Low Regularity A Priori Estimates For The Fourth Order Cubic Nonlinear Schrodinger Equation

We consider the low regularity behavior of the fourth order cubic nonlinear Schrodinger equation (4NLS){i partial derivative tu + partial derivative(4)(x)u = +/-vertical bar u vertical bar 2u, (t, x) is an element of R x Ru(x, 0) = u(0)(x) is an element of H-s (R) .In [29], the author showed that this equation is globally well-posed in H-s (R),s >= -1/2 and mildly ill-posed in the sense that the solution map fails to be locally uniformly continuous for -15/14 < s < -1/2 Therefore, s = -1/2 is the lowest regularity that can be handled by the contraction argument. In spite of this mild ill-posedness result, we obtain an a priori bound below s < -1/2. This an a priori estimate guarantees the existence of a weak solution for -3/4 < s < -1/2. Our method is inspired by Koch-Tataru [17]. We use the U-P and V-P based spaces adapted to frequency dependent time intervals on which the nonlinear evolution can still be described by linear dynamics.