on the structural stability of the euler voigt and navier stokes voigt models abstract we consider the euler voigt equations and the navier stokes voigt equations which are obtained by an inviscid regularization from the corresponding equations the main result we show is the structural stability of the system in terms of the variations of both viscosity and regularization parameters

local well posedness of the incompressible euler equations in b 11 and the inviscid limit of the navier stokes equations abstract we prove the inviscid limit of the incompressible navier stokes equations in the same topology of besov spaces as the initial data the proof is based on proving the continuous dependence of the navier stokes equations uniformly with respect to the viscosity to show the latter we rely on some bona smith type argument in the l p setting our obtained result implies a new result that the cauchy problem of the euler equations is locally well posed in the borderline besov space b 1 1 r d d 2 in the sense of hadmard which is an open problem left in recent works by bourgain and li in 3 4 and by misiolek and yoneda in 12 13 14

on the vanishing viscosity limit for the 3d navier stokes equations with a slip boundary condition where and below and denote the div and curl operators respectively n is the outward normal and is the unit tangential vector of the investigation of vanishing viscosity limit of solutions of the navier stokes equations both in the two and three spacial dimensional cases is a classical issue there are two related questions arising from here one is how to describe the inviscid limiting behavior of the navier stokes equation and the other is that does the euler equation can be approximated by the navier stokes equations in the case that the solution to the this research is supported in part by zheng ge ru foundation and hong kong rgc earmarked research grants cuhk 4028 04p cuhk 4040 02p and cuhk 4279 00p

a characterization of bipartite distance regular graphs abstract it is well known that the halved graphs of a bipartite distance regular graph are distance regular examples are given to show that the converse does not hold thus a natural question is to find out when the converse is true in this paper we give a quasi spectral characterization of a connected bipartite weighted 2 punctually distance regular graph whose halved graphs are distance regular in the case the spectral diameter is even we show that the graph characterized above is distance regular

on the stationary solutions and inviscid limit for the generalized proudman johnson equation with o 1 forcing abstract we consider the stationary problem for the generalized proudman johnson equation x x x a x x x x x x x f where a r and 0 are parameters and and f satisfy the periodic boundary conditions on we establish the existence of a time dependent solution and the uniqueness of the stationary solution for some large then we investigate the behavior of solution sequences in the limit 0 fixing a and f where bifurcations appear and new solutions emerge in particular we analyze if there is convergence towards stationary solutions for the corresponding inviscid equation

initial boundary value problem for two dimensional viscous boussinesq equations we study the initial boundary value problem of two dimensional viscous boussinesq equations over a bounded domain with smooth boundary we show that the equations have a unique classical solution for h3 initial data and the no slip boundary condition in addition we show that the kinetic energy is uniformly bounded in time

nonlinear evolution equations and the euler flow abstract a new theorem on abstract nonlinear equations of evolution is proved as an application the existence uniqueness regularity and continuous dependence on the data are proved for the solution of the euler equation for incompressible fluids in a bounded domain in r m

on the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half plane we consider the navier stokes equations for viscous incompressible flows in the half plane under the no slip boundary condition by using the vorticity formulation we prove the local in time convergence of the navier stokes flows to the euler flows outside a boundary layer and to the prandtl flows in the boundary layer in the inviscid limit when the initial vorticity is located away from the boundary 2014 wiley periodicals inc

boundary layers associated with incompressible navier stokes equations the noncharacteristic boundary case abstract the goal of this article is to study the boundary layer of wall bounded flows in a channel at small viscosity when the boundaries are uniformly noncharacteristic i e there is injection and or suction everywhere at the boundary following earlier work on the boundary layer for linearized navier stokes equations in the case where the boundaries are characteristic no slip at the boundary and non permeable we consider here the case where the boundary is permeable and thus noncharacteristic the form of the boundary layer and convergence results are derived in two cases linearized equation and full nonlinear equations we prove that there exists a boundary layer at the outlet downwind of the form e uz e where u is the speed of injection suction at the boundary z is the distance to the outlet of the channel and e is the kinematic viscosity we improve an earlier result of s n alekseenko 1994 siberian math j 35 no 2 209 230 where the convergence in l 2 of the solutions of the navier stokes equations to that of the euler equations at vanishing viscosity was established in the two dimensional case we are able to derive the physically relevant uniform in space l norm estimates of the boundary layer the uniform in space estimate is derived by properly developing our previous idea of better control on the tangential derivative and the use of an anisotropic sobolev imbedding to the best of our knowledge this is the first rigorously proved result concerning boundary layers for the full nonlinear navier stokes equations for incompressible fluids

