On the free boundary problem of magnetohydrodynamics

In the paper, the solvability of the free boundary problem of magnetohydrodynamics for a viscous incompressible fluid in a simply connected domain is proved. The solution is obtained in the Sobolev–Slobodetskii spaces W₂^{2 + l,1 + l/2},1/2 < l < 1 W_2^{2 + l,1 + l/2},1/2 < l < 1 . Bibliography: 15 titles.     

On a problem of magnetohydrodynamics in a multi-connected domain

We consider the following problem in the MHD approximation: the vessel Ω₁⊂Ω is filled with an incompressible, electrically conducting fluid, and is surrounded by a dielectric or by vacuum, occupying the bounded domain Ω₂=Ω?Ω₁. In Ω we have a magnetic and electric field and the external surface S=∂Ω is an ideal conductor. The emphasis in the paper is on when Ω is not simply connected, in which case the MHD system is degenerate. We use Hodge-type decomposition theorems to obtain strong solutions locally in time or global for small enough initial data, and a linearization principle for the stability of a stationary solution.     

Some boundary value problems in three-dimensional domains

Some nonstandard boundary value problems are studied for the stationary Poisson system, Stokes system, and Navier–Stokes system. The problems under consideration are “intermediate” between the Dirichlet problem and Neumann problem. The well-posedness of these problems is proved.     

Elliptic boundary value problems in hybrid domains

We derive asymptotic transmission conditions at points where segments are attached to a three-dimensional body. These conditions result in a formally self-adjoint problem on a hybrid set with properties similar to those of standard boundary value problems. In particular, the problem has a zero index and possesses a variational statement. If the systems of differential equations have a special form, then the operator of the problem is realized as a self-adjoint extension of the decoupled operators of the problems on the body and the segments. From this viewpoint, we interpret the results of asymptotic analysis of coupled thin and solid bodies.Dedicated to Victor Borisovich Lidskii on the occasion of his jubileeTranslated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 55–72, 2004Original Russian Text Copyright © by S. A. NazarovSupported by the Russian Foundation for Basic Research (grant No. 00-01-00835).     

Two related free boundary problems

Two related free boundary problems are solved: the first isthe viscous film coating of wedges of arbitrary angle; the secondis the rectangular dam problem with evaporation (or fluid removal)from the free surface. Both problems are of practical interestand explicit solutions are given. The two examples treated aregeneralizations of problems solved using Polubarinova-Kochina's(1962) analytic differential equation method and conformal mappingsinvolving elliptic modular functions to an intermediate plane.Here conformal mappings involving Legendre functions are usedto generalize these results.     

The indicatrix method in free boundary problems

Summary An approximate method for nonlinear problems with functional constraints is considered, in which the constraint in the whole domain is replaced by the constraint on a manifold of lower dimension. The stability criterion is introduced, and convergence theorems are proved for the onedimensional problem. Numerical results for the elastic-plastic torsion problem are given.     

On free boundary problems in two dimensions

Let L be a linear elliptic operator in two dimensions with analytic coefficients and of second order, andu(x, y) a solution of Lu=0 in a simply connected domain ω with rectifiable boundary Γ. Suppose ψ(x, y) analytic on ω∪Γ and L ψ≠0 there.H is shown that ifu and ψ coincide with first derivatives on an open portion Γ₀ of Γ, then Γ₀ permits the representation λ=x (θ),y=y (θ) withx(θ),y(θ)analytic functions of a real parameter θ.     

Homogenization of elliptic boundary value problems in Lipschitz domains

In this paper we study the L ^p boundary value problems for \({\mathcal{L}(u)=0}\) in \({\mathbb{R}^{d+1}_+}\) , where \({\mathcal{L}=-{\rm div} (A\nabla )}\) is a second order elliptic operator with real and symmetric coefficients. Assume that A is periodic in x _d+1 and satisfies some minimal smoothness condition in the x _d+1 variable, we show that the L ^p Neumann and regularity problems are uniquely solvable for 1 < p < 2 + δ. We also present a new proof of Dahlberg’s theorem on the L ^p Dirichlet problem for 2 ? δ < p < ∞ (Dahlberg’s original unpublished proof is given in the Appendix). As the periodic and smoothness conditions are imposed only on the x _d+1 variable, these results extend directly from \({\mathbb{R}^{d+1}_+}\) to regions above Lipschitz graphs. Consequently, by localization techniques, we obtain uniform L ^p estimates for the Dirichlet, Neumann and regularity problems on bounded Lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization. The results on the Neumann and regularity problems are new even for smooth domains.     

Factorization of linear elliptic boundary value problems in non-cylindrical domains

We present a method of factorization for linear elliptic boundary value problems considered in non-cylindrical domains. We associate a control problem to the boundary value problem which regularizes it. The technique of change of variables is used to study this problem.