Error Bounds for Semi-Galerkin Approximations of Nonhomogeneous Incompressible Fluids

We consider spectral semi-Galerkin approximations for the strong solutions of the nonhomogeneous Navier–Stokes equations. We derive an optimal uniform in time error bound in the H¹ norm for approximations of the velocity. We also derive an error estimate for approximations of the density in some spaces L^r. P. Braz e Silva was supported for this work by FAPESP/Brazil, #02/13270-1 and is currently supported in part by CAPES/MECD-DGU Brazil/Spain, #117/06. M. Rojas-Medar is partially supported by CAPES/MECD-DGU Brazil/Spain, #117/06 and project BFM2003-06446-CO-01, Spain.     

On the Capillary Problem for Compressible Fluids

Classical capillarity theory is based on a hypothesis that virtual motions of fluid particles distinct from those on a surface interface have no effect on the form of the interface. That hypothesis cannot be supported for a compressible fluid. A heuristic reasoning suggests that even small amounts of compressibility could have significant effect on surface behavior. In an earlier work, Finn took a partial account of compressibility, and formulated a variant of the classical capillarity equation for fluid surface height in a vertical capillary tube; he was led to a necessary condition for existence of a solution with prescribed mass in a tube closed at the bottom. For a circular tube, he proved that the condition also suffices, and that solutions are uniquely determined for any contact angle γ. Later Finn took more complete account of compressibility and obtained a new equation of highly nonlinear character but for which the same necessary condition holds. In the present work we consider that equation for circular tubes. We prove that the necessary condition again suffices for existence when 0 ≤ γ < π, and we establish uniqueness when 0 ≤ γ ≤ π/2. Our result is put into relief by the observation that for the unconstrained problem of a tube dipped into an infinite liquid bath, solutions do not in general exist when γ > π/2. Presumably an actual fluid would in that case descend to the bottom of the tube. This kind of singular behavior does not occur for the equation previously considered, nor does it occur in the present case under the presence of a mass constraint.     

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.     

The Pressure Singularity for Compressible Stokes Flows in a Concave Polygon

A compressible Stokes system is studied in a polygon with one concave vertex. A corner singularity expansion is obtained up to second order. The expansion contains the usual corner singularity functions for the velocity plus an “associated” velocity singular function, and a pressure singular function. In particular the singularity of pressure is not local but occurs along the streamline emanating from the incoming concave vertex. It is observed that certain first derivatives of the pressure become infinite along the streamline of the ambient flow emanating from the concave vertex. Higher order regularity is shown for the remainder. This work was supported by the Com2MaC-SRC/ERC program of MOST/KOSEF (grant R11-1999-054), and by the U.S. National Science Foundation.     

Global Existence of Solutions for Shear Flow of Certain Viscoelastic Fluids

We prove the global existence in time of solutions to time-dependent shear flows for certain viscoelastic fluids. The essential point in the proof is an a priori estimate for the shear stress. Positive definiteness constraints for the stress play a crucial role in obtaining such estimates. This research was supported by the National Science Foundation under Grant DMS-0405810.     

On non-Newtonian Fluids with Energy Transfer

The model combining incompressible Navier–Stokes’ equation in a non-Newtonian p-power-law modification and the nonlinear heat equation is considered. Existence of its (very) weak solutions is proved for p > 11/5 under mild assumptions of the temperature-dependent stress tensor by careful successive limit passage in a Galerkin approximation.      

Some Global Stability Results for Shear Flows of Viscoelastic Fluids

We consider plane shear flows of viscoelastic fluids. For a number of constitutive models, we prove stability of the rest state for perturbations of arbitrary size. We also consider stability of plane Poiseuille flow in a few special cases. This research was supported by the National Science Foundation under Grant DMS-0405810.     

Stability of a Steady Viscous Incompressible Flow Past an Obstacle

We derive a sufficient condition for stability of a steady solution of the Navier–Stokes equation in a 3D exterior domain Ω. The condition is formulated as a requirement on integrability on the time interval (0, +∞) of a semigroup generated by the linearized problem for perturbations, applied to a finite family of certain functions. The norm of the semigroup is measured in a bounded sub-domain of Ω. We do not use any condition on “smallness” of the basic steady solution.      