geometric regularity criteria for incompressible navier stokes equations with navier boundary conditions abstract we study the regularity criteria for weak solutions to the 3d incompressible navier stokes equations in terms of the direction of vorticity taking into account the boundary conditions a boundary regularity theorem is proved on regular curvilinear domains with a family of oblique derivative boundary conditions provided that the directions of vorticity are coherently aligned up to the boundary as an application we establish the boundary regularity for weak solutions to navier stokes equations in round balls half spaces and right circular cylindrical ducts subject to the classical navier and kinematic boundary conditions

suitable weak solutions to the 3d navier stokes equations are constructed with the voigt approximation abstract in this paper we consider the navier stokes equations supplemented with either the dirichlet or vorticity based navier slip boundary conditions we prove that weak solutions obtained as limits of solutions of the navier stokes voigt model satisfy the local energy inequality and we also prove certain regularity results for the pressure moreover in the periodic setting we prove that if the parameters are chosen in an appropriate way then we can construct suitable weak solutions through a fourier galerkin finite dimensional approximation in the space variables

reducing slip boundary value problems from the half to the whole space applications to inviscid limits and to non newtonian fluids abstract the study of a very large class of linear and non linear stationary and evolutive partial differential problems in the half space or similar under the slip boundary condition is reduced here to the much simpler study of the corresponding results for the same problem in the whole space the approach is particularly suitable for proving new results in strong norms to determine whether this extension is available turns out to be a simple exercise the verification depends on a few general features of the functional space x related to the space variables hence we present an approach as much as possible independent of the particular space x we appeal to a reflection technique hence a crucial assumption is to be in the presence of flat boundaries see below instead of stating general theorems we rather prefer to illustrate how to apply our results by considering a couple of interesting problems as a main example we show that sharp vanishing viscosity limit results that hold for the evolution navier stokes equations in the whole space can be extended to the slip boundary value problem in the half space we also show some applications to non newtonian fluid problems

on the roughness induced effective boundary conditions for an incompressible viscous flow abstract we consider the laminar viscous channel flow with the lateral surface of the channel containing surface irregularities it is supposed that a uniform pressure gradient is maintained in the longitudinal direction of the channel after studying the corresponding boundary layers we obtain rigorously the navier friction condition it is valid when the size and amplitude of the imperfections tend to zero furthermore the coefficient in the law is determined through an auxiliary boundary layer type problem and the tangential drag force and the effective mass flow are determined up to order o e3 2 the value of the effective coefficient is shown to be independent with respect to the position of the mean surface in the range of o e

the limit of vanishing viscosity for the incompressible 3d navier stokes equations with helical symmetry abstract in this paper we are concerned with the vanishing viscosity problem for the three dimensional navier stokes equations with helical symmetry in the whole space we choose viscosity dependent initial u 0 with helical swirl an analogue of the swirl component of axisymmetric flow of magnitude o in the l 2 norm we assume u 0 u 0 in h 1 the new ingredient in our analysis is a decomposition of helical vector fields through which we obtain the required estimates

a well posedness theorem for non homogeneous inviscid fluids via a perturbation theorem on applique un theoreme de perturbation pour montrer le caractere bien pose d equations hyperboliques non lineaires application a une equation d ecoulement

local in time existence and uniqueness of solutions to the prandtl equations by energy methods we prove local existence and uniqueness for the two dimensional prandtl system in weighted sobolev spaces under oleinik s monotonicity assumption in particular we do not use the crocco transform or any change of variables our proof is based on a new nonlinear energy estimate for the prandtl system this new energy estimate is based on a cancellation property that is valid under the monotonicity assumption to construct the solution we use a regularization of the system that preserves this nonlinear structure this new nonlinear structure may give some insight into the convergence properties from the navier stokes system to the euler system when the viscosity goes to 0 2015 wiley periodicals inc

