The Initial Value Problem for a Third Order Dispersive Equation on the Two-dimensional Torus

|α|=2 2! α! aσ(α)(x)∂ α +~b(x) · ∇+ c(x), ∂t = ∂/∂t, ∂j = ∂/∂xj , ∂ = ∇ = (∂1, ∂2), p(ξ) = ξ1ξ2(ξ1 + ξ2), α = (α1, α2) is a multi-index, |α| = α1 + α2, α! = α1!α2!, σ(α) = (α1 − α2)/2, ~b(x) = (b1(x), b2(x)), and aσ(α), bj(x), c(x) are real-valued smooth functions on T2. Such operators arise in the study of gravity wave of deep water. See [1], [3], [4] and the references therein. The purpose of this paper is to present the necessary and sufficient condition of the existence of a unique solution to (1)-(2). To state our results, we introduce notation and the definition of L2-wellposedness. C∞(T2) is the set of all smooth functions on T2. L2(T2) is the set of all square-integrable functions on T2. For g∈L2(T2), set ‖g‖ = ( ∫