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

The top-order energy of quasilinear wave equations in two space dimensions is uniformly bounded