Solvability of the linearized problem on the motion of a drop in a liquid flow

In the paper one establishes the solvability in the anisotropic Sobolev-Slobodetskii spaces of the linear problem, generated by the problem of the nonstationary motion of a drop in a fluid medium. In the formulation of the problem one takes into account the surface tension, which occurs in the noncoercive integral term in the conditions for the jump of the normal stresses. In the general case the velocity vector need not be solenoidal but its divergence must be represented in a special form. The proof of the solvability is carried out first in the Sobolev-Slobodetskii spaces and is based on a priori estimates for the solutions in these spaces.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 171, pp. 53–65, 1989.     

Classical Solvability of a Model Problem in a Half-Space, Related to the Motion of an Isolated Mass of a Compressible Fluid

For an arbitrary finite time interval, the unique solvability of a linear half-space problem is obtained in Hölder classes of functions. The problem arises as the result of the linearization of a free boundary problem for the Navier--Stokes system governing the unsteady motion of a finite mass of a compressible fluid. The boundary conditions in the linear problem are noncoercive because of the surface tension acting on the free boundary. This fact presents the main difficulty in the problem, while the differential system in itself is parabolic in the sense of Petrovskii. The principal idea of the investigation is to reduce the noncoercive problem to a coercive one with zero coefficient of the surface tension. Bibliography: 6 titles.     

Weighted estimates of a solution to the linear problem connected with the one-phase Stefan problem in the case where the specific heat tends to zero

We prove estimates in weighted Hölder norms for a solution to the model linear problem related with the one-phase Stefan problem with a small multiplier ε at time derivative in the heat equation. These estimates are uniform with respect to parameter ε and are significant in justification of passage to the limit in the one-phase Stefan problem as the specific heat tends to zero. Bibliography: 8 titles.     

Solvability of the classical initial-boundary-value problems for the heat-conduction equation in a dihedral angle

One proves the unique solvability of the fundamental initial-boundary-value problems for the heat-conduction equation in an -dimensional infinite dihedral angle and one obtains coercive estimates of their solutions in weighted Hölder norms. The Neumann problem is investigated in detail; for the Dirichlet problem and for the problem with mixed boundary conditions, one gives the formulation of the basic result.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 146–180, 1984.     

An estimate of the maximum modulus of a solution of the linear,time-dependent system of Navier-Stokes equations

It is proved that the maximum modulus of a solution of the initial boundary-value problem for the time-dependent Stokes system with zero boundary data is bounded in terms of the maximum modulus of the initial data with a constant which depends only on the domain.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 69, pp. 34–44, 1977.     

Investigation of a problem for the Laplace equation with a boundary condition of a special form in a plane angle

The explicit solution of the equation u=f in a plane infinite angle, satisfying on one side of the angle a Neumann condition and on the other one the condition u/n + hu/r + u= (/r is the tangential derivative, C, 0), is constructed and estimated in weighted Sobolev spaces. The obtained estimates are sharp with respect to the differential order and are uniform with respect to . The construction of the solution reduces to the investigation of a finite difference equation in the complex plane, arising after a Mellin transform.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 182, pp. 149–167, 1990.