the laplacian spectral excess theorem for distance regular graphs the spectral excess theorem states that in a regular graph the average excess which is the mean of the numbers of vertices at maximum distance from a vertex is bounded above by the spectral excess a number that is computed by using the adjacency spectrum of and is distance regular if and only if equality holds in this note we prove the corresponding result by using the laplacian spectrum without requiring regularity of

inviscid limit for damped and driven incompressible navier stokes equations in mathbb r 2 we consider the zero viscosity limit of long time averages of solutions of damped and driven navier stokes equations in mathbb r 2 we prove that the rate of dissipation of enstrophy vanishes stationary statistical solutions of the damped and driven navier stokes equations converge to renormalized stationary statistical solutions of the damped and driven euler equations these solutions obey the enstrophy balance

initial boundary layer associated with the nonlinear darcy brinkman oberbeck boussinesq system abstract in this paper we study the vanishing darcy number limit of the nonlinear darcy brinkman oberbeck boussinesq system dbob this singular perturbation problem involves singular structures both in time and in space giving rise to initial layers boundary layers and initial boundary layers we construct an approximate solution to the dbob system by the method of multiple scale expansions the convergence with optimal convergence rates in certain sobolev norms is established rigorously via the energy method

a review on some contributions to perturbation theory singular limits and well posedness abstract in the beginning of the 1990s we devoted a sequence of papers to perturbation theory singular limits and well posedness problems in particular the strong well posedness of the initial boundary value problem for the compressible euler equations was demonstrate for the first time our method also allowed singular limit results in the strong norm even under assumptions weaker than the current ones in the literature where the strong norm is not reached it is worth noting that until now the above method and results have not been substantially improved hence an introduction to it still looks timely actually in a forthcoming paper by returning to this method we improve in a very substantial way some important results recently appeared in the literature

perturbation theorems for linear hyperbolic mixed problems and applications to the compressible euler equations the main result of this paper which is completely new apart from our previous and less general result proved in reference 9 states that the nonlinear system of equations 1 11 or equivalently 1 10 that describes the motion of an inviscid compressible barotropic fluid in a bounded domain gives rise to a strongly well posed problem in the hadamard classical sense in spaces hk k 3 see theorem 1 4 below roughly speaking if an n a in hk hk and if fn f in 2 0 t hk then vn gn v g in 0 t hk hk the method followed here see also 8 also applies to the non barotropic case p p s see 10 and to other nonlinear problems these results are based upon an improvement of the structural stability theorem for linear hyperbolic equations see theorem 1 2 below added in proof the reader is referred to 29 part i for a concise explanation of some fundamental points in the method followed here 1993 john wiley sons inc

zero viscosity limit for analytic solutions of the navier stokes equation on a half space i existence for euler and prandtl equations this is the first of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in this paper we prove short time existence theorems for the euler and prandtl equations with analytic initial data in either two or three spatial dimensions the main technical tool in this analysis is the abstract cauchy kowalewski theorem for the euler equations the projection method is used in the primitive variables to which the cauchy kowalewski theorem is directly applicable for the prandtl equations cauchy kowalewski is applicable once the diffusion operator in the vertical direction is inverted

interaction of vortices in weakly viscous planar flows we consider the inviscid limit for the two dimensional incompressible navier stokes equation in the particular case where the initial flow is a finite collection of point vortices we suppose that the initial positions and the circulations of the vortices do not depend on the viscosity parameter and we choose a time t 0 such that the helmholtz kirchhoff point vortex system is well posed on the interval 0 t under these assumptions we prove that the solution of the navier stokes equation converges as 0 to a superposition of lamb oseen vortices whose centers evolve according to a viscous regularization of the point vortex system convergence holds uniformly in time in a strong topology which allows us to give an accurate description of the asymptotic profile of each individual vortex in particular we compute to leading order the deformations of the vortices due to mutual interactions this makes it possible to estimate the self interactions which play an important role in the convergence proof

