Well-posedness for the fourth-order Schrödinger equations with quadratic nonlinearity

Abstract

This paper is concerned with one-dimensional quadratic semilinear fourth-order Schrödinger equations. Motivated by the quadratic Schrödinger equations in the pioneering work of Kenig-Ponce-Vega [12], three bilinearities $uv,$ $\overline{uv},$ and $\overline{u}v $ for functions $u,v: \mathbb{R}\times[0,T]\mapsto \mathbb{C} $ are sharply estimated in function spaces $X_{s,b}$ associated to the fourth-order Schrödinger operator $i\partial_t+\Delta^2-\varepsilon\Delta$. These bilinear estimates imply local wellposedness results for fourth-order Schrödinger equations with quadratic nonlinearity. To establish these bilinear estimates, we derive a fundamental estimate on dyadic blocks for the fourth-order Schrödinger from the $[k,Z]$-multiplier norm argument of Tao [20].