On a Rigorous Resolvent Estimate for Plane Couette Flow

We derive a rigorous bound of the solution of the resolvent equation for plane Couette flow in three space dimensions. We combine analytical techniques with numerical computations. Compared to earlier results, our analytical techniques cover a larger part of the parameter domain consisting of wave numbers in two space directions and the Reynolds number. Numerical computations are needed only in a compact subset of the parameter domain. This research was supported by the Swedish Research Council grant 2003-5443.     

Second Order Adaptive Boundary Conditions for Exterior Flow Problems: Non-Symmetric Stationary Flows in Two Dimensions

We consider the problem of solving numerically the stationary incompressible Navier–Stokes equations in an exterior domain in two dimensions. For numerical purposes we truncate the domain to a finite sub-domain, which leads to the problem of finding so called “artificial boundary conditions” to replace the boundary conditions at infinity. To solve this problem we construct – by combining results from dynamical systems theory with matched asymptotic expansion techniques based on the old ideas of Goldstein and Van Dyke – a smooth divergence free vector field depending explicitly on drag and lift and describing the solution to second and dominant third order, asymptotically at large distances from the body. The resulting expression appears to be new, even on a formal level. This improves the method introduced by the authors in a previous paper and generalizes it to non-symmetric flows. The numerical scheme determines the boundary conditions and the forces on the body in a self-consistent way as an integral part of the solution process. When compared with our previous paper where first order asymptotic expressions were used on the boundary, the inclusion of second and third order asymptotic terms further reduces the computational cost for determining lift and drag to a given precision by typically another order of magnitude. Peter Wittwer: Supported in part by the Fonds National Suisse.     

On flows of third-grade fluids with non-linear slip boundary conditions

We prove the existence and uniqueness of steady flows of incompressible fluids of grade three subject to slip and no-slip boundary conditions in bounded domains. The slip boundary condition is a non-linear generalization of the Navier slip boundary condition and permits situations in which the solid boundary undergoes non-rigid tangential motion. The existence proof is based on a fixed point method in which the boundary-value problem is decomposed into four linear problems.     

Instabilities in an open top horizontal porous layer subjected to pulsating thermal boundary conditions

   Received October 30, 2000 / Published online January 23, 2001     

Existence of a Rigid Core in the Flow of a Compressible Bingham Fluid under the Action of a Homogeneous Force

A compressible one-dimensional plain Bingham flow starting in equilibrium under the action of a time-increasing spatially homogeneous mass force is investigated. A lower estimate for the width of a rigid zone is obtained. The estimate shows that the rigid zone converges to the whole interval for t tends to zero. In other words, existence of a rigid core is established. As a supplementary result, additional smoothness of solutions to the system considered is established.     

On a Serrin-Type Regularity Criterion for the Navier–Stokes Equations in Terms of the Pressure

We prove a Serrin-type regularity result for Leray–Hopf solutions to the Navier–Stokes equations, extending a recent result of Zhou [28].     

Quasi-Stability of the Primary Flow in a Cone and Plate Viscometer

We investigate the flow between a shallow rotating cone and a stationary plate. This cone and plate device is used in rheometry, haemostasis as well as in food industry to study the properties of the flow w.r.t. shear stress. Physical experiments and formal computations show that close to the apex the flow is approximately azimuthal and the shear-stress is constant within the device, the quality of the approximation being controlled essentially by the single parameter Re ², where Re is the Reynolds number and the thinness of the cone-plate gap. We establish this fact by means of rigorous energy estimates and numerical simulations. Surprisingly enough, this approximation is valid though the primary flow is not itself a solution of the Navier-Stokes equations, and it does not even fulfill the correct boundary conditions, which are in this particular case discontinuous along a line, thus not allowing for a usual Leray solution. To overcome this difficulty we construct a suitable corrector.     

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.      

On Compactness, Domain Dependence and Existence of Steady State Solutions to Compressible Isothermal Navier–Stokes Equations

The steady state system of isothermal Navier–Stokes equations is considered in two dimensional domain including an obstacle. The shape optimisation problem of minimisation of the drag with respect to the admissible shape of the obstacle is defined. The generalized solutions for the Navier–Stokes equations are introduced. The existence of an optimal shape is proved in the class of admissible domains. In general the solutions are not unique for the problem under considerations.     

On the Existence of Strong Solutions to a Coupled Fluid-Structure Evolution Problem

We consider here a model of fluid-structure evolution problem which, in particular, has been largely studied from the numerical point of view. We prove the existence of a strong solution to this problem.     

Global Strong Solutions for the Two-Dimensional Motion of an Infinite Cylinder in a Viscous Fluid

In this paper, we consider a two-dimensional fluid-rigid body problem. The motion of the fluid is modelled by the Navier-Stokes equations, whereas the dynamics of the rigid body is governed by the conservation laws of linear and angular momentum. The rigid body is supposed to be an infinite cylinder of circular cross-section. Our main result is the existence and uniqueness of global strong solutions.     

