  Alinhac solved a long-standing open problem in 2001 and established that
quasilinear wave equations in two space dimensions with quadratic null
nonlinearities admit global-in-time solutions, provided that the initial data
are compactly supported and sufficiently small in Sobolev norm. In this work,
Alinhac obtained an upper bound with polynomial growth in time for the
top-order energy of the solutions. A natural question then arises whether the
time-growth is a true phenomena, despite the possible conservation of basic
energy. Analogous problems are also of central importance for Schr\"odinger
equations and the incompressible Euler equations in two space dimensions, as
studied by Bourgain, Colliander-Keel-Staffilani-Takaoka-Tao, Kiselev-Sverak,
and others. In the present paper, we establish that the top-order energy of the
solutions in Alinhac theorem remains globally bounded in time, which is
opposite to Alinhac's blowup-at-infinity conjecture.
