Sharp local well-posedness for the “good” Boussinesq equation

Abstract

In the present article, we prove the sharp local well-posedness and ill-posedness results for the “good” Boussinesq equation on 1d torus; the initial value problem is locally well-posed in H−1/2(T) and ill-posed in Hs(T) for s<−12. Well-posedness result is obtained from reduction of the problem into a quadratic nonlinear Schrödinger equation and the contraction argument in suitably modified Xs,b spaces. The proof of the crucial bilinear estimates in these spaces, especially in the lowest regularity, rely on some bilinear estimates for periodic functions in Xs,b spaces, which are generalization of the bilinear refinement of the L4 Strichartz estimate on R. Our result improves the known local well-posedness in Hs(T) with s>−38 given by Oh, Stefanov (2012). Similar ideas also establish the sharp local well-posedness in H−1/2(R) and ill-posedness below H−1/2 for the nonperiodic case, which improves the result of Tsugawa and the author (2010) in Hs(R) with s>−12.