The second initial boundary value problem for the gravitation-gyroscopic wave equation in exterior domains

Abstract

ABSTRACT. We study the existence and uniqueness of the solution to the second initial boundary value problem for the gravitation-gyroscopic wave equation in an exterior multiply connected domain with various types of conditions at infinity.KEY WORDS: equations of composite type, existence and uniqueness theorem, conditions at infinity.The gravitation-gyroscopic wave equation is a fourth-order evolution equation of composite type with two space variables. It has the form ~u ~u ~u 2 02uwhere Wl, w2 > 0 are constants. The solvability of exterior initial boundary value problems for this equation was studied in [1-3] without taking into account the conditions at infinity and without analyzing the uniqueness of the solution. Problems in half-space for equations of composite type were studied in [4]. Various problems in the case of three space variables for equations of these types were studied in [5][6]. In [7][8][9] solutions of some initial boundary value problems for the gravitation-gyroscopic wave equation were obtained in the exterior multiply connected domains formed by systems of cuts on the plane. Such problems describe nonstationary interior waves excited by the motion of several bodies in a stratified rotating fluid.We study the existence and uniqueness of the solution to the second initial boundary value problem for the gravitation-gyroscopic wave equation in an exterior multiply connected domain with various types of conditions at infinity. The role of the conditions at infinity for the correct statement of this problem is clarified. The proof of the solvability makes use of a new function-theoretic method which has never been used earlier in such problems.w Statement of the problem and uniqueness theorem Consider an exterior connected domain D with boundary F E C 2'~ on a plane with Cartesian coordinates x = (Xl, z2). Let F consist of N simple closed curves rl, ... , FN having no common points. We specify the direction of the circuit as positive if the domain D is on its right. Suppose that the boundary r is parametrized, z(s) = (xl(s), zu(S)) is the arc length s measured from some fixed point on each closed part of r. Suppose further that s increases in the positive direction and that there is a one-to-one correspondence between the values of s and the points of F (except for the above-mentioned fixed points). By n we denote the outward normal on F with respect to D. Let us introduce some notation and definitions which we shall need in what follows. Let u(t, z) = u(t, za, z~) be a sufficiently smooth function defined for t > 0 and x E D. Assuming that n is the outward normal at a point x(s) E F, we define the differential operator 2q't~ at the points E D by setting

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