Maximal L p – L q -Estimates for the Stokes Equation: a Short Proof of Solonnikov’s Theorem

Introducing a new localization method involving Bogovskiĭ's operator we give a short and new proof for maximal L^p – L^q-estimates for the solution of the Stokes equation. Moreover, it is shown that, up to constants, the Stokes operator is an R{\mathcal{R}}-sectorial operator in L^p_s(W)L^{p}_{\sigma}(\Omega), 1 < p < ￥1 < p < \infty, of R{\mathcal{R}}-angle 0, for bounded or exterior domains of Ω.     

On the Navier�CStokes equations with rotating effect and prescribed outflow velocity

We consider the equations of Navier–Stokes modeling viscous fluid flow past a moving or rotating obstacle in \mathbb R^d{\mathbb R^d} subject to a prescribed velocity condition at infinity. In contrast to previously known results, where the prescribed velocity vector is assumed to be parallel to the axis of rotation, in this paper we are interested in a general outflow velocity. In order to use L ^p -techniques we introduce a new coordinate system, in which we obtain a non-autonomous partial differential equation with an unbounded drift term. We prove that the linearized problem in \mathbb R^d{\mathbb R^d} is solved by an evolution system on L^p_s(\mathbb R^d){L^p_{\sigma}(\mathbb R^d)} for 1 < p < ∞. For this we use results about time-dependent Ornstein–Uhlenbeck operators. Finally, we prove, for p ≥ d and initial data u₀ ? L^p_s(\mathbb R^d){u_0\in L^p_{\sigma}(\mathbb R^d)}, the existence of a unique mild solution to the full Navier–Stokes system.     

The Fundamental Solution of the Linearized Navier–Stokes Equations for Spinning Bodies in Three Spatial Dimensions – Time Dependent Case

Explicit formulae for the fundamental solution of the linearized time dependent Navier–Stokes equations in three spatial dimensions are obtained. The linear equations considered in this paper include those used to model rigid bodies that are translating and rotating at a constant velocity. Estimates extending those obtained by Solonnikov in [23] for the fundamental solution of the time dependent Stokes equations, corresponding to zero translational and angular velocity, are established. Existence and uniqueness of solutions of these linearized problems is obtained for a class of functions that includes the classical Lebesgue spaces L^p(R³), 1 < p < ∞. Finally, the asymptotic behavior and semigroup properties of the fundamental solution are established.     

Optimal Decay Rate of the Compressible Navier–Stokes–Poisson System in {\mathbb {R}^3}

The compressible Navier–Stokes–Poisson (NSP) system is considered in ${\mathbb {R}^3}The compressible Navier–Stokes–Poisson (NSP) system is considered in \mathbb R³{\mathbb {R}^3} in the present paper, and the influences of the electric field of the internal electrostatic potential force governed by the self-consistent Poisson equation on the qualitative behaviors of solutions is analyzed. It is observed that the rotating effect of electric field affects the dispersion of fluids and reduces the time decay rate of solutions. Indeed, we show that the density of the NSP system converges to its equilibrium state at the same L ²-rate (1+t)^{-\frac 34}{(1+t)^{-\frac {3}{4}}} or L ^∞-rate (1 + t)^−3/2 respectively as the compressible Navier–Stokes system, but the momentum of the NSP system decays at the L ²-rate (1+t)^{-\frac 14}{(1+t)^{-\frac {1}{4}}} or L ^∞-rate (1 + t)⁻¹ respectively, which is slower than the L ²-rate (1+t)^{-\frac 34}{(1+t)^{-\frac {3}{4}}} or L ^∞-rate (1 + t)^−3/2 for compressible Navier–Stokes system [Duan et al., in Math Models Methods Appl Sci 17:737–758, 2007; Liu and Wang, in Comm Math Phys 196:145–173, 1998; Matsumura and Nishida, in J Math Kyoto Univ 20:67–104, 1980] and the L ^∞-rate (1 + t)^−p with p ? (1, 3/2){p \in (1, 3/2)} for irrotational Euler–Poisson system [Guo, in Comm Math Phys 195:249–265, 1998]. These convergence rates are shown to be optimal for the compressible NSP system.     

