V. A. Solonnikov

Lectures on Evolution Free Boundary Problems: Classical Solutions

1 Introduction 2 The model problem 3 Transformation of the problem (1) 4 Proof of Theorem 3.1 5 Introduction 6 Linear and model problems 7 Lagrangean coordinates and local existence theorems 8 Proof of Theorem 5.3 9 Scheme of the proof of Theorem 5.4 References

Estimate of the Generalized Energy in a Free-Boundary Problem for a Viscous Incompressible Fluid

We introduce and estimate the generalized energy in the problem on evolution of an isolated fluid mass bounded by a free surface under the surface tension force. Bibliography: 9 titles.

On the Problem of Evolution of an Isolated Liquid Mass

The paper is concerned with the problem of stability of equilibrium figures of a uniformly rotating, viscous, incompressible, self-gravitating liquid subjected to capillary forces at the boundary. It is shown that a rotationally symmetric equilibrium figure F is exponentially stable if the functional G defined on the set of domains Ω close to F and satisfying the conditions of volume invariance (|Ω|=|F|) and the barycenter position attains its minimum for Ω=F. The proof is based on the direct analysis of the corresponding evolution problem with initial data close to the regime of a rigid rotation.

Estimates of Solutions of the Second Initial Boundary-Value Problem for the Stokes System in the Spaces of Functions Having Hölder Continuous Derivatives with Respect to Spatial Variables

It is known that the solutions of parabolic initial boundary-value problems can be estimates in the Hölder norms with respect to the spatial variables for any fixed positive value of time in terms of similar norms of the data. We obtain estimate of this type for the solutions of nonstationary Stokes equations with stresses prescribed on the boundary. Bibliography: 12 titles.

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.

The Initial Boundary-Value Problem for a Generalized Stokes System in a Half-Space

AbstractThe paper contains the construction of a solution of the Cauchy–Dirichlet problem in the half-space

Recent Advances in Partial Differential Equations and Applications

\mathbb{R}_ + ^3

On the free boundary problem of magnetohydrodynamics

for a family of systems of differential equations that includes a Stokes system. For this solution, coercive estimates in the Hölder spaces of functions are obtained. Bibliography: 12 titles.