Space-time estimates for null forms and the local existence theorem. (English) Zbl 0803.35095

For nonlinear wave equations of the type \((\partial^ 2_ t - \Delta) \Phi^ i = {\mathcal F}^ i (\Phi,D \Phi)\), \(i=1, \dots,N\), \(\Phi (0,x) = f_ 0(x)\), \(\Phi_ t(0,x) = f_ 1(x)\), \(f_ 0 \in H^{s + 1} (\mathbb{R}^ n)\), \(f_ n \in H^ s (\mathbb{R}^ n)\), a local existence theorem in a low regularity class is presented. For \({\mathcal F}^ i = \Gamma^ i_{jk} (\Phi) B^ i_{jk} (D \Phi^ j,D \Phi^ k)\), \(D\) denoting first derivatives, and \(B^ i_{jk}\) denoting any of the null forms \(Q_ 0 = \partial_ \alpha \Phi \partial^ \alpha \psi\), \(Q_{\alpha \beta} = \partial_ \alpha \Phi \partial_ \beta \psi - \partial_ \beta \Phi \partial_ \alpha \psi\), the well-posedness for \(s=1\), \(n=3\) is proved. The essential ingredients are space-time estimates in \(L^ 2([0,T] \times \mathbb{R}^ 3)\) for \(DQ (\Phi, \psi)\), \(Q\) representing \(B^ i_{jk}\) for solutions \(\Phi, \psi\) to linear, inhomogeneous wave equations. Further results and interesting remarks deal with the optimality of the results, the validity for general bilinear forms \(B^ i_{jk}\) and for spherically symmetric solutions, as well as with corresponding results in other space dimensions, and with extensions to estimates for \(\Delta^{-1/2} Q\) replacing \(DQ\).
Reviewer: R.Racke (Konstanz)


35L70 Second-order nonlinear hyperbolic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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