The effective boundary conditions for vector fields on domains with rough boundaries: Applications to fluid mechanics

The Navier-Stokes system is studied on a family of domains with rough boundaries formed by oscillating riblets. Assuming the complete slip boundary conditions we identify the limit system, in particular, we show that the limit velocity field satisfies boundary conditions of a mixed type depending on the characteristic direction of the riblets.     

Two-dimensional slices of non-pseudoconvex open sets

Let D be a non-pseudoconvex open set in ${\mathbb{C}^n}$ and S be the union of all two-dimensional planes with non-empty and non-pseudoconvex intersection with D. Sufficient conditions are given for ${\mathbb{C}^3{\setminus} S}$ to belong to a complex line. Moreover, in the ${{\mathcal{C}}^2}$ -smooth case, it is shown that ${S=\mathbb{C}^n}$ .     

Maximal operators with rough kernels on product domains

In this paper, we study the Lp boundedness of certain maximal operators on product domains with rough kernels in L(logL). We prove that our operators are bounded on Lp for all 2?p<∞. Moreover, we show that our condition on the kernel is optimal in the sense that the space L(logL) cannot be replaced by Lr(logL) for any r<1. Our results resolve a problem left open in [Y. Ding, A note on a class of rough maximal operators on product domains, J. Math. Anal. Appl. 232 (1999) 222-228].     

Navier-stokes equations on unbounded domains with rough initial data

We consider the Navier-Stokes equations in unbounded domains Ω ⊆ ℝ ⁿ of uniform C ^1,1-type. We construct mild solutions for initial values in certain extrapolation spaces associated to the Stokes operator on these domains. Here we rely on recent results due to Farwig, Kozono and Sohr, the fact that the Stokes operator has a bounded H ^∞-calculus on such domains, and use a general form of Kato’s method. We also obtain information on the corresponding pressure term.     

The Dirichlet-to-Neumann operator on rough domains

We consider a bounded connected open set Ω⊂Rd whose boundary Γ has a finite (d−1)-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator D₀ on L₂(Γ) by form methods. The operator −D₀ is self-adjoint and generates a contractive C₀-semigroup S=(St)_t>0 on L₂(Γ). We show that the asymptotic behaviour of St as t→∞ is related to properties of the trace of functions in H¹(Ω) which Ω may or may not have.     

Highly rotating fluids in rough domains

We consider a rotating fluid in a domain with rough horizontal boundaries. The Rossby number, kinematic viscosity and roughness are supposed of characteristic size ε. We prove a strong convergence theorem on solutions of Navier–Stokes–Coriolis equations, as ε goes to 0, in the well-prepared case. We show in particular that the limit system is a two-dimensional Euler equation with a nonlinear damping term due to boundary layers. We thus give a substantial refinement of the results obtained on flat boundaries with the classical Ekman layers.     

Stokes' theorem for domains with fractal boundary

In this paper, integrals of second kind over a rectifiable curve or a piecewise smooth surface are extended to continuous fractal curves and surfaces. Theorems for the existence of these integrals are proved. Green's, Gauss' and Stokes' theorems are developed for domains with fractal boundaries.     

Multiplicative inequalities in domains with noncompact boundary

Exact embedding theorems of the multiplicative type are established for functions of Sobolev spaces defined in a domain R ⁿ,n2, whose boundary is not compact. The main condition on the domain is of the isoperimetric type.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 260–268, February, 1992.     

Cauchy-type integrals in domains with smooth boundary

Let be a conformal mapping of the unit circle onto the domain bounded by a curve of classC ¹. There exist curves such that the function (f·) is not necessarily a Cauchy type integral in the unit circle when f is a Cauchy type integral in Int .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 173–174, 1987.In conclusion, I express my gratitude to S. A. Vinogradov and V. P. Khavin for advice and for their interest in this note.     

Two-dimensional Navier-Stokes flow in unbounded domains

Dedicated to Professor Hiroki Tanabe on the occasion of his sixtieth birthday     

Boundary homogenization in domains with randomly oscillating boundary

We consider a model homogenization problem for the Poisson equation in a domain with a rapidly oscillating boundary which is a small random perturbation of a fixed hypersurface. A Fourier boundary condition with random coefficients is imposed on the oscillating boundary. We derive the effective boundary condition, prove a convergence result, and establish error estimates.     

Linearly convex domains with singularities on the boundary

We present a topological classification of linearly convex domains with almost smooth boundary whose singularities lie in a hyperplane. We investigate sets with linearly convex boundary and the closures of linearly convex domains.     

Diffusions in perforated domains with mixed boundary conditions

This article investigates averaging effects associated with a fine-grained boundary. A simple diffusion occurs everywhere except at a large number of small “holes” in the medium, at which an appropriately scaled mixed boundary condition is applied. The scaling considered is fitting for boundary conditions resulting from thin layer approximations in which the layer thickness scales with the diameter of the hole. Probabilistic methods associated with the Feynman-Kac formula are applied to find the limiting behavior, and the perforated domain and complex boundary condition are replaced by a straightforward attenuating term.     

Spectral boundary homogenization in domains with oscillating boundaries

We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with a rapidly oscillating boundary. We consider both cases where the eigenvalues of the limit problem are simple and multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics.     

Coarse grid spaces for domains with a complicated boundary

It is shown that, with homogeneous Dirichlet boundary conditions, the condition number of finite element discretization matrices remains uniformly bounded independent of the size of the boundary elements provided that the size of the elements increases with their distance to the boundary. This fact allows the construction of simple multigrid methods of optimal complexity for domains of nearly arbitrary shape. This revised version was published online in June 2006 with corrections to the Cover Date.