A Criterion for Global Existence in Case of Ordinary Differentia!

Abstract

The non-local existence problem for an ordinary differential equation (D)(d/dt)y = v(t, y) is investigated, the field of directions v being continuously defined for all (üj)eüxZ, where ü denotes a compact real interval, Z the closure of a bounded open set Z ⊂R"n≧2. We suppose v satisfies a boundary condition on J x σZ. which is a quite familiar and natural one in fluid mechanics: v(t,y) has to fall into the tangent space of σZ at y for all (t, y) ejx σZ, σZ being smooth. Then through each point of σZ there passes at least one global solution of (D) which stays on dZ (Theorem 1). From this, v being continuous on üxZ, the corresponding statement of global existence on Z follows by simple argumentation (Theorem 2). Thus far, uniqueness being not required, separation of integral curves of (D) from the part Jx σZ of the boundary σ(JxZ) might occur. If additionally we suppose a uniqueness condition, a stronger result holds (the final corollary). The results established in this note are required for applications in fluid mechanics, [14]