Effect of Density Dependent Viscosities on Multiphasic Incompressible Fluid Models

In this paper, we look at the influence of the choice of the Reynolds tensor on the derivation of some multiphasic incompressible fluid models, called Kazhikhov–Smagulov type models. We show that a compatibility condition between the viscous tensor and the diffusive term allows us to obtain similar models without assuming a small diffusive term as it was done for instance by A. Kazhikhov and Sh. Smagulov. We begin with two examples: The first one concerning pollution and the last one concerning a model of combustion at low Mach number. We give the compatibility condition that provides a class of models of the Kazhikhov–Smagulov type. We prove that these models are globally well posed without assumptions between the density and the diffusion terms.     

Asymptotic Properties of Axis-Symmetric D-Solutions of the Navier–Stokes Equations

We consider asymptotic behavior of Leray’s solution which expresses axis-symmetric incompressible Navier–Stokes flow past an axis-symmetric body. When the velocity at infinity is prescribed to be nonzero constant, Leray’s solution is known to have optimum decay rate, which is in the class of physically reasonable solution. When the velocity at infinity is prescribed to be zero, the decay rate at infinity has been shown under certain restrictions such as smallness on the data. Here we find an explicit decay rate when the flow is axis-symmetric by decoupling the axial velocity and the horizontal velocities. The first author was supported by KRF-2006-312-C00466. The second author was supported by KRF-2006-531-C00009.     

The Oberbeck–Boussinesq Approximation as a Singular Limit of the Full Navier–Stokes–Fourier System

We consider the asymptotic limit for the complete Navier–Stokes–Fourier system as both Mach and Froude numbers tend to zero. The limit is investigated in the context of weak variational solutions on an arbitrary large time interval and for the ill-prepared initial data. The convergence to the Oberbeck–Boussinesq system is shown.      

On the W 2,q -Regularity of Incompressible Fluids with Shear-Dependent Viscosities: The Shear-Thinning Case

In this paper we improve the results stated in Reference [2], in this same Journal, by using -basically- the same tools. We consider a non Newtonian fluid governed by equations with p-structure and we show that second order derivatives of the velocity and first order derivatives of the pressure belong to suitable Lebesgue spaces.      

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.     

Nonhomogeneous Boundary Value Problems for Equations of Viscous Heat-Conducting Gas in Time-Decreasing Non-Rectangular Domains

In this article the global solvability of the initial-boundary value problems for the system of equations describing non-stationary flow of the viscous heat-conducting one-dimensional gas in time-decreasing non-rectangular domains is proved.      

On the Asymptotic Limit of Flows Past a Ribbed Boundary

We consider a stationary Navier–Stokes flow in a bounded domain supplemented with the complete slip boundary conditions. Assuming the boundary of the domain is formed by a family of unidirectional asperities, whose amplitude as well as frequency is proportional to a small parameter ε, we shall show that in the asymptotic limit the motion of the fluid is governed by the same system of the Navier–Stokes equations, however, the limit boundary conditions are different. Specifically, the resulting boundary conditions prevent the fluid from slipping in the direction of asperities, while the motion in the orthogonal direction is allowed without any constraint. The work of Š. N. supported by Grant IAA100190505 of GA ASCR in the framework of the general research programme of the Academy of Sciences of the Czech Republic, Institutional Research Plan AV0Z10190503.     

Global Solvability in W 2 2,1 -Weighted Spaces of the Two-Dimensional Navier–Stokes Problem in Domains with Strip-Like Outlets to Infinity

The time-dependent Navier–Stokes system is studied in a two-dimensional domain with strip-like outlets to infinity in weighted Sobolev function spaces. It is proved that under natural compatibility conditions there exists a unique solution with prescribed fluxes over cross-sections of outlets to infinity which tends in each outlet to the corresponding time-dependent Poiseuille flow. The obtained results are proved for arbitrary large norms of the data (in particular, for arbitrary fluxes) and globally in time. The authors are supported by EC FP6 MC–ToK programme SPADE2, MTKD–CT–2004–014508.     

