed to solve the global problem on a coarse grid the online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues the computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re used for solving the forward problem with any forcing and boundary condition compared with the other approaches where global snapshots are used the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse grid level we present various examples in the paper and some numerical results to demonstrate the effectiveness of our method

regularity criterion for 3d mhd fluid passing through the porous medium in terms of gradient pressure abstract the aim of the paper is to establish regularity criteria for the weak solution of fluid passing through the porous media in r 3 our results show that if p l with 2 3 3 1 then the weak solution is regular and unique if p l with 2 3 2 3 2 then the weak solution is regular and unique

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

on weighted norm estimates for nonstationary incompressible navier stokes flows in a 3d exterior domain abstract we study the time decay of weighted norms of weak and strong solutions to the navier stokes equations in a 3d exterior domain moment estimates for weak solutions and weighted l q estimates for strong solutions are deduced both of which seem to be optimal the relation is discussed between the space time decay and the vanishing of the total net force exerted by the fluid to the body a class of initial data is given so that the total net force associated to the corresponding fluid flows does not vanish

boundedness almost periodicity and stability of certain navier stokes flows in unbounded domains abstract we investigate the existence uniqueness and stability of bounded and almost periodic mild solutions to several navier stokes flow problems in our strategy we propose a general framework for studying the semi linear evolution equations with certain smoothing properties of the linear part and with the local lipschitz continuity of the nonlinear operator our method is based on interpolation functors combined with differential inequalities our abstract results are applied to navier stokes oseen equations describing flows of incompressible viscous fluid passing a translating and rotating obstacle and to navier stokes equations on aperture domains and or in besov spaces

on the uniqueness of time periodic solutions to the navier stokes equations in unbounded domains we present a uniqueness theorem for time periodic solutions to the navier stokes equations in unbounded domains thus far results on the uniqueness of time periodic solutions to the navier stokes equations in unbounded domain roughly speaking have only found that a small time periodic ln solution is unique within the class of solutions which have sufficiently small l ln norm in this paper we show that a small time periodic ln solution is unique within the class of all time periodic ln solutions which contains large solutions we also consider the uniqueness of solutions in weak ln space the proof of the present uniqueness theorem is based on the method of dual equations

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

optimal weighted estimates of the flows in exterior domains abstract in this paper we prove some decay properties of global solutions for the navier stokes equations in an exterior domain r n n 2 3 when a domain has a boundary the pressure term is troublesome since we do not have enough information on the pressure near the boundary to overcome this difficulty by multiplying a special form of test functions we obtain an integral equation he xin 2000 12 first introduced this method and then bae jin 2006 2007 1 13 modified their method to obtain better decay rates also bae roh 2009 11 improved bae jin s results unfortunately their results were not optimal because there exists an unpleasant positive small in their rates in this paper we obtain the following optimal rate without x u t l p c t n 2 1 r 1 p 2 for the initial condition u 0 with x u 0 l r and for any n

on time periodic solutions to the boussinesq equations in exterior domains abstract we construct time periodic solutions to the boussinesq equations in a three dimensional exterior domain to this end we estimate the l 1 norm of buoyancy and give some estimates of the heat semigroup in the lorentz space

sobolev multipliers maximal functions and parabolic equations with a quadratic nonlinearity abstract we develop a general framework to describe global mild solutions to a cauchy problem with small initial values concerning a general class of semilinear parabolic equations with a quadratic nonlinearity this class includes the navier stokes equations the subcritical dissipative quasi geostrophic equation and the parabolic elliptic keller segel system

multiscale finite element method for heat transfer problem during artificial ground freezing abstract in this work we consider the heat transfer problem during artificial ground freezing we present the mathematical model and the fine grid approximation for heterogeneous porous media where freezing pipes are considered as line source terms mathematical model is described by a two phase stefan problem fine grid approximation is performed using a finite element method the main goal of the work is the construction of the coarse grid approximation using the generalized multiscale finite element method gmsfem in gmsfem we solve local spectral problem in order to determine main heat flow characteristics in the heterogeneous porous media and construct special additional multiscale basis functions in order to perform an accurate approximation of the line source terms freezing pipes we present numerical results for two heat transfer model problems in the two and three dimensional formulations and investigate accuracy of the proposed multiscale methods for different numbers of the multiscale basis functions

global smooth solutions of mhd equations with large data abstract in this paper we establish the global existence of smooth solutions of the three dimensional mhd system for a class of large initial data both the initial velocity and magnetic field can be arbitrarily large in the critical norm

global solutions of the navier stokes equations in thin three dimensional domains this paper is concerned with the spatially periodic navier stokes equations in a thin three dimensional domain it is shown that the classical solution exists globally when the initial velocityu0and the external forcefare chosen in the set significantly larger than those presented in earlier works what is more it is shown that these global solutions which seem large in a function space are small in nature