initial boundary layer associated with the nonlinear darcy brinkman system abstract we study the interaction of initial layer and boundary layer in the nonlinear darcy brinkman system in the vanishing darcy number limit in particular we show the existence of a function of corner layer type so called initial boundary layer in the solution of the nonlinear darcy brinkman system an approximate solution is constructed by the method of multiple scale expansion in space and in time we establish the optimal convergence rates in various sobolev norms via energy method

vanishing viscosity limit for the 3d magnetohydrodynamic system with a slip boundary condition abstract this work investigates the solvability regularity and vanishing viscosity limit of the 3d viscous magnetohydrodynamic system in a class of bounded domains with a slip boundary condition

the inviscid limit in the cauchy problem of the inhomogeneous incompressible navier stokes equations abstract applying similar arguments as introduced in bona smith 4 we mainly study the inviscid limit of the inhomogeneous incompressible navier stokes equations in r d some results of local uniform estimates on solutions of the navier stokes equations independent of the viscosity are also obtained

navier stokes equations green s matrices vorticity direction and regularity up to the boundary abstract we consider the initial boundary value problem for the 3d navier stokes equations the physical domain is a bounded open set with a smooth boundary on which we assume a condition of free boundary type we show that if a suitable hypothesis on the vorticity direction is assumed then weak solutions are regular the main tool we use in the proof is an explicit representation of the velocity in terms of the vorticity by means of green s matrices

boundary layer analysis of the navier stokes equations with generalized navier boundary conditions we study the weak boundary layer phenomenon of the navier stokes equations with generalized navier friction boundary conditions u n 0 s u n tan au 0 in a bounded domain in r3 when the viscosity e 0 is small here s u is the symmetric gradient of the velocity u and a is a type 1 1 tensor on the boundary when a i we obtain navier boundary conditions and when a is the shape operator we obtain the conditions u n curlu n 0 by constructing an explicit corrector we prove the convergence as e tends to zero of the navier stokes solutions to the euler solution both in the natural energy norm and uniformly in time and space

couette flows over a rough boundary and drag reduction we consider the couette flow between two plates the lower plate is fixed and has periodically placed riblets of the characteristic size on it in the limit 0 we find the effective couette navier flow as an o 2 approximation for the effective mass flow and an o 2 l 1 approximation for the velocity in the effective solution the effect of roughness enters through the navier slip condition with the matrix coefficient in front of the effective shear stress calculated using a boundary layer problem furthermore an o 2 approximation for the tangential drag force is found in all estimates explicit dependence on the kinematic viscosity the velocity of the upper plate and the distance between the plates l 3 is kept also the uniqueness of the solution is expressed through a non linear algebraic condition linking and l 3 then the result is applied to the viscous sub layers around immersed bodies strictly containing the surface riblets it is found that for the riblets of the characteristic size being of the order smaller or equal to the approximation obtained for the tangential drag could be applied we compare and for realistic data and our results lead to the conclusion that the riblets reduce significantly tangential drag which may explain their presence on the skin of nektons

existence and uniqueness of two dimensional euler poisson system and wkb approximation to the nonlinear schrodinger poisson system in this paper we study a dispersive euler poisson system in two dimensional euclidean space our aim is to show unique existence and the zero dispersion limit of the time local weak solution since one may not use dispersive structure in the zero dispersion limit when reducing the regularity lack of critical embedding h1 l becomes a bottleneck we hence employ an estimate on the best constant of the gagliardo nirenberg inequality by this argument a reasonable convergence rate for the zero dispersion limit is deduced with a slight loss we also consider the semiclassical limit problem of the schrodinger poisson system in two dimensions

on the euler equations of incompressible fluids euler equations of incompressible fluids use and enrich many branches of mathematics from integrable systems to geometric analysis they present important open physical and mathematical problems examples include the stable statistical behavior of ill posed free surface problems such as the rayleightaylor and kelvin helmholtz instabilities the paper describes some of the open problems related to the incompressible euler equations with emphasis on the blowup problem the inviscid limit and anomalous dissipation some of the recent results on the quasigeostrophic model are also mentioned