Stability for the 3D Navier–Stokes Equations with Nonzero far Field Velocity on Exterior Domains

Concerning to the non-stationary Navier–Stokes flow with a nonzero constant velocity at infinity, just a few results have been obtained, while most of the results are for the flow with the zero velocity at infinity. The temporal stability of stationary solutions for the Navier–Stokes flow with a nonzero constant velocity at infinity has been studied by Enomoto and Shibata (J Math Fluid Mech 7:339–367, 2005), in L ^p spaces for p ≥ 3. In this article, we first extend their result to the case \frac32 < p{\frac{3}{2} < p} by modifying the method in Bae and Jin (J Math Fluid Mech 10:423–433, 2008) that was used to obtain weighted estimates for the Navier–Stokes flow with the zero velocity at infinity. Then, by using our generalized temporal estimates we obtain the weighted stability of stationary solutions for the Navier–Stokes flow with a nonzero velocity at infinity.     

Stability of Discrete Stokes Operators in Fractional Sobolev Spaces

Using a general approximation setting having the generic properties of finite-elements, we prove uniform boundedness and stability estimates on the discrete Stokes operator in Sobolev spaces with fractional exponents. As an application, we construct approximations for the time-dependent Stokes equations with a source term in L ^p (0, T; L ^q (Ω)) and prove uniform estimates on the time derivative and discrete Laplacian of the discrete velocity that are similar to those in Sohr and von Wahl [20]. On long leave from LIMSI (CNRS-UPR 3251), BP 133, 91403, Orsay, France.     

Temporal and Spatial Decays for the Stokes Flow

We estimate the time decay rates in L ¹, in the Hardy space and in L ^∞ of the gradient of solutions for the Stokes equations on the half spaces. For the estimates in the Hardy space we adopt the ideas in [7], and also use the heat kernel and the solution formula for the Stokes equations. We also estimate the temporal-spatial asymptotic estimates in L ^q , 1 < q < ∞, for the Stokes solutions. This work was supported by grant No. (R05-2002-000-00002-0(2002)) from the Basic Research Program of the Korea Science & Engineering Foundation.     

The modulational theory of nonlinear gravity-wave in fluid with varying depth and nonuniform current

By means of WKB expansions, new fourth order evolution equations are derived for two-dimensional Stokes waves over the bottom with arbitrary depth. The effects of slowly varying depthh=h(ε ²x) and currentU=U(ε ²x,ε²t,ε⁴z) on the evolution of a packet of Stokes waves are considered as well. In addition, numerical simulation is performed for the evolution of single envelope by finite-difference method. Project supported by National Natural Science Foundation of China and Centre of Advanced Academic Research of Zhongshan University.     

Concerning the W k,p -Inviscid Limit for 3-D Flows Under a Slip Boundary Condition

We consider the 3-D evolutionary Navier–Stokes equations with a Navier slip-type boundary condition, see (1.2), and study the problem of the strong convergence of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. We prove here, in the flat boundary case, convergence in Sobolev spaces W ^k, p (Ω), for arbitrarily large k and p (for previous results see Xiao and Xin in Comm Pure Appl Math 60:1027–1055, 2007 and Beir?o da Veiga and Crispo in J Math Fluid Mech, 2009, doi:). However this problem is still open for non-flat, arbitrarily smooth, boundaries. The main obstacle consists in some boundary integrals, which vanish on flat portions of the boundary. However, if we drop the convective terms (Stokes problem), the inviscid, strong limit result holds, as shown below. The cause of this different behavior is quite subtle. As a by-product, we set up a very elementary approach to the regularity theory, in L ^p -spaces, for solutions to the Navier–Stokes equations under slip type boundary conditions.     

