Solvability of an inhomogeneous boundary value problem for steady MHD equations

In this paper, we consider the steady MHD equations with inhomogeneous boundary conditions for the velocity and the tangential component of the magnetic field. Using a new construction of the magnetic lifting, we obtain existence of weak solutions under sharp assumption on boundary data for the magnetic field.     

On an initial boundary value problem for the equations of ideal magneto-hydrodynamics

We study an initial boundary value problem for the symmetric hyperbolic quasilinear system of the equations of ideal magneto-hydrodynamics with a perfectly conducting wall boundary condition. Existence of a unique classical solution is proved inside a suitable class of functions of Sobolev type. Moreover, solutions inside the above class are shown to depend continuously in strong norm on the data.     

The initial boundary value problem for Navier-Stokes equations

By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability, we establish the global existence theorem of strong solutions for Navier-Stokes equatios in arbitrary three dimensional domain with uniformlyC ³ boundary, under the assumption that |a| _L ²(Θ) + |f| _L ¹(0,∞;L ²(Θ)) or |∇a| _L ²(Θ) + |f| _L ²(0,∞;L ²(Θ)) small or viscosityv large. Herea is a given initial velocity andf is the external force. This improves on the previous results. Moreover, the solvability of the case with nonhomogeneous boundary conditions is also discussed. This work is supported by foundation of Institute of Mathematics, Academia Sinica     

共振情形下四点泛函边值问题解的存在性

运用Mawhin重合度理论,讨论了一类二阶四点泛函边值问题解的存在性和多解性.分别在非线性项f有界和无界的条件下,获得了此类泛函边值问题解的存在性结果.     

Solvability of a boundary value problem for operator-differential equations of mixed type

We study boundary value problems for operator-differential equations of mixed type, prove existence of generalized solutions, and establish their smoothness in weighted Sobolev spaces. The results are applied to odd order forward-backward equations.     

Solvability of boundary value problem at resonance for third-order functional differential equations

This paper is devoted to the study of boundary value problem of third-order functional differential equations. We obtain some existence results for the problem at resonance under the condition that the nonlinear terms is bounded or generally unbounded. In this paper we mainly use the topological degree theory.     

Solvability of an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid

We study an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid corresponding to the case in which the tangential component of the magnetic field is specified on the boundary and the Dirichlet condition is posed for the velocity. We derive sufficient conditions on the input data for the global solvability of the problem and the local uniqueness of the solution.     

On the two-dimensional initial boundary value problem for the Navier-Stokes equations with discontinuous boundary data

We consider the initial boundary value problem for the Navier-Stokes equations with boundary conditions . We assume that may have jump discontinuities at finitely many points ξ1;. . .,ξm of the boundary ϖΩ of a bounded domain Ω ⊂ ℝ². We prove that this problem has a unique generalized solution in a finite time interval or for small initial and boundary data. The solution is found in a class of vector fields with infinite energy integral. The case of a moving boundary is also considered. Bibliography: 11 titles. Dedicated to O. A. Ladyzhenskaya on the occasion of her 70^th birthday. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 159–178, 1992. Translated by E. V. Frolova.     

First boundary value problem for the equations of hydrodynamics in a domain with a piecewise-smooth boundary

One considers the first boundary value problem for the Stokes and Navier-Stokes equations in a bounded domain with a piecewise-smooth boundary without zero angles. In Theorems 1 and 2 one gives solvability conditions in certain weighted spaces of Sobolev or Hölder type. Pointwise estimates for the Green tensor are formulated. Theorem 3 represents a generalization of the Miranda-Agmon maximum principle, while in Theorem 4 one gives conditions under which one has an estimate in W² ₂ ().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematlcheskogo Instituta im. V. A. Steklova Akad Nauk SSSR, Vol. 96, pp. 179–186, 1980.     

A boundary value problem for a class of mixed-type equations in an unbounded domain

We consider a boundary value problem for an equation of the mixed type with a singular coefficient in an unbounded domain. The uniqueness of the solution of the problem is proved with the use of the extremum principle. In the proof of the existence of a solution of the problem, we use the method of integral equations.     

Solvability of the boundary value problem for stationary magnetohydrodynamic equations under mixed boundary conditions for the magnetic field

The global solvability of the boundary value problem for stationary magnetohydrodynamic equations under the Dirichlet boundary condition for the velocity and mixed boundary conditions for the magnetic field is proved.     

Solvability of a singular boundary value problem in a domain of the Lobachevskii plane

Under study is some singular elliptic boundary value problem in a domain of the Lobachevskii plane with an isolated boundary point. We introduce the new function spaces that coincide with the spaces of Sobolev-Nikol’skii-Besov type outside the singular point, and the concept of σ-trace at the singular point. The main result is a proof that the singular boundary value problem is uniquely solvable.     

The initial boundary value problem for the nonlinear parabolic equation in unbounded domain

In this paper we consider the mixed problem for the equation u _tt  + A ₁ u + A ₂(u _t ) + g(u _t ) = f(x, t) in unbounded domain, where A ₁ is a linear elliptic operator of the fourth order and A ₂ is a nonlinear elliptic operator of the second order. Under natural assumptions on the equation coefficients and f we proof existence of a solution. This result contains, as a special case, some of known before theorems of existence. Essentially, in difference up to previous results we prove theorems of existence without the additional assumption on behavior of solution at infinity.