Uniform Energy Bounds Of Solutions To3D Nonlinear Wave Equations With Null Condition

This article concerns the time growth of Sobolev norms of classical solutions to the3D quasi-linear wave equations with the null condition. Given initial data belonging to Hs x Hs-1with compact supports, the global well-posedness theory has been established independently by Klainerman [22] and Christodoulou [5], respectively, for a relatively large integer s after introducting the well-known null condition which ensures the solution behaving like a solution of the linear wave equation for large time. The result in [22] depends on the full range of F multipliers(Translation,Spatial rotation,Lorentz boost and Scaling), and the higher order Sobolev energy, namely, the Hs energy of the solution proved there may have a polynomial growth in time.Though this method was utilized extensively to illustrate the global well-posed or lifespan results in the following papers,the usage of the full range of F multipliers could only be possible when considering problems concerning the wave equation with one single speed due to the bad commutation property of the Lorentz boosts with other kinds of wave equations. In subsequent papers [24,32,33,34,7],the method without using the Lorentz boost has been ex-ploited and extended for various kinds of equations.However,all these papers distinguished two different levels energy, and from deriving an ODE system the lower order Sobolev energy of must remain small, while the higher Sobolev en-ergy will grow polynomially in time. Recently,Wang F.[37] proved the uniform energy bound for the wave equation with one single speed using the full range of F multipliers.However,the proof there strongly relies on the Klainerman-Sobolev inequality and the relatively better decomposition of ordinary derivatives with F vector fields to get good decay.In this paper, we show that the Hs energy of solutions is also uniformly bounded for s≧9without using the Lorentz boosts or deriving a couple of differential equations for both high energy and low energy. The proof employs the generalized energy method of Klainerman and Sideris [24], enhanced by weighted L2estimates and the ghost weight introduced by Alinhac [2].