## 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)

### MSC:

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

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