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.     

Density-Dependent Incompressible Fluids in Bounded Domains

This paper is devoted to the study of the initial value problem for density dependent incompressible viscous fluids in a bounded domain of with boundary. Homogeneous Dirichlet boundary conditions are prescribed on the velocity. Initial data are almost critical in term of regularity: the initial density is in W^1,q for some q > N, and the initial velocity has fractional derivatives in L^r for some r > N and arbitrarily small. Assuming in addition that the initial density is bounded away from 0, we prove existence and uniqueness on a short time interval. This result is shown to be global in dimension N = 2 regardless of the size of the data, or in dimension N ≥ 3 if the initial velocity is small. Similar qualitative results were obtained earlier in dimension N = 2, 3 by O. Ladyzhenskaya and V. Solonnikov in [18] for initial densities in W^1,∞ and initial velocities in with q > N.     

A Non-Newtonian Fluid with Navier Boundary Conditions

We consider in this paper the equations of motion of third grade fluids on a bounded domain of or with Navier boundary conditions. Under the assumption that the initial data belong to the Sobolev space H ², we prove the existence of a global weak solution. In dimension two, the uniqueness of such solutions is proven. Additional regularity of bidimensional initial data is shown to imply the same additional regularity for the solution. No smallness condition on the data is assumed.     

Compressible Flow in a Half-Space with Navier Boundary Conditions

We prove the global existence of weak solutions of the Navier–Stokes equations of compressible flow in a half-space with the boundary condition proposed by Navier: the velocity on the boundary is proportional to the tangential component of the stress. This boundary condition allows for the determination of the scalar function in the Helmholtz decomposition of the acceleration density, which in turn is crucial in obtaining pointwise bounds for the density. Initial data and solutions are small in energy-norm with nonnegative densities having arbitrarily large sup-norm. These results generalize previous results for solutions in the whole space and are the first for solutions in this intermediate regularity class in a region with a boundary.     

Compressible Navier–Stokes Model with Inflow-Outflow Boundary Conditions

In the paper [7], author gives a definition of weak solution to the nonsteady Navier–Stokes system of equations which describes compressible and isentropic flows in some bounded region Ω with influx of fluid through a part of the boundary ∂Ω. Here, we present a way for proving existence of such solutions in the same situation as in [7] under the sole hypothesis γ > 3/2 for the adiabatic constant.     

Navier–Stokes Equations with First Order Boundary Conditions

On the basis of semigroup and interpolation-extrapolation techniques we derive existence and uniqueness results for the Navier–Stokes equations. In contrast to many other papers devoted to this topic, we do not complement these equations with the classical Dirichlet (no-slip) condition, but instead consider stress-free or slip boundary conditions. We also study various regularity properties of the solutions obtained and provide conditions for global existence.     

The Euler Limit of the Navier‐Stokes Equations, and Rotating Fluids with Boundary

In this paper we study the convergence of weak solutions of the Navier-Stokes equations in some particular domains, with different horizontal and vertical viscosities, when they go to zero with different speeds. The difficulty here comes from the Dirichlet boundary conditions. Precisely we show that if the ratio of the vertical viscosity to the horizontal viscosity also goes to zero, then the solutions converge to the solution of the Euler system. We study the same limit for rotating fluids with Rossby number also going to zero. (Accepted March 20, 1997)     

Necessary Conditions at the Boundary for Minimizers in Incompressible Finite Elasticity

New necessary conditions for energy-minimizing states of an incompressible, elastic body are found. These must hold at boundary points at which dead-load traction data is prescribed.     

Shear Thinning Viscous Fluids in Cylindrical Domains. Regularity up to the Boundary

We consider the motion of a non-Newtonian fluid with shear dependent viscosity between two cylinders. We prove regularity results for the second derivatives of the velocity and the first derivatives of the pressure up to the boundary. A similar problem is studied in reference [2] in the case of a flat boundary. Here we extend the techniques applied in [2] to cylindrical coordinates.      

Rotating Fluids with Self-Gravitation in Bounded Domains

In this paper, we study the steady solutions of Euler-Poisson equations in bounded domains with prescribed angular velocity. This models a rotating Newtonian star consisting of a compressible perfect fluid with given equation of state P=e^S. When the domain is a ball and the angular velocity is constant, we obtain both existence and non-existence theorems, depending on the adiabatic gas constant . In addition we obtain some interesting properties of the solutions; e.g., monotonicity of the radius of the star with both angular velocity and central density. We also prove that the radius of a rotating spherically symmetric star, with given constant angular velocity and constant entropy, is uniformly bounded independent of the central density. This is physically striking and in sharp contrast to the case of the non-rotating star. For general domains and variable angular velocities, both an existence result for the isentropic equations of state and non-existence result for the non-isentropic equation of state are also obtained.Part of this work was completed when Tao Luo was an assistant professor at the University of Michigan. Joel Smoller was supported in part by the NSF, contract number DMS-010-3998. We are grateful to the referee for his very interesting remarks and comments, which enabled a new section, Section 6, to be added in the final version of the paper.     

