  In this paper, we study the global well-posedness and scattering theory for
the defocusing fourth-order nonlinear Schr\"odinger equation (FNLS)
$iu_t+\Delta^2 u+|u|^pu=0$ in dimension $d\geq9$. We prove that if the solution
$u$ is apriorily bounded in the critical Sobolev space, that is, $u\in
L_t^\infty(I;\dot H^{s_c}_x(\R^d))$ with all $s_c:=\frac{d}2-\frac4p\geq1$ if
$p$ is an even integer or $s_c\in[1,2+p)$ otherwise, then $u$ is global and
scatters. The impetus to consider this problem stems from a series of recent
works for the energy-supercritical and energy-subcritical nonlinear
Schr\"odinger equation (NLS) and nonlinear wave equation (NLW). We will give a
uniform way to treat the energy-subcritical, energy-critical and
energy-supercritical FNLS, where we utilize the strategy derived from
concentration compactness ideas to show that the proof of the global
well-posedness and scattering is reduced to exclude the existence of three
scenarios: finite time blowup; soliton-like solution and low to high frequency
cascade. Making use of the No-waste Duhamel formula, we deduce that the energy
or mass of the finite time blow-up solution is zero and so get a contradiction.
Finally, we adopt the double Duhamel trick, the interaction Morawetz estimate
and interpolation to kill the last two scenarios.

Analysis of PDEs