Floquet’s Theorem and Stability of Periodic Solitary Waves

This paper is concerned with the spectrum the Hill operator L(y) = −y′′ + Q(x) y in L²_per[0, p]{L^{2}_{\rm per}[0, \pi]} . We show that the eigenvalues of L can be characterized by knowing one of its eigenfunctions. Applications are given to nonlinear stability of a class of periodic problems.     

Probabilistic Analysis of the Upwind Scheme for Transport Equations

We provide a probabilistic analysis of the upwind scheme for d-dimensional transport equations. We associate a Markov chain with the numerical scheme and then obtain a backward representation formula of Kolmogorov type for the numerical solution. We then understand that the error induced by the scheme is governed by the fluctuations of the Markov chain around the characteristics of the flow. We show, in various situations, that the fluctuations are of diffusive type. As a by-product, we recover recent results due to Merlet and Vovelle (Numer Math 106: 129–155, 2007) and Merlet (SIAM J Numer Anal 46(1):124–150, 2007): we prove that the scheme is of order 1/2 in L^￥([0,T],L¹(\mathbb R^d)){L^{\infty}([0,T],L^1(\mathbb R^d))} for an integrable initial datum of bounded variation and of order 1/2−ε, for all ε > 0, in L^￥([0,T] ×\mathbb R^d){L^{\infty}([0,T] \times \mathbb R^d)} for an initial datum of Lipschitz regularity. Our analysis provides a new interpretation of the numerical diffusion phenomenon.     

A Global Existence Result for the Compressible Navier–Stokes Equations in the Critical L p Framework

The present paper is dedicated to the global well-posedness issue for the barotropic compressible Navier–Stokes system in the whole space \mathbbR^d{\mathbb{R}^d} with d ≧ 2. We aim at extending the work by Danchin (Inventiones Mathematicae 141(3):579–614, 2000) to a critical framework which is not related to the energy space. For small perturbations of a stable equilibrium state in the sense of suitable L ^p -type Besov norms, we establish the global existence. As a consequence, like for incompressible flows, one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true. In passing, we obtain new estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows.     

Erratum to: Floquet’s Theorem and Stability of Periodic Solitary Waves

This paper is concerned with the spectrum the Hill operator L(y) = −y′′ + Q(x) y in L²_per[0, p]{L^2_{{\rm per}}[0, \pi]}. We show that the eigenvalues of L can be characterized by knowing one of its eigenfunctions. Applications are given to nonlinear stability of a class of periodic problems.     

Nonlocal Effects in Two-Dimensional Conductivity

The paper deals with the asymptotic behaviour as ε → 0 of a two-dimensional conduction problem whose matrix-valued conductivity a _ε is ε-periodic and not uniformly bounded with respect to ε. We prove that only under the assumptions of equi-coerciveness and L ¹-boundedness of the sequence a _ε , the limit problem is a conduction problem of same nature. This new result points out a fundamental difference between the two-dimensional conductivity and the three-dimensional one. Indeed, under the same assumptions of periodicity, equi-coerciveness and L ¹-boundedness, it is known that the high-conductivity regions can induce nonlocal effects in three (or greater) dimensions.     

Incompressible Fluids in Thin Domains with Navier Friction Boundary Conditions (I)

This study is motivated by problems arising in oceanic dynamics. Our focus is the Navier–Stokes equations in a three-dimensional domain Ω_ɛ, whose thickness is of order O(ɛ) as ɛ → 0, having non-trivial topography. The velocity field is subject to the Navier friction boundary conditions on the bottom and top boundaries of Ω_ɛ, and to the periodicity condition on its sides. Assume that the friction coefficients are of order O(ɛ^3/4) as ɛ → 0. It is shown that if the initial data, respectively, the body force, belongs to a large set of H¹(Ω_ɛ), respectively, L²(Ω_ɛ), then the strong solution of the Navier–Stokes equations exists for all time. Our proofs rely on the study of the dependence of the Stokes operator on ɛ, and the non-linear estimate in which the contributions of the boundary integrals are non-trivial.     