note on loss of regularity for solutions of the 3 d incompressible euler and related equations one of the central problems in the mathematical theory of turbulence is that of breakdown of smooth indefinitely differentiable solutions to the equations of motion in 1934 j leray advanced the idea that turbulence may be related to the spontaneous appearance of singularities in solutions of the 3 d incompressible navier stokes equations the problem is still open we show in this report that breakdown of smooth solutions to the 3 d incompressible slightly viscous i e corresponding to high reynolds numbers or highly turbulent navier stokes equations cannot occur without breakdown in the corresponding solution of the incompressible euler ideal fluid equation we prove then that solutions of distorted euler equations which are equations closely related to the euler equations for short term intervals do breakdown

viscous boundary layers for the navier stokes equations with the navier slip conditions we tackle the issue of the inviscid limit of the incompressible navier stokes equations when the navier slip with friction conditions are prescribed on impermeable boundaries we justify an asymptotic expansion which involves a weak amplitude boundary layer with the same thickness as in prandtl s theory and a linear behavior this analysis holds for general regular domains in both dimensions two and three

boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems abstract in this paper we study viscous perturbations of quasilinear hyperbolic systems in several dimensions as the viscosity goes to zero the boundary is noncharacteristic for the hyperbolic system we in particular describe the boundary layer which arises near the boundary and give a sufficient condition for the convergence of the solution to the solution of some mixed hyperbolic problem with some nonlinear maximal dissipative boundary conditions a counterexample is given when this condition is not satisfied and the solution blows up as the viscosity goes to 0

a new boundary condition for the three dimensional navier stokes equation and the vanishing viscosity limit in this paper we propose a new vorticity boundary condition for the three dimensional incompressible navier stokes equation for a general smooth domain in r3 this boundary condition is motivated by the generalized navier slip boundary condition involving the vorticity it is shown first that such an initial boundary value problem is well posed at least local in time furthermore more importantly we obtain some estimates on rate of convergence in c 0 t h1 and c 0 t h2 of the solutions to the corresponding solutions of the ideal euler equations with the standard slip boundary condition

a variational approach for the vector potential formulation of the stokes and navier stokes problems in three dimensional domains abstract a few well posed variational problems are constructed whose solutions are vector potentials of the velocity field of the three dimensional stokes problem a choice of the adequate boundary conditions to be imposed is performed in a systematic manner even in the case of not simply connected domains the test functions do not have to be divergence free in the formulations given here it is then seen that a similar approach is well suited to the treatment of the non linear navier stokes problem

a simple proof of the spectral excess theorem for distance regular graphs the spectral excess theorem provides a quasi spectral characterization for a regular graph with d 1 distinct eigenvalues to be distance regular graph in terms of the excess number of vertices at distance d of each of its vertices the original approach due to fiol and garriga in 1997 was obtained by using a local approach so giving a characterization of the so called pseudo distance regularity around a vertex in this paper we present a new simple projection method based in a global point of view and where the mean excess plays an essential role

the euler limit of the navier stokes equations and rotating fluids with boundary in this paper we study the convergence of weak solutions of the navier stokes equations in some particular domains with different horizontal and vertical viscosities when they go to zero with different speeds the difficulty here comes from the dirichlet boundary conditions precisely we show that if the ratio of the vertical viscosity to the horizontal viscosity also goes to zero then the solutions converge to the solution of the euler system we study the same limit for rotating fluids with rossby number also going to zero

inviscid limit for the damped boussinesq equation abstract in this paper we consider the limit behavior of solutions to the cauchy problem for damped boussinesq equation in the regime of small viscosity in r n it is shown that the strong solutions of damped boussinesq equation converge to a strong solution of boussinesq equation in any time interval 0 t the key idea is that we only take into account the effect of dispersion and overlook the effect of dissipation in the damped boussinesq equation which is a kind of new view we analyze the dispersion by the techniques of high low frequency decomposition and littlewood paley dyadic decomposition stationary phase estimate and some properties of bessel function

