brezis gallouet wainger type inequalities and blow up criteria for navier stokes equations in unbounded domains we shall find the weakest norm that satisfies the brezis gallouet wainger type inequality under some conditions as an application of the brezis gallouet wainger type inequality we shall establish beale kato majda type blow up criteria of smooth solutions to the 3 d navier stokes equations in unbounded domains

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

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

analyticity of the stokes semigroup in spaces of bounded functions the analyticity of the stokes semigroup with the dirichlet boundary condition is established in spaces of bounded functions when the domain occupied with fluid is bounded or more generally admissible which admits a special estimate for the helmholtz decomposition the proof is based on a blow up argument this is the first proof of the analyticity in spaces of bounded functions which was left open more than thirty years

global ln solution and its decay property for the navier stokes equations in half space r n soit r n x 1 x 2 x n 1 x n x n 0 n 2 dans q r n 0 on considere les equations de navier stokes u t u u u 0 dans q divu 0 dans q u x u n 0 0 u 0 a on demontre l existence et l unicite d une l n solution globale forte pour des donnees initiales petites

accounting for acoustic damping in a helmholtz solver thermoacoustic helmholtz solvers provide a cheap and efficient way of predicting combustion instabilities however because they rely on the inviscid euler equations at zero mach number they cannot properly describe the regions where aerodynamics may interact with acoustic waves in the vicinity of dilution holes and injectors for example a methodology is presented to incorporate the effect of non purely acoustic mechanisms into a three dimensional thermoacoustic helmholtz solver the zones where these mechanisms are important are modeled as two port acoustic elements and the corresponding matrices which notably contain the dissipative effects due to acoustic hydrodynamic interactions are used as internal boundary conditions in the helmholtz solver the rest of the flow domain where dissipation is negligible is solved by the classical helmholtz equation with this method the changes in eigenfrequency and eigenmode structure introduced by the acoustic hydrodynamic effects are captured while keeping the simplicity and efficiency of the helmholtz solver the methodology is successfully applied on an academic configuration first with a simple diaphragm then with an industrial swirler with matrices measured from experiments and large eddy simulation

total decoupling of general quadratic pencils part i theory abstract the notion of quadratic pencils 2 m c k where m c and k are n n real matrices with or without some additional properties such as symmetry or positive definiteness plays critical roles in many important applications it has been long desirable yet with very limited success to reduce a complicated high degree of freedom system to some simpler low degree of freedom subsystems recently garvey friswell and prells proposed a promising approach by which under some mild assumptions a general quadratic pencils can be converted by real valued isospectral transformations into a totally decoupled system this approach if numerically feasible would reduce the original n degree of freedom second order system to n totally independent single degree of freedom second order subsystems such a claim would be a striking breakthrough in the common knowledge that generally no three matrices m c and k can be diagonalized simultaneously this paper intends to serve three purposes to clarify some of the ambiguities in the original proposition to simplify some of the computational details and most importantly to complete the theory of existence by matrix polynomial factorization tactics

on stokes operators with variable viscosity in bounded and unbounded domains we consider a generalization of the stokes resolvent equation where the constant viscosity is replaced by a general given positive function such a system arises in many situations as linearized system when the viscosity of an incompressible viscous fluid depends on some other quantities we prove that an associated stokes like operator generates an analytic semi group and admits a bounded h calculus which implies the maximal l q regularity of the corresponding parabolic evolution equation the analysis is done for a large class of unbounded domains with w 2 frac1r r boundary for some r d with r q q in particular the existence of an l q helmholtz projection is assumed

liouville theorems for the stokes equations with applications to large time estimates abstract we study liouville theorems for the non stationary stokes equations in exterior domains in r n under decay conditions for spatial variables as applications we prove that the stokes semigroup is a bounded analytic semigroup on l of angle 2 for n 3 we also prove large time estimates for n 2 with zero net force

on a class of stable steady flows to the exterior convection problem abstract a class of stable steady convection flows in an exterior domain is given it is proved that the class is l 3 w l m w with m m m 1 when the gravitational field is in l m w for some 1 m

a well posed problem for the exterior stokes equations in two and three dimensions this paper treats the stokes problem in exterior lipschitz continuous domains of 2 and 3 using the weighted sobolev spaces of hanouzet in 3 and giroire in 2 we establish the inf sup condition between the velocity and pressure spaces this fundamental result shows that the variational stokes problem is well posed in those spaces in the last paragraph we obtain additional regularity of the solution when the data are smoother

l p theory of the navier stokes flow in the exterior of a moving or rotating obstacle in this paper we describe two recent approaches for the lp theory of the navier stokes flow in the exterior of a moving or rotating obstacle

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

the navier stokes equations in exterior lipschitz domains lp theory abstract we show that the stokes operator defined on l p for an exterior lipschitz domain r n n 3 admits maximal regularity provided that p satisfies 1 p 1 2 1 2 n e for some e 0 in particular we prove that the negative of the stokes operator generates a bounded analytic semigroup on l p for such p in addition l p l q mapping properties of the stokes semigroup and its gradient with optimal decay estimates are obtained this enables us to prove the existence of mild solutions to the navier stokes equations in the critical space l 0 t l 3 locally in time and globally in time for small initial data

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

l q l r estimates for solutions of the nonstationary stokes equations in an exterior domain and the navier stokes initial value problems inl q spaces