Estimates for null forms and the spaces \(\mathbb{H}_{s,\delta}\). (English) Zbl 0909.35095

From the introduction: The authors continue their investigations [see S. Klainerman and M. Machedon, Commun. Pure Appl. Math. 46, 1221-1268 (1993; Zbl 0803.35095); Duke Math. J. 81, 99-133 (1995; reviewed above); Duke Math. J. 87, 553-589 (1997; Zbl 0878.35075)] concerning the optimal local regularity properties of nonlinear wave equations verifying some form of the “null condition”. They restrict their attention to equations with quadratic nonlinearities of the type \[ \square \phi^I+ F^I(\phi, \partial\phi)= 0, \] where \(\square =-\partial^2_t+\Delta\) denotes the standard d’Alembertian in \(\mathbb{R}^{3+1}\) and the nonlinear terms \(F\) have the form \(F^I= \sum_{J,K} \Gamma^I_{JK} B^I_{JK}(D\phi^J, D\phi^K)\), where \(\Gamma^I_{JK}\) is constant and \(B^I_{JK}\) is any of the null forms: \[ Q_0(\phi,\psi)= \partial_\alpha\phi\cdot \partial^\alpha\psi= -\partial_t\phi \partial_t\psi+ \sum^n_{i= 1} \partial_i\phi\partial_i\psi, \]
\[ Q_{\alpha,\beta}(\phi, \psi)= \partial_\alpha\phi\partial_\beta\psi- \partial_\beta \phi\partial_\alpha\psi\qquad 0\leq\alpha< \beta\leq n. \] The goal of the paper is to prove the main estimate, which implies that the above equations are well posed in the sharp space \(H^s\) for \(s> 3/2\).


35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35B65 Smoothness and regularity of solutions to PDEs
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