Incompressible Limit for Compressible Fluids with Stochastic Forcing

We study the asymptotic behavior of the isentropic Navier–Stokes system driven by a multiplicative stochastic forcing in the compressible regime, where the Mach number approaches zero. Our approach is based on the recently developed concept of a weak martingale solution to the primitive system, uniform bounds derived from a stochastic analogue of the modulated energy inequality, and careful analysis of acoustic waves. A stochastic incompressible Navier–Stokes system is identified as the limit problem.     

Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks

We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}We prove a regularity result for the anisotropic linear elasticity equationP u : = div ( C ·?u) = f{P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f} , with mixed (displacement and traction) boundary conditions on a curved polyhedral domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} in weighted Sobolev spaces K^m+1_a+1(W){\mathcal {K}^{m+1}_{a+1}(\Omega)} , for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of K^m_a{\mathcal {K}^{m}_{a}} -regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are no neighboring traction boundary conditions and |a| < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions.     

Sharp Inviscid Limit Results under Navier Type Boundary Conditions. An L p Theory

We consider the evolutionary Navier–Stokes equations with a Navier slip-type boundary condition, and study the convergence of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. We obtain quite sharp results in the 2-D and 3-D cases. However, in the 3-D case, we need to assume that the boundary is flat.     

On Regularity and Uniqueness of the Solutions of a Free Boundary Problem of Heat Transfer in Incompressible Fluids

In this paper we show some regularity and uniqueness results of weak solutions of the Stefan problem with convection, in two and three-dimensional cases. The mathematical formulation adopted is based on the enthalpy method and the convection is described by the Navier-Stokes system.     

Existence of Strong Solutions for Incompressible Fluids with Shear Dependent Viscosities

Certain rheological behavior of non-Newtonian fluids in engineering sciences is often modeled by a power law ansatz with p ∈ (1, 2]. In the present paper the local in time existence of strong solutions is studied. The main result includes also the degenerate case (δ = 0) of the extra stress tensor and thus improves previous results of [L. Diening and M. Růžička, J. Math. Fluid Mech., 7 (2005), pp. 413–450].     

Steady Flows of Viscoelastic Fluids in Domains with Outlets to Infinity

The equations governing the motion of incompressible viscoelastic fluids of Rivlin—Ericksen and Oldroyd type are investigated in domains with cylindrical and paraboloidal outlets to infinity. For sufficiently small fluxes, prescribed in each outlet, existence and uniqueness of solutions are proven in weighted Hölder spaces. In domains with paraboloidal outlets the solution is obtained as a perturbation of the corresponding Navier—Stokes solution and in domains with cylindrical outlets as a perturbation of a flux carrier, constructed by joining together the exact solutions found in each outlet. These exact solutions are shown to be either rectilinear flows of Poiseuille type or flows composed of a rectilinear and of a transverse secondary component.     

四边任意支承条件下弹性矩形薄板弯曲问题的解析解

利用辛几何法推导出了四边为任意支承条件下矩形薄板弯曲的解析解。在分析过程中首先把矩形薄板弯曲问题表示成Hamilton正则方程，然后利用辛几何方法对全状态相变量进行分离变量，求出其本征值后，再按本征函数展开的方法求出四边为任意支承条件下矩形薄板弯曲的解析解。由于在求解过程中并不需要人为的事先选取挠度函数，而是从弹性矩形薄板弯曲的基本方程出发，直接利用数学的方法求出问题的解析解，使得这类问题的求解更加理论化和合理化。文中的最后还给出了计算实例来验证本文方法的正确性。     

Incompressible Viscous Fluid Flows in a Thin Spherical Shell

Linearized stability of incompressible viscous fluid flows in a thin spherical shell is studied by using the two-dimensional Navier–Stokes equations on a sphere. The stationary flow on the sphere has two singularities (a sink and a source) at the North and South poles of the sphere. We prove analytically for the linearized Navier–Stokes equations that the stationary flow is asymptotically stable. When the spherical layer is truncated between two symmetrical rings, we study eigenvalues of the linearized equations numerically by using power series solutions and show that the stationary flow remains asymptotically stable for all Reynolds numbers.      

无网格局部Petrov-Galerkin方法中本质边界条件的处理

鉴于无网格局部Petrov-Galerkin方法(MLPG)形函数的非插值性质,将一种新的本质边界处理方案——完全变换法与MLPG结合,通过变换矩阵修正形函数,使其满足Kronecker-δ条件,实现了本质边界的精确实施。进一步与MLPG中通常处理边界的罚方法作了比较研究,数值结果表明新方法的可靠性与精确度。