Existence of Solutions with the Prescribed Flux of the Navier–Stokes System in an Infinite Cylinder

The existence and uniqueness of a solution to the nonstationary Navier–Stokes system having a prescribed flux in an infinite cylinder is proved. We assume that the initial data and the external forces do not depend on x₃ and find the solution (u, p) having the following form

Variants of the Stokes Problem: the Case of Anisotropic Potentials

We investigate the smoothness properties of local solutions of the nonlinear Stokes problem$\begin{eqnarray*}-\diverg \{T(\eps(v))\} + \nabla \pi &=& g \msp \mbox{on $\Omega$,}\\\diverg v&\equiv & 0 \msp \mbox{on $\Omega$,}\end{eqnarray*}$where v: ⁿ is the velocity field, $\pi$: $ denotes the pressure function, and g: ⁿ represents a system of volume forces, denoting an open subset of ⁿ . The tensor T is assumed to be the gradient of some potential f acting on symmetric matrices. Our main hypothesis imposed on f is the existence of exponents 1 < p q < \infty such that\lambda (1+|\eps|^{2})^{\frac{p-2}{2}} |\sigma|^{2} \leq D^{2}f(\eps)(\sigma ,\sigma) \leq \Lambda (1+|\eps|^{2})^{\frac{q-2}{2}} |\sigma|^{2}holds with suitable constants , > 0, i.e. the potential f is of anisotropic power growth. Under natural assumptions on p and q we prove that velocity fields from the space W ¹ _{p, loc} (; ⁿ ) are of class C ^1, on an open subset of with full measure. If n = 2, then the set of interior singularities is empty.Dedicated to O. A. Ladyzhenskaya on the occasion of her 80th birthday     

Effect of Vertical Modulation on the Onset of Filtration Convection

The present paper examines the effect of vertical harmonic vibration on the onset of convection in an infinite horizontal layer of fluid saturating a porous medium. A constant temperature distribution is assigned on the rigid boundaries, so that there exists a vertical temperature gradient. The mathematical model is described by equations of filtration convection in the Darcy–Oberbeck–Boussinesq approximation. The linear stability analysis for the quasi-equilibrium solution is performed using Floquet theory. Employment of the method of continued fractions allows derivation of the dispersion equation for the Floquet exponent σ in an explicit form. The neutral curves of the Rayleigh number Ra versus horizontal wave number α for the synchronous and subharmonic resonant modes are constructed for different values of frequency Ω and amplitude A of vibration. Asymptotic formulas for these curves are derived for large values of Ω using the method of averaging, and, for small values of Ω, using the WKB method. It is shown that, at some finite frequencies of vibration, there exist regions of parametric instability. Investigations carried out in the paper demonstrate that, depending on the governing parameters of the problem, vertical vibration can significantly affect the stability of the system by increasing or decreasing its susceptibility to convection.      

Joint Analyticity and Analytic Continuation of Dirichlet–Neumann Operators on Doubly Perturbed Domains

In this paper we take up the question of analyticity properties of Dirichlet–Neumann operators (DNO) which arise in boundary value and free boundary problems from a wide variety of applications (e.g., fluid and solid mechanics, electromagnetic and acoustic scattering). More specifically, we consider DNO defined on domains inspired by the simulation of ocean waves over bathymetry, i.e. domains perturbed independently at both the top and bottom. Our analysis shows that the DNO, when perturbed from an arbitrary smooth domain, is parametrically analytic (as a function of deformation height/slope) for profiles of finite smoothness. Additionally, we extend these results to joint spatial and parametric analyticity when the perturbations are real analytic. This analysis is novel not only in that it accounts for the doubly perturbed nature of the geometry, but also in that the technique of proof establishes the full joint analyticity from an arbitrary smooth profile simultaneously.