Klainerman, Sergiu; Selberg, Sigmund Remark on the optimal regularity for equations of wave maps type. (English) Zbl 0884.35102 Commun. Partial Differ. Equations 22, No. 5-6, 901-918 (1997). 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. Reviewer: L.Skrzypczak (Poznań) Cited in 2 ReviewsCited in 48 Documents MSC: 35L70 Second-order nonlinear hyperbolic equations 35L15 Initial value problems for second-order hyperbolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) Keywords:modified Sobolev-type spaces; pointwise multiplication properties PDFBibTeX XMLCite \textit{S. Klainerman} and \textit{S. Selberg}, Commun. Partial Differ. Equations 22, No. 5--6, 901--918 (1997; Zbl 0884.35102) Full Text: DOI References: [1] Bourgain J., Funct. Anal. 3 pp 107– (1993) · Zbl 0787.35097 · doi:10.1007/BF01896020 [2] Klainerman S., Comm. Pure Appl. Matah. 46 pp 1221– (1993) · Zbl 0803.35095 · doi:10.1002/cpa.3160460902 [3] Klainerman S., Duke Math. J. 81 pp 99– (1995) · Zbl 0909.35094 · doi:10.1215/S0012-7094-95-08109-5 [4] Klainerman S., \(au:1 (1996)\) [5] Klainerman S., Duke Math. J [6] Zhou Y., Math. Z This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.