On the solvability of a nonstationary problem describing the dynamics of an incompressible viscoelastic fluid

We study the local solvability of the Cauchy-Dirichlet problem for the system which describes the dynamics of an incompressible viscoelastic Kelvin-Voigt fluid. The configuration space of the problem is described. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 442–450, March, 1998.     

Free surface motion of an incompressible ideal fluid

We consider the free boundary problem for an incompressible ideal fluid in the two-dimensional space. We show the unique existence of the solution, locally in time, even if the initial surface and the bottom are uneven.     

Unique solvability of an initial- and boundary-value problem for viscous incompressible nonhomogeneous fluids

In a bounded domain of the space R ⁿ (n=2 or 3) we consider the initialand boundary-value problem regarding the determination of the velocity vector of the fluid, the pressure, and the density from the system of Navier-Stokes equation and the continuity equations, as well as from the initial conditions for the velocity and from the adherence boundary conditions. It is proved that the three-dimensional problem is uniquely solvable on some finite time interval and, in the case of a small initial velocity vector and a small volume force, also on an infinite interval; however, the two-dimensional problem is uniquely solvable for all t0 without any smallness restrictions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 52, pp. 52–109, 1975.     

Dynamics of a rotating layer of an ideal electrically conducting incompressible fluid

A system of nonlinear partial differential equations is considered that models perturbations in a layer of an ideal electrically conducting rotating fluid bounded by spatially and temporally varying surfaces with allowance for inertial forces. The system is reduced to a scalar equation. The solvability of initial boundary value problems arising in the theory of waves in conducting rotating fluids can be established by analyzing this equation. Solutions to the scalar equation are constructed that describe small-amplitude wave propagation in an infinite horizontal layer and a long narrow channel.     

Flow problem for the equations of an ideal fluid

One considers the problem of the flow of an ideal fluid through a bounded domain with prescribed velocity vector at the entry and prescribed pressure on the part of the outflow. One proves the solvability of the problem, locally with respect to time, in the case of a plane-parallel or axisymmetric motion.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 96, pp. 84–96, 1980.The authors express their sincere gratitude to O. A. Ladyzhenskaya for the helpful discussion of the results of the paper.     

Leray's problem for a viscous incompressible micropolar fluid

This work concerns the steady motion of a viscous incompressible micropolar fluid in unbounded domains having cylindrical outlets to infinity. We prove the existence of a solution that approaches prescribed parallel solutions along the outlets of the domain. We also study the uniqueness, the regularity and the asymptotic behavior of the solution.     

On solvability of a space-periodic problem for the nonstationary VT-equation of transonic gas dynamics

We consider the problem of existence and uniqueness of a solution to the nonstationary VT-equation of transonic gas dynamics which is periodic in an appropriate space variable and study the question of smoothness of this solution in time.     

Local solvability of the capillary problem

This paper studies conditions for local (in time) solvability of a qualitatively new singularllimit problem, the free (unknown) boundary problem appearing recently. In fact, there are not so many different free boundary problems, which corresponds to not so large a variety of principally different phase transitions of the first and second kinds. Therefore, the appearance of principally new problems elicits interest. This paper studies structural features of a certain problem on the basis of a certain method developed previously, precisely, the localization method [1, 3, 9].     

Free boundary problem for a two-layer inviscid incompressible fluid

The existence of a local (in time) classical solution of a free boundary problem for a two-layer inviscid incompressible fluid is shown. The method of successive approximations and the novel approach to Lagrangian coordinates of Solonnikov are used.     

Controlled motion of a rigid body with internal mechanisms in an ideal incompressible fluid

We consider the controlled motion in an ideal incompressible fluid of a rigid body with moving internal masses and an internal rotor in the presence of circulation of the fluid velocity around the body. The controllability of motion (according to the Rashevskii–Chow theorem) is proved for various combinations of control elements. In the case of zero circulation, we construct explicit controls (gaits) that ensure rotation and rectilinear (on average) motion. In the case of nonzero circulation, we examine the problem of stabilizing the body (compensating the drift) at the end point of the trajectory. We show that the drift can be compensated for if the body is inside a circular domain whose size is defined by the geometry of the body and the value of circulation.     

Unique global solvability of an axisymmetric convection problem of a viscous,thermically nonhomogeneous fluid

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 7, pp. 885–893, July, 1989.     

Local solvability of a problem of nonlinear elasticity theory

One considers an integral functional depending only on the trace of the metric tensor induced by a mapping of ann-dimensional domain into ⁿ. One seeks a mapping which minimizes this functional. Under a well-defined smallness in the boundary conditions, one proves the existence of an infinite set of critical points of the functional. Under additional restrictions, one discusses an existence theorem and the character of the extremum. The convexity of the functional is not assumed. Such functionals are encountered in elasticity theory.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 163–173, 1981.     

Local solvability of a one-phase problem with free boundary

A certain one-phase problem with free boundary is studied. The local (in time) solvability of this problem is proved; moreover, the general method elaborated is applied in a more concrete case. For this purpose, a new change of variables and the parametrization of the boundary are introduced, and the problem studied is reduced to a problem in a constant domain.     

Local solvability of one-dimensional problem of thermoviscoelasticity

The local solvability of the initial-boundary value problem in spaces of summable functions for some one-dimensional system of equations of thermoviscoelasticity is established. The nonlinearities in equations are determined by the difference between Lagrangian and Eulerian coordinates. The coercive approach implies a necessary condition on the initial data.     

Kawahara-BO方程Cauchy问题的局部可解性

研究Kawahara-BO方程在低正则Sobolev空间Hr上的局部适定性和极限行为.证明了:当r>-7/5时,对任意的初始值uo ∈Hr,Kawahara-BO方程的Cauchy问题存在唯一的解u∈C([0,T],Hr)∩Xr,s;当BO项系数λ趋向于O时,Kawahara-BO方程的解收敛到Kawahara方程的解.     

Dynamics of a rotating layer of an ideal electrically conducting incompressible inhomogeneous fluid in an equatorial region

The equations describing the three-dimensional equatorial dynamics of an ideal electrically conducting inhomogeneous rotating fluid are studied. The magnetic and velocity fields are represented as superpositions of unperturbed steady-state fields and those induced by wave motion. As a result, after introducing two auxiliary functions, the equations are reduced to a special scalar one. Based on the study of this equation, the solvability of initial-boundary value problems arising in the theory of waves propagating in a spherical layer of an electrically conducting density-inhomogeneous rotating fluid in an equatorial zone is analyzed. Particular solutions of the scalar equation are constructed that describe small-amplitude wave propagation.     

On a mathematical problem of the dynamics of a gas microbubble in an incompressible fluid

We consider the oscillations of a gas microbubble in an incompressible fluid. We give an analytic construction of the solution and exhibit the characteristic modes of vibration of the bubble.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 95–98.     

On existence of two-dimensional nonstationary flows of an ideal incompressible liquid admitting a curl nonsummable to any power greater than 1

Rostov-on-Don. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 5, pp. 209–212, September–October, 1992.