Global solution of the Cauchy problem in nonlinear thermodi.usion in solid body

We prove a theorem about global existence (in time) of the solution to the initial‐value problem for a nonlinear hyperbolic parabolic system of coupled partial differential equation of second order describing the process of thermodiffusion in solid body. The corresponding global existence theorems has been proved using the L^p ‐ L^q time decay estimates for the solution of the associated linearized problem. Next, we proved the energy estimate in the Sobolev space with constant independent of time. Such an energy estimate allows us to apply the standard (continuation argument and to continue the local solution to one de.ned for all t ∈ 〈0, ∞)).     

Continuity equations for a deformable solid

For three-dimensional problems of the mechanics of a deformable solid in a Cartesian coordinate system, we show that there are only three continuity equations in terms of deformations and not six, as was considered earlier. We obtain continuity equations of the integro-differential type. They are reduced to the corresponding three of the six well-known Saint-Venant continuity differential equations only if the necessary conditions of consistency between displacements and deformations at the domain boundary are satisfied. As a results, the problem of closedness of the equations of elasticity and thermoelasticity in terms of stresses for three-dimensional problems as well as the equations of the mechanics of a deformable solid is finally solved as a whole. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Matermatychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 117–123, April–June, 1998.     

The Euler problem in solid body dynamics and the Jacobi problem about geodesics on an ellipsoid are not topologically conjugate

Two integrable problems are considered: the geodesic flow of an ellipsoid (the Jacobi problem) and the rotation of a solid about its center of mass (the Euler problem). It is proved that transforming the dynamical system of the Euler problem into the dynamical system of the Jacobi problem by a continuous change of coordinates is impossible. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 252–258, February, 1997. Translated by M. A. Shishkova     

On the Cauchy problem for elliptic equations in a disk

Letφ, ψ be smooth functions on the boundary of the unit diskB ₁. A second order uniformly elliptic operatorL and a functionu with second order derivatives inL ^p (1<p<2) are constructed with the following properties:u solvesLu=0 inB ₁ and satisfies the Cauchy dataφ, ψ on∂B ₁.     

Initial-boundary value problem and integrodifferential equations of electrodynamics for an inhomogeneous conductive body

The initial-boundary value problem of determining the electromagnetic field in an inhomogeneous conducting sample and the surrounding external medium is solved under certain assumptions on the sample, the external medium, and the external current density. The existence of a classical solution to this problem is proved. The electromagnetic field under small variations in the sample’s electric conductivity is computed by applying perturbation theory.     

Optimal cooling of a section of a solid body

The article considers the cooling of a section of a solid body subject to constant internal energy. The numerical method proposed for the solution of the problem is a generalization of Plotnikov's method for the solution of optimal control problems of parabolic systems.Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 3, pp. 58–64, 1993.     

On convection in a fluid filling the cavity of a moving solid body

The problem of simultaneous motion about a fixed point O of a solid body and unevenly heated viscous incompressible fluid completely filling a finite cavity of the body is considered in linear formulation. The center of mass of the system body plus fluid in the state of mechanical equilibrium is assumed to coincide with point O. The theorem on solvability of the Cauchy problem for small unsteady equilibrium perturbations is proved, and normal perturbations and the spectrum of the problem arising in the analysis of such perturbations are investigated. It is shown that the whole spectrum consists of normal eigenvalues and lies in some half-band containing the real axis. It is shown that the respective system of root vectors is complete. Properties of the spectrum and the dependence on Rayleigh numbers are investigated. Rayleigh numbers for which the real parts of eigenvalues are positive,i.e. when the generated oscillating normal perturbations are damped in time, are evaluated in the case of the fluid being heated from below and above.     

Thermodynamic analysis of the equations of a linear viscoelastic anisotropic solid

Constraints on the coefficients of the equations of state of anisotropic linear viscoelasticity are obtained using the positive definiteness of two quadratic forms—the potential energy and the energy dissipation rate.Mekhanika Polimerov, Vol. 3, No. 2, pp. 227–235, 1967     

Cauchy problem for a class of integro-differential equations

We consider the Cauchy problem for integro-differential equations containing Wiener-Hopf operators. We define the characteristic polynomial of the problem. We show that if all roots of the characteristic polynomial have negative imaginary parts and the corresponding strong moment problem is solvable, then the problem is equivalent to a Wiener-Hopf integral equation. We consider an example of an application of the result to a problem related to nonlocal wave interaction.     

Multipoint problem for a class of evolution equations

We prove the well-posed solvability of a nonlocal problem for evolution equations with nonnegative self-adjoint operators that have discrete spectrum and with boundary conditions in the space of linear continuous functionals of the type of ultra-distributions identified with formal Fourier series.     

Families of stable manifolds for one-dimensional parabolic equations

We prove that to any invariant subset of the dynamical system generated by a one-dimensional quasilinear parabolic equation there corresponds an invariant family of stable manifolds of finite codimension. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 11–23, July, 1996.     

Completeness of the Floquet solutions to the problem of a solid oscillating in a fluid

We prove the completeness of the Floquet solutions to the parabolic equation describing small oscillations of a fluid-solid system. The symmetry axis of the solid is fixed inside a container of an arbitrary shape which is filled with an incompressible viscous fluid. The solid oscillates torsionally under the action of an elastic force with time periodic rigidity.