Heat Convection of Compressible Viscous Fluids. I.

The stationary problem for the heat convection of compressible fluid is considered around the equilibrium solution with the external forces in the horizontal strip domain z_0 < z < z_0 + 1 and it is proved that the solution exists uniformly with respect to z_0 larger than Z_0 . The limit system as z_0 goes to infinity is the Oberbeck-Boussinesq equation

A Remark on a Non-linear Hyperbolic Equation of First Order

This remark is concerned with an existence theorem of the generalized solution of the Canchy problem in the large uₜ+(u²/2)ₓ=g(x)u² u(0, x)=u₀(x) t≥0, -∞<x<+∞, where g(x) is any continuously differentiable function with compact support and u₀(x) is any bounded measurable function

A Note on the Nonlinear Vibrations of the Elastic String

The nonlinear vibrations of an elastic string which are described by the equation ρh∂²w/∂t²＝(P₀+Eh/2L·(∂w/∂x)²dx)∂²w/∂x² were investigated. A proof for the existence and uniqueness of a solution with finite harmonics in the large in time and a conditionally periodic behavior is given for this equation

A Note on an Existence of Conditionally Periodic Oscillation in a one-dimensional Anharmonic Lattice

The oscillations of a one-dimensional anharmonic lattice is investigated sufficiently near the equilibrium state, which is described by the equation d²yᵢ/dt²＝y₊₁-2yᵢ+yᵢ₋₁+α{(yᵢ₊₁－yᵢ)³－(yᵢ－yᵢ₋₁)³}, －∞＜t＜+∞, where i＝1, 2, ・・・ n, α＝constant, boundary conditions are y₀＝yₙ₊₁＝O and initial conditions yᵢ(0), dyᵢ(O)/dt are given. A large part of the oscillations starting from initial data sufficiently near the origin of the phase space of the system is proved to be a conditionally periodic motion under the incommensurable condition on linear frequencies mentioned later