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On the regularity properties of the wave equation. (English) Zbl 0833.35093
Flato, M. (ed.) et al., Physics on manifolds. Proceedings of the international colloquium analysis, manifolds and physics in honour of Yvonne Choquet-Bruhat, Paris, France, June 3-5, 1992. Dordrecht: Kluwer Academic Publishers. Math. Phys. Stud. 15, 177-191 (1994).
The authors report on the regularity of solutions to systems of the form \[ \square \varphi^i= F^i(\varphi, D\varphi), \]
\[ \varphi(0, x)= f_0(x),\quad \partial_t \varphi(0, x)= f_1(x),\quad x\in \mathbb{R}^n, \] where \(F^i\) is a smooth function of \(\varphi\), \(D\varphi\), quadratic relative to \(D\varphi\). Recalling the classical existence theorem with \(f_0\in H^{m+ 1}(\mathbb{R}^n)\), \(f_1\in H^m(\mathbb{R}^n)\), \(m= n/2+ \varepsilon\), \(\varepsilon> 0\), recent improvements of the authors and others with respect to the regularity assumptions on the data are described, in particular the necessary space- time estimates. Three conjectures are formulated, also for the special situation that \(F\) satisfies the null condition. Illustrations are given by the Maxwell and by the Dirac equations.
For the entire collection see [Zbl 0818.00018].
Reviewer: R.Racke (Konstanz)

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