boundary layer associated with the darcy brinkman boussinesq model for convection in porous media abstract we study the asymptotic behavior of the infinite darcy prandtl number darcy brinkman boussinesq system for convection in porous media at a small brinkman darcy number the existence of a boundary layer with thickness proportional to the square root of the brinkman darcy number for the velocity field is established in both the l h 1 norm in 2 and 3 d and the l l norm in 2d this improves in several respects an earlier result of payne and straughan 1998 60 where the vanishing brinkman darcy number limit is studied without resolving the boundary layer

on the vanishing viscosity limit of 3d navier stokes equations under slip boundary conditions in general domains we consider the vanishing viscosity limit for the navier stokes equations with certain slip without friction boundary conditions in a bounded domain with non flat boundary in particular we are able to show convergence in strong norms for a solution starting with initial data belonging to the special subclass of data with vanishing vorticity on the boundary the proof is obtained by smoothing the initial data and by a perturbation argument with quite precise estimates for the equations of the vorticity and for that of the curl of the vorticity

effective boundary conditions for laminar flows over periodic rough boundaries effective boundary conditions or wall laws are proposed for a laminar flow over a rough wall with periodic roughness elements these effective conditions are posed on a regularized boundary which allows the details of the wall to be avoided and dramatically reduces the computational cost the effective conditions stem from an asymptotic expansion of the solution which is presented here both first and second order conditions are discussed and tested numerically on bidimensional cases

zero viscosity limit for analytic solutions of the navier stokes equation on a half space ii construction of the navier stokes solution this is the second of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in either 2d or 3d under the assumption of analytic initial data we construct solutions of navier stokes for a short time which is independent of the viscosity the navier stokes solution is constructed through a composite asymptotic expansion involving the solutions of the euler and prandtl equations which were constructed in the first paper plus an error term this shows that the navier stokes solution goes to an euler solution outside a boundary layer and to a solution of the prandtl equations within the boundary layer the error term is written as a sum of first order euler and prandtl corrections plus a further error term the equation for the error term is weakly nonlinear its linear part is the time dependent stokes equation this error equation is solved by inversion of the stokes equation through expressing the solution as a regular euler like part plus a boundary layer prandtl like part the main technical tool in this analysis is the abstract cauchy kowalewski theorem

zero viscosity limit for analytic solutions of the primitive equations the aim of this paper is to prove that the solutions of the primitive equations converge in the zero viscosity limit to the solutions of the hydrostatic euler equations we construct the solution of the primitive equations through a matched asymptotic expansion involving the solution of the hydrostatic euler equation and boundary layer correctors as the first order term and an error that we show to be o sqrt nu the main assumption is spatial analyticity of the initial datum

uniform regularity for the navier stokes equation with navier boundary condition we prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the navier stokes equation with the navier boundary condition has a strong solution this solution is uniformly bounded in a conormal sobolev space and has only one normal derivative bounded in l this allows us to obtain the vanishing viscosity limit to the incompressible euler system from a strong compactness argument

weak strong uniqueness for measure valued solutions we prove the weak strong uniqueness for measure valued solutions of the incompressible euler equations these were introduced by diperna and majda in their landmark paper commun math phys 108 4 667 689 1987 where in particular global existence to any l2 initial data was proven whether measure valued solutions agree with classical solutions if the latter exist has apparently remained open

zygmund spaces inviscid limit and uniqueness of euler flows the paper improves the classical uniqueness result for the incompressible euler system in the n dimensional case assuming that nabla u e in l 1 0 t it bmo omega only moreover the rate of the convergence for the inviscid limit of solutions to the navier stokes equations is obtained under the same regularity of the limit eulerian flow a key element of the proof is a logarithmic inequality between the hardy and l 1 spaces which is a consequence of the basic properties of the zygmund space l ln l

global regularity for vortex patches we present a proof of chemin s 4 result which states that the boundary of a vortex patch remains smooth for all time if it is initially smooth

a class of posets with integral laplacian spectrum abstract for any positive integers k r and s we present a class of posets denoted by p r s k we constructively prove that the laplacian matrices for them have integral spectrum

