Initial-Boundary Value Problems for an Equation of Internal Waves in a Stratified Fluid

Abstract

In the plane x = (x 1, x 2) ∈ R 2 we consider the internal or external multiply connected domain D bounded by closed curves Γ ∈ C 2,0. The following PDE of composite type $$\frac{{{\partial ^4}u}}{{\partial {t^2}\partial x_1^2}} + \frac{{{\partial ^4}u}}{{\partial {t^2}\partial x_2^2}} + \omega _1^2\frac{{{\partial ^2}u}}{{\partial x_1^2}} + \omega _2^2\frac{{{\partial ^2}u}}{{\partial x_2^2}} = 0;{\omega _1},{\omega _2}0,$$ (1) describes internal waves in the ocean [1]. The potential theory has been constructed for eq.(1) recently. Some applications of potentials to solving problems are presented in [1–8]. In particular, explicit solutions of some problems in canonical domains were obtained in [1–4], [9–10]. In the present note we study the solvability of the initial-boundary value problems with either Dirichlet or Neumann boundary condition in arbitrary domains with the help of the potential technique and the boundary integral equation method [5–8]. Boundary value problems for equations of composite type in multiply connected domains were not treated before.