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Remark on the optimal regularity for equations of wave maps type. (English) Zbl 0884.35102
The authors investigate the equations of wave maps type i.e. the systems of equations of the form $\square \varphi^I+ \Gamma^I_{JK} (\varphi) Q_0(\varphi^J, \varphi^K) =0,$ where $$\square$$ denotes the standard D’Alambertian and $$Q_0(\varphi,\psi) =-\partial_t \varphi \partial_t \psi+ \sum_{i=1}^n \partial_i \varphi \partial_i\psi$$ is a null form. The modified Sobolev-type spaces $$H^{[s,\delta]} (\mathbb{R}^{n+1})$$ are introduced and the bilinear estimates for the null form $$Q_0(\varphi,\psi)$$ in $$H^{[s,\delta]}$$ is proved. For these estimates the author investigates the pointwise multiplication properties of the scale $$H^{[s,\delta]}$$. The estimates imply that for real analytic $$\Gamma (\varphi)$$ the initial value problem for the system of equations, subject to the initial conditions $$\varphi (0,x)\in H^s(\mathbb{R}^n)$$ and $$\partial_t \varphi (0,x)\in H^{s-1} (\mathbb{R}^n)$$, $$s>n/2$$, is well posed.

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35L15 Initial value problems for second-order hyperbolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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##### References:
 [1] Bourgain J., Funct. Anal. 3 pp 107– (1993) · Zbl 0787.35097 [2] Klainerman S., Comm. Pure Appl. Matah. 46 pp 1221– (1993) · Zbl 0803.35095 [3] Klainerman S., Duke Math. J. 81 pp 99– (1995) · Zbl 0909.35094 [4] Klainerman S., $$au:1 (1996)$$ [5] Klainerman S., Duke Math. J [6] Zhou Y., Math. Z
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