on the blow up criterion of 3d navier stokes equations abstract in this paper we prove some properties of the maximal solution of navier stokes equations if the maximum time t is finite we establish that the growth of u t h s is at least of the order of t t s 3 see eq 1 4 also we give some new blow up results specific properties and standard techniques are used

vanishing viscosity limit for homogeneous axisymmetric no swirl solutions of stationary navier stokes equations abstract 1 homogeneous axisymmetric no swirl solutions of three dimensional incompressible stationary navier stokes equations which are smooth on the unit sphere minus the north and south poles have been classified in this paper we study the vanishing viscosity limit of sequences of these solutions as the viscosity tends to zero some sequences of solutions c l o c m converge to solutions of euler equations on the sphere minus the poles while for other sequences of solutions transition layer behaviors occur for every latitude circle there are sequences which c l o c m converge respectively to different solutions of the euler equations on the spherical caps above and below the latitude circle we give detailed analysis of these convergence and transition layer behaviors

zero viscosity limit of the incompressible navier stokes equations with sharp vorticity gradient abstract it is well known that the 3d incompressible euler equations admit some local in time solutions for which the vorticity is piecewise smooth and discontinuous across a smooth time dependent hypersurface which evolves with the flow in this paper we prove that such a solution can be obtained as zero viscosity limit of strong solutions to the navier stokes equations whose vorticity has sharp variations near the hypersurface associated with the inviscid limit indeed we exhibit some sequences of exact solutions to the navier stokes equations with vanishing viscosity which are given by multi scale asymptotic expansions involving some characteristic boundary layers given by some linear pdes the convergence and the validity of the expansion are guaranteed on the time interval associated with the solution to the euler equations

vanishing diffusivity limit for the 3d heat conductive boussinesq equations with a slip boundary condition abstract in this paper we consider the 3d heat conductive boussinesq equations with a slip boundary condition for the velocity field and neumann boundary condition for the temperature we prove that there exists an interval of time which is uniform for the heat conductivity coefficient and for which the 3d heat conductive boussinesq equations have a strong solution the solution is uniformly bounded in some spaces with respect to based on these uniform estimates we establish the vanishing diffusivity limit for the boussinesq equations and also obtain the convergence rates for the velocity field and the temperature

uniform regularity and vanishing viscosity limit for the free surface navier stokes equations we study the inviscid limit of the free boundary navier stokes equations we prove the existence of solutions on a uniform time interval by using a suitable functional framework based on sobolev conormal spaces this allows us to use a strong compactness argument to justify the inviscid limit our approach does not rely on the justification of asymptotic expansions in particular we get a new existence result for the euler equations with free surface from the one for navier stokes

attractors of the velocity vorticity voigt model of the 3d navier stokes equations with damping abstract in this paper we prove the existence of global and exponential attractors of optimal regularity of a regularization for the three dimensional navier stokes equations with damping which is called the three dimensional velocity vorticity voigt vvv system proposed by larios pei and rebholz larios et al 2019 differently there is no voigt term 2 a w t in the second equation of system 1 1 below the proof is based on some energy estimates in sobolev spaces and the semigroup decomposition

local and global existence of smooth solutions for the stochastic euler equations with multiplicative noise we establish the local existence of pathwise solutions for the stochastic euler equations in a three dimensional bounded domain with slip boundary conditions and a suitable nonlinear multiplicative noise in the two dimensional case we obtain the global existence of these solutions with additive or linear multiplicative noise finally we show that in the three dimensional case the addition of linear multiplicative noise provides a regularizing effect the global existence of solutions occurs with high probability if the initial data is sufficiently small or if the noise coefficient is sufficiently large

from the boltzmann equation to the stokes fourier system in a bounded domain we prove that the renormalized solutions of the boltzmann equation considered in a bounded domain with different types of kinetic boundary conditions converge to the stokes fourier system with different types of fluid boundary conditions when the main free path goes to zero this extends the work of f golse and d levermore 9 to the case of a bounded domain 2003 wiley periodicals inc