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

existence and generalized gevrey regularity of solutions to the kuramoto sivashinsky equation in rn abstract motivated by the work of foias and temam c foias r temam gevrey class regularity for the solutions of the navier stokes equations j funct anal 87 1989 359 369 we prove the existence and gevrey regularity of local solutions to the kuramoto sivashinsky equation in r n with initial data in the space of distributions the control on the gevrey norm provides an explicit estimate of the analyticity radius in terms of the initial data in the particular case when n 1 our analysis allows for initial data that are less smooth than that considered by grujic and kukavica z grujic i kukavica space analyticity for the navier stokes and related equations with initial data in l p j funct anal 152 1998 447 466

commutator estimate and its application to regularity criteria of the dissipative quasi geostrophic equation abstract a new commutator estimate with respect to a nonlinear convection upper bounded by a single partial derivative component in hilbert spaces is obtained as an application regularity criteria on the supercritical quasi geostrophic equation are obtained provided that solution growth conditions are assumed to involve a single partial derivative component

initial and boundary values for lαq lp solution of the navier stokes equations in the half space abstract in this paper we study the initial and boundary value problem of the navier stokes equations in the half space we prove the existence of weak solution u l q 0 l p r n 1 2 1 n p 2 q 0 n p with u l loc q 2 0 l loc p 2 r n for the solenoidal initial data h b p q 1 n p r n and the boundary data g l q 0 b p p 1 p r n 1 when h b p q 1 n p r n g l q 0 b p p 1 p r n 1 is small enough moreover the solution is unique in the class l q 0 t l p r n for any t if 0 and for some t if 0

forward self similar solutions of the fractional navier stokes equations abstract we study forward self similar solutions to the 3 d navier stokes equations with the fractional diffusion first we construct a global time forward self similar solutions to the fractional navier stokes equations with 5 6 1 for arbitrarily large self similar initial data by making use of the so called blow up argument moreover we prove that this solution is smooth in r 3 0 in particular when 1 we prove that the solution constructed by korobkov and tsai 2016 16 satisfies the decay estimate by establishing regularity of solution for the corresponding elliptic system which implies this solution has the same properties as a solution which was constructed in jia and sverak 2014 13

on the role of the besov spaces for the solutions of the generalized burgers equation in homogeneous sobolev spaces abstract the generalized burgers equation tu xxu xuk 1 0 with initial data u0 in homogeneous sobolev spaces are investigated the starting point of this work is the construction of solutions in l 0 b 1 k for small size initial data in b 1 k if in addition the initial data belongs to l p s then the obtained solution is actually in l 0 l p s

decay estimates of gradient of a generalized oseen evolution operator arising from time dependent rigid motions in exterior domains let us consider the motion of a viscous incompressible fluid past a rotating rigid body in three dimensions where the translational and angular velocities of the body are prescribed but time dependent in a reference frame attached to the body we have the navier stokes system with the drift and one half of the coriolis terms in a fixed exterior domain the existence of the evolution operator t t s in the space l q generated by the linearized non autonomous system was proved by hansel and rhandi j reine angew math 694 1 26 2014 and the large time behavior of t t s f in l r for t s rightarrow infty was then developed by hishida math ann 372 915 949 2018 when f is taken from l q with q leqq r the contribution of the present paper concerns such l q l r decay estimates of nabla t t s with optimal rates which must be useful for the study of stability attainability of the navier stokes flow in several physically relevant situations our main theorem completely recovers the l q l r estimates for the autonomous case stokes and oseen semigroups those semigroups with rotating effect in three dimensional exterior domains which were established by hishida and shibata arch ration mech anal 193 339 421 2009 iwashita math ann 285 265 288 1989 kobayashi and shibata math ann 310 1 45 1998 maremonti and solonnikov ann sc norm super pisa 24 395 449 1997 and shibata in amann arendt hieber neubrander nicaise von below eds functional analysis and evolution equations the gunter lumer volume birkhauser basel pp 595 611 2008

on l3 stability of the navier stokes system in exterior domains abstract this paper studies the stability of a stationary solution of the navier stokes system with a constant velocity at infinity in an exterior domain more precisely this paper considers the stability of the navier stokes system governing the stationary solution which belongs to the weak l 3 space l 3 under the condition that the initial datum belongs to a solenoidal l 3 space we prove that if both the l 3 norm of the initial datum and the l 3 norm of the stationary solution are sufficiently small then the system admits a unique global in time strong l 3 solution satisfying both l 3 asymptotic stability and l asymptotic stability moreover we investigate l 3 r asymptotic stability of the global in time solution using l p l q type estimates for the oseen semigroup and applying an equivalent norm on the lorentz space are key ideas to establish both the existence of a unique global in time strong or mild solution of our system and the stability of our solution

a general approach to time periodic incompressible viscous fluid flow problems this article develops a general approach to time periodic incompressible fluid flow problems and semilinear evolution equations it yields on the one hand a unified approach to various classical problems in incompressible fluid flow and on the other hand gives new results for periodic solutions to the navier stokes oseen flow the navier stokes flow past rotating obstacles and in the geophysical setting for ornstein uhlenbeck and various diffusion equations with rough coefficients the method is based on a combination of interpolation and topological arguments as well as on the smoothing properties of the linearized equation