Roughness induced forced convective laminar-transitional micropipe flow: energy and exergy analysis

Variable fluid property continuity, Navier–Stokes and energy equations are solved for roughness induced forced convective laminar-transitional flow in a micropipe. Influences of Reynolds number, heat flux and surface roughness, on the momentum-energy transport mechanisms and second-law of thermodynamics, are investigated for the ranges of Re = 1–2,000, Q = 5–100 W/m² and ε = 1–50 μm. Numerical investigations put forward that surface roughness accelerates transition with flatter velocity profiles and increased intermittency values (γ); such that a high roughness of ε = 50 μm resulted in transitional character at Re _tra = 450 with γ = 0.136. Normalized friction coefficient (C _f^*) values showed augmentation with Re, as the evaluated C _f^* are 1.006, 1.028 and 1.088 for Re = 100, 500 and 1,500, respectively, at ε = 1 μm, the corresponding values rise to C _f^* = 1.021, 1.116 and 1.350 at ε = 50 μm. Heat transfer rates are also recorded to rise with Re and ε; moreover the growing influence of ε on Nusselt number with Re is determined by the Nu _ε=50 μm/Nu _ε=1 μm ratios of 1.086, 1.168 and 1.259 at Re = 500, 1,000 and 1,500. Thermal volumetric entropy generation values decrease with Re and ε in heating; however the contrary is recorded for frictional volumetric entropy generation data, where the augmentations in are more considerable when compared with the decrease rates of      

Existence and Regularity of Very Weak Solutions of the Stationary Navier–Stokes Equations

We consider the stationary Navier–Stokes equations in a bounded domain Ω in R ⁿ with smooth connected boundary, where n = 2, 3 or 4. In case that n = 3 or 4, existence of very weak solutions in L ⁿ (Ω) is proved for the data belonging to some Sobolev spaces of negative order. Moreover we obtain complete L ^q -regularity results on very weak solutions in L ⁿ (Ω). If n = 2, then similar results are also proved for very weak solutions in with any q ₀ > 2. We impose neither smallness conditions on the external force nor boundary data for our existence and regularity results.     

Existence of Singular Self-Similar Solutions of the Three-Dimensional Euler Equations in a Bounded Domain

A self-similar solution of the three-dimensional (3d) incompressible Euler equations is defined byu(x,t) =U(y)/t*-t) ^{α, y = x/(t* ~ t)β,α,β>} 0, whereU(y) satisfiesζU + βy. ΔU + U. VU + VP = 0,divU = 0. For α = β = 1/2, which is the limiting case of Leray’s self-similar Navier—Stokes equations, we prove the existence of(U,P) ε H³(Ω,R³ X R) in a smooth bounded domain Ω, with the inflow boundary data of non-zero vorticity. This implies the possibility that solutions of the Euler equations blow up at a timet = t*, t* < +∞.     

Conditional Stability of Unstable Viscous Shock Waves in Compressible Gas Dynamics and MHD

Extending our previous work in the strictly parabolic case, we show that a linearly unstable Lax-type viscous shock solution of a general quasilinear hyperbolic–parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small L ¹ ∩ H ³ perturbations, converging time asymptotically to a translate of the unperturbed wave. That is, for a shock with p unstable eigenvalues, we establish conditional stability on a codimension-p manifold of initial data, with sharp rates of decay in all L ^p . For p = 0, we recover the result of unconditional stability obtained by Mascia and Zumbrun. The main new difficulty in the hyperbolic–parabolic case is to construct an invariant manifold in the absence of parabolic smoothing.     

Lp-Lq Estimate of the Stokes Operator and Navier–Stokes Flows in the Exterior of a Rotating Obstacle

We consider the motion of a viscous fluid filling the whole three-dimensional space exterior to a rotating obstacle with constant angular velocity. We develop the L _p -L _q estimates and the similar estimates in the Lorentz spaces of the Stokes semigroup with rotation effect. We next apply them to the Navier–Stokes equation to prove the global existence of a unique solution which goes to a stationary flow as t → ∞ with some definite rates when both the stationary flow and the initial disturbance are sufficiently small in L _3,∞ (weak-L ₃ space).