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Smoothing estimates for null forms and applications. (English) Zbl 0832.35096
In this announcement, the authors present results for the classical local existence of solutions to nonlinear wave equations which satisfy the null condition, which improve previous results by $$1/2- \varepsilon$$ if the nonlinearities are restricted to wave maps type. In this case, for $$x\in \mathbb{R}^3$$, the initial value problem is shown to be well posed for initial data in $$H^{3/2+ \varepsilon}(\mathbb{R}^3)$$, $$H^{1/2+ \varepsilon}(\mathbb{R}^3)$$, $$\varepsilon> 0$$. Sketches of the proofs are given. Space-time estimates – being of own interest – in norms that shall capture the gain of regularity of the solution $$\Phi$$ to $$\square\Phi= F$$, are the essential tools.
Reviewer: R.Racke (Konstanz)

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35L05 Wave equation
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