WELL-POSEDNESS FOR THE FOURTH-ORDER SCHRODINGER EQUATIONS WITH QUADRATIC NONLINEARITY

This paper is concerned with one-dimensional quadratic semilinear fourth-order Schrodinger equations. Motivated by the quadratic Schrodinger equations in the pioneering work of Kenig-Ponce-Vega [12], three bilinearities uv, (uv) over bar, and (uv) over bar for functions u, v : R x [0,T] -> C are sharply estimated in function spaces X(s,b) associated to the fourth-order Schrodinger operator i partial derivative(t) + Delta(2) - epsilon Delta. These bilinear estimates imply local wellposedness results for fourth-order Schrodinger equations with quadratic nonlinearity. To establish these bilinear estimates, we derive a fundamental estimate on dyadic blocks for the fourth-order Schrodinger from the [k, Z]-multiplier norm argument of Tao [20].