Weighted Sobolev Spaces for a Scalar Model of the Stationary Oseen Equations in $$\mathbb{R}^{3}$$

This paper is devoted to a scalar model of the Oseen equations, a linearized form of the Navier–Stokes equations. To control the behavior of functions at infinity, the problem is set in weighted Sobolev spaces including anisotropic weights. In a first step, some weighted Poincaré-type inequalities are obtained. In a second step, we establish existence, uniqueness and regularity results.     

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.     

An Example of Finite-time Singularities in the 3d Euler Equations

Let be the exterior of the closed unit ball. Consider the self-similar Euler system

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.      

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

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.     

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.     

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.      

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.     

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 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.     

Navier–Stokes Equations with Shear-Thickening Viscosity. Regularity up to the Boundary

In this article we prove some sharp regularity results for the stationary and the evolution Navier–Stokes equations with shear dependent viscosity, see (1.1), under the no-slip boundary condition(1.4). We are interested in regularity results for the second order derivatives of the velocity and for the first order derivatives of the pressure up to the boundary, in dimension n ≥ 3. In reference [4] we consider the stationary problem in the half space \mathbbR₊ⁿ{\mathbb{R}}_+^n under slip and no-slip boundary conditions. Here, by working in a simpler context, we concentrate on the basic ideas of proofs. We consider a cubic domain and impose our boundary condition (1.4) only on two opposite faces. On the other faces we assume periodicity, as a device to avoid unessential technical difficulties. This choice is made so that we work in a bounded domain Ω and, at the same time, with a flat boundary. In the last section we provide the extension of the results from the stationary to the evolution problem.     

A Navier–Stokes Approximation of the 3D Euler Equation with the Zero Flux on the Boundary

Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T ₀ > 0, ν ₀ > 0 and a unique continuous family of strong solutions u ^ν (0 ≤ ν < ν ₀) of the Euler or Navier–Stokes initial-boundary value problem on the time interval (0, T ₀). In addition to the condition of the zero flux, the solutions of the Navier–Stokes equation satisfy certain natural boundary conditions imposed on curl u ^ν and curl ² u ^ν .      

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.      

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.      

Nearly-Hamiltonian Structure for Water Waves with Constant Vorticity

We show that the governing equations for two-dimensional gravity water waves with constant non-zero vorticity have a nearly-Hamiltonian structure, which becomes Hamiltonian for steady waves.      

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.      

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.