remarks about the inviscid limit of the navier stokes system in this paper we prove two results about the inviscid limit of the navier stokes system the first one concerns the convergence in hs of a sequence of solutions to the navier stokes system when the viscosity goes to zero and the initial data is in hs the second result deals with the best rate of convergence for vortex patch initial data in 2 and 3 dimensions we present here a simple proof which also works in the 3d case the 3d case is new

uniform regularity and relaxation limit for the outer pressure problem of gas dynamics with several thermal nonequilibrium modes abstract we study in this paper the uniform regularity and relaxation limit for the outer pressure problem of one dimensional gas dynamics with several thermal nonequilibrium modes the uniform regularity is established for the initial data satisfying the compatible conditions on the boundary the relaxation limit to equilibrium flow or frozen flow is also investigated by a compactness argument

on the convergence rate of the euler α an inviscid second grade complex fluid model to the euler equations we study the convergence rate of the solutions of the incompressible euler an inviscid second grade complex fluid equations to the corresponding solutions of the euler equations as the regularization parameter approaches zero first we show the convergence in hs s n 2 1 in the whole space and that the smooth euler solutions exist at least as long as the corresponding solution of the euler equations next we estimate the convergence rate for two dimensional vortex patch with smooth boundaries

vanishing viscosity limit to the planar rarefaction wave for the two dimensional full compressible navier stokes equations abstract the vanishing viscosity limit for the multi dimensional compressible navier stokes equations to the corresponding euler equations is a difficult and challenging problem in the mathematics recently l li d wang and y wang 17 verified that the solutions for the 2d compressible isentropic navier stokes equations converge to the planar rarefaction wave solution for the corresponding 2d euler equations as viscosity vanishes with a convergence rate 1 6 ln in this paper the vanishing viscosity limit of 2d non isentropic compressible navier stokes equations is studied in contrast to the work 17 the convergence rate for the 2d full compressible navier stokes equations is improved to 2 7 ln 2 by choosing a different scaling argument and performing more detailed energy estimates

boundary layer effects for the 2 d boussinesq equations with vanishing diffusivity limit in the half plane nsfc 40890150 10801111 10971171 national basic research program 2005cb321700 natural science foundation of fujian province of china 2010j05011 fundamental research funds for the central universities 2010121006

some properties on the surfaces of vector fields and its application to the stokes and navier stokes problems with mixed boundary conditions abstract in this paper we are concerned with the stationary and non stationary stokes and navier stokes problems with mixed boundary conditions involving velocity pressure rotation stress and normal derivative of velocity together relying on the relations among strain rotation normal derivative of velocity and shape of boundary surface we get variational formulations of the stokes and navier stokes problems with the mixed boundary conditions also we study existence and uniqueness of solutions to the problems

global regularity for the 2d boussinesq equations with partial viscosity terms abstract in this paper we prove the global in time regularity for the 2d boussinesq system with either the zero diffusivity or the zero viscosity we also prove that as diffusivity viscosity tends to zero the solutions of the fully viscous equations converge strongly to those of zero diffusion viscosity equations our result for the zero diffusion system in particular solves the problem no 3 posed by moffatt in r l ricca ed kluwer academic publishers dordrecht the netherlands 2001 pp 3 10

regularity criteria for the three dimensional magnetohydrodynamic equations nsfc 11171154 11271184 11271305 10976026 papd fundamental research funds for the central universities ncet 11 0227

uniform regularity and vanishing dissipation limit for the full compressible navier stokes system in three dimensional bounded domain in the present paper we study the uniform regularity and vanishing dissipation limit for the full compressible navier stokes system whose viscosity and heat conductivity are allowed to vanish at different orders the problem is studied in a three dimensional bounded domain with navier slip type boundary conditions it is shown that there exists a unique strong solution to the full compressible navier stokes system with the boundary conditions in a finite time interval which is independent of the viscosity and heat conductivity the solution is uniformly bounded in w 1 infty and is a conormal sobolev space based on such uniform estimates we prove the convergence of the solutions of the full compressible navier stokes to the corresponding solutions of the full compressible euler system in l infty 0 t l 2 l infty 0 t h 1 and l infty 0 t times omega with a rate of convergence