Interior Differentiability of Weak Solutions to the Equations of Stationary Motion of a Class of Non-Newtonian Fluids

In this paper, we consider weak solutions to the equations of stationary motion of a class of non-Newtonian fluids the constitutive law of which includes the power law model as special case. We prove the existence of second order derivatives of weak solutions to these equations.     

On Estimates of the Boltzmann Collision Operator with Cut-off

We present new estimates of the Boltzmann collision operator in weighted Lebesgue and Bessel potential spaces. The main focus is put on hard potentials under the assumption that the angular part of the collision kernel fulfills some weighted integrability condition. In addition, the proofs for some previously known -estimates have been considerably shortened and carried out by elementary methods. For a class of metric spaces, the collision integral is seen to be a continuous operator into the same space. Furthermore, we give a new pointwise lower bound as well as asymptotic estimates for the loss term without requiring that the entropy is finite.     

The Damped-Driven 2D Navier–Stokes System on Large Elongated Domains

The Navier–Stokes system with damping, which is motivated by Stommel–Charney model of ocean circulation, is considered in a large elongated periodic rectangular domain with area of the order α⁻¹, as α → 0. We obtain estimates for the dimension of the global attractor that are sharp as both α → 0 and ν → 0, where ν is the viscosity coefficient. This work was supported in part by the US Civilian Research and Development Foundation, grant no. RUM1-2654-MO-05 (A.A.I. and E.S.T.). The work of A.A.I. was supported in part by the Russian Foundation for Fundamental Research, grants no. 06-001-0096 and no. 05-01-429, and by the RAS Programme no. 1 ‘Modern problems of theoretical mathematics’. The work of E.S.T. was supported in part by the NSF, grant no. DMS-0204794, the MAOF Fellowship of the Israeli Council of Higher Education, and by the BSF, grant no. 200423.     

About the Linear Stability of the Spherically Symmetric Solution for the Equations of a Barotropic Viscous Fluid under the Influence of Self-Gravitation

This paper is concerned with the question of linear stability of motionless, spherically symmetric equilibrium states of viscous, barotropic, self-gravitating fluids. We prove the linear asymptotic stability of such equilibria with respect to perturbations which leave the angular momentum, momentum, mass and the position of the center of gravity unchanged. We also give some decay estimates for such perturbations, which we derive from resolvent estimates by means of analytic semigroup theory.     

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.      

An Evolution Principle and the Existence of Entropy. First Order Systems

We formulate a basic principle, called evolution principle, for a given set of physical processes. Then we consider a set given by solutions of first order systems without reaction terms. We show that for strictly hyperbolic systems and for the Euler system the evolution principle is equivalent to the entropy principle. Received May 20, 1997     

Stopping a Viscous Fluid by a Feedback Dissipative Field: I. The Stationary Stokes Problem

We consider a planar stationary flow of an incompressible viscous fluid in a semiinfinite strip governed by the Stokes system with a body forces field. We show how this fluid can be stopped at a finite distance of the entrance of the semi-infinite strip by means of a feedback field depending in a sub-linear way on the velocity field. This localization effect is proved reducing the problem to a non-linear bi-harmonic type one for which the localization of solutions is obtained by means of the application of a suitable energy method. Since the presence of the non-linear terms defined through the body forces field is not standard in the fluid mechanics literature, we establish also some results about the existence and uniqueness of weak solutions for this problem.     

A Note on the Existence of Solutions to the Oseen System in Lipschitz Domains

We study the boundary-value problem associated with the Oseen system in the exterior of m Lipschitz domains of an euclidean point space We show, among other things, that there are two positive constants and α depending on the Lipschitz character of Ω such that: (i) if the boundary datum a belongs to L^q(∂Ω), with q ∈ [2,+∞), then there exists a solution (u, p), with and u ∈ L^∞(Ω) if a ∈ L^∞(∂Ω), expressed by a simple layer potential plus a linear combination of regular explicit functions; as a consequence, u tends nontangentially to a almost everywhere on ∂Ω; (ii) if a ∈ W^1-1/q,q(∂Ω), with then ∇u, p ∈ L^q(Ω) and if a ∈ C^0,μ(∂Ω), with μ ∈ [0, α), then also, natural estimates holds.     

Existence of Weak Solutions for the Unsteady Interaction of a Viscous Fluid with an Elastic Plate

The purpose of this work is to study the existence of solutions for an unsteady fluid-structure interaction problem. We consider a three-dimensional viscous incompressible fluid governed by the Navier–Stokes equations, interacting with a flexible elastic plate located on one part of the fluid boundary. The fluid domain evolves according to the structure’s displacement, itself resulting from the fluid force. We prove the existence of at least one weak solution as long as the structure does not touch the fixed part of the fluid boundary. The same result holds also for a two-dimensional fluid interacting with a one-dimensional membrane.