New singular solutions of Protter's problem for the3D wave equation

In 1952, for the wave equation,Protter formulated some boundary value problems (BVPs), which are multidimensional analogues of Darboux problems on the plane. He studied these problems in a3D domainΩ0,bounded by two characteristic conesΣ1andΣ2,0and a plane regionΣ0. What is the situation around these BVPs now after 50 years? It is well known that, for the infinite number of smooth functions in the right-hand side of the equation, these problems do not have classical solutions. Popivanov and Schneider (1995) discovered the reason of this fact for the cases of Dirichlet's or Neumann's conditions onΣ0. In the present paper, we consider the case of third BVP onΣ0and obtain the existence of many singular solutions for the wave equation. Especially, for Protter's problems inℝ3, it is shown here that for anyn∈ℕthere exists aCn(Ω¯0)- right-hand side function, for which the corresponding unique generalized solution belongs toCn(Ω¯0\O),but has a strong power-type singularity of ordernat the pointO. This singularity is isolated only at the vertexOof the characteristic coneΣ2,0and does not propagate along the cone.

Inertial oscillations in a rigid axisymmetric container

Free oscillations of a fluid rotating with constant angular velocity Ω in a rigid axisymmetric container are considered. Approximations are sought for modes that vary rapidly in each axial plane, on the premise that the pressure at the axis varies with axial distance z as Re [A( z )e iino(z )], where n ≫ 1, o' is real, and A (z) and o (z) >do not vary rapidly with z The pattern made by the characteristic cones of Poincaré’s equation after repeated reflexions at the boundary proves pertinent. Modes are evaluated, with a proportional error o(n -1 ), for a class of containers that has special symmetries and for eigenfrequencies that produce reflexion patterns with topologies like those found in a sphere. The largest velocities in the modes considered occur near the circles where a characteristic cone touches a boundary.

Pathological oscillations of a rotating fluid

A theoretical study is made of the free periods of oscillation of an incompressible inviscid fluid, bounded by two rigid concentric spheres of radii a, b (a > b), and rotating with angular velocity Ω about a common diameter. An attempt is made to use the Longuet-Higgins solution of the Laplace tidal equation as the first term of an expansion in powers of the parameter ε = (a − b)/(a + b), of the solution to the full equations governing oscillations in a spherical shell. This leads to a singularity in the second-order terms at the two critical circles where the characteristic cones of the governing equation touch the shell boundaries.A boundary-layer type of argument is used to examine the apparent non-uniformity in the neighbourhood of these critical circles, and it is found that, in order to remove the singularity in the pressure, an integrable singularity in the velocity components must be introduced on the characteristic cone which touches the inner spherical boundary. Further integrable singularities are introduced by repeated reflexion at the shell boundaries, and so, even outside the critical region the velocity terms contain what may reasonably be described as a pathological term, generally of order ε½ compared to that found by Longuet-Higgins, periodic with wavelength O(εa) in the radial and latitudinal directions.Some consequences of this result are discussed.