energy decay for a weak solution of the navier stokes equation with slowly varying external forces we consider the navier stokes system with slowly decaying external forces formula we show that the energy norm of a weak solution has non uniform decay u t 2 0 t under suitable conditions on the data f and a which make the energy of solution bounded in time also we show the exact rate of the decay uniform decay of the energy u t 2 c 1 t e for external forces with a given explicit decay rate

time periodic strong solutions to the incompressible navier stokes equations with external forces of non divergence form abstract we discuss the time periodic problem to the incompressible navier stokes equations on the whole space r n n 3 with the external forces of non divergence form firstly we consider the existence of time periodic solutions in b c r l n r n assuming the smallness of external forces in b c r l 1 r 3 and b c r l n 3 r n in the case n 4 next we show that the mild solution above becomes a strong solution in the topology of l n r n with a natural condition of the external force derived from the strong solvability of the inhomogeneous stokes equations in l n r n for this aim we re construct a strong solvability of an abstract evolution equation where the associated semigroup is not strongly continuous at t 0

the navier stokes equations in the weak l n space with time dependent external force abstract we consider the navier stokes equations with time dependent external force either in the whole time or in positive time with initial data with domain either the whole space the half space or an exterior domain of dimension n ge 3 we give conditions on the external force sufficient for the unique existence of small solutions in the weak l n space bounded for all time in particular this result gives sufficient conditions for the unique existence and the stability of small time periodic solutions or almost periodic solutions this result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak l n space with time independent external force

generalized multiscale discontinuous galerkin method for solving the heat problem with phase change abstract in this work we consider a numerical solution of a heat transfer problem with phase change in heterogeneous domains for simulation of heat transfer processes with phase transitions we use a classic stefan model computational implementation is based on generalized multiscale discontinuous galerkin method gmsdgm in this method the interior penalty discontinuous galerkin method is used for the global coupling on a coarse grid the main idea of these methods is to construct a small dimensional local solution space that can provide an efficient calculation on coarse grid level we present numerical results for different geometries to demonstrate an accuracy of the method

ill posedness for the navier stokes equations in critical besov spaces b q 1 abstract we study the cauchy problem for the incompressible navier stokes equations in two and higher spatial dimensions 0 1 u t u u u p 0 div u 0 u 0 x u 0 for arbitrarily small 0 we show that the solution map u 0 u in critical besov spaces b q 1 q 1 2 is discontinuous at origin it is known that the navier stokes equation is globally well posed for small data in bmo 1 20 taking notice of the embedding b q 1 bmo 1 q 2 we see that for sufficiently small 0 u 0 b q 1 q 2 can guarantee that 0 1 has a unique global solution in bmo 1 however this solution is instable in b q 1 and the solution can have an inflation in b q 1 for certain initial data

note on prodi serrin ladyzhenskaya type regularity criteria for the navier stokes equations in this article we prove new regularity criteria of the prodi serrin ladyzhenskaya type for the cauchy problem of the three dimensional incompressible navier stokes equations our results improve the classical lr 0 t ls regularity criteria for both velocity and pressure by factors of certain negative powers of the scaling invariant norms u l 3 and u h 1 2

asymptotic behavior of solutions to viscous conservation laws with slowly varying external forces this paper considers the existence and large time behavior of solutions to the convection diffusion equation ut u b x u u q 1 f x t in n 0 where f x t is slowly decaying and q 1 1 n or in some particular cases q 1 the initial condition u0 is supposed to be in an appropriate lp space uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term

existence of strong solutions and decay of turbulent solutions of navier stokes flow with nonzero dirichlet boundary data abstract recently leray s problem of the l 2 decay of a special weak solution to the navier stokes equations with nonhomogeneous boundary values was studied by the authors exploiting properties of the approximate solutions converging to this solution in this paper this result is generalized to the case of an arbitrary weak solution satisfying the strong energy inequality

space time regularity of the koch tataru solutions to navier stokes equations abstract in koch tataru 2001 the authors have proved local and global well posedness of the navier stokes equations with small initial data u 0 b m o 1 and then the spatial analyticity of the koch tataru solution stated as t k 2 k u x t for any positive integer k has been presented by germain pavlovic staffilani 2007 where x t is the space defined by koch tataru 2001 see also definition 1 2 in this paper we shall present the space time regularity more precisely for any positive integer m k we have t m k 2 t m k u x t

minimal initial data for potential navier stokes singularities abstract assuming some initial data u 0 h 1 2 r 3 lead to a singularity for the 3d navier stokes equations we show that there are also initial data with the minimal h 1 2 norm which will produce a singularity

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

well posedness for the navier stokes nernst planck poisson system in triebel lizorkin space and besov space with negative indices abstract this paper is concerned with the well posedness of the navier stokes nerst planck poisson system nsnpp let s p 2 n p we prove that the nsnpp has a unique local solution u v w e u t e v t e v t for u 0 v 0 w 0 in a subspace i e v u 1 v v 1 v v 1 of f 1 2 b p s p b p s p with u 0 0 we also prove that there exists a unique small global solution u v w e u e v e v for any small initial data u 0 v 0 w 0 f 1 2 b p s p b p s p with u 0 0