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Smoothing estimates for null forms and applications. (English) Zbl 0909.35094
The authors continue their work [Commun. Pure Appl. Math. 46, 1221-1268 (1993; Zbl 0803.35095)]. The main results concern the estimates of a null form as a right-hand side of the considered nonlinear hyperbolic equation. The estimations are formulated for space-time integral type norms with exponent $$s>1/2$$ and for $$1/2< s<1$$ and with a weight function (for the details see the original text). For such forms, the authors prove that the initial value problem $-\partial^2_t\Phi^I+ \Delta\Phi^I+ \sum_{J,K} \Gamma^I_{J,K}(\Phi) Q_0(\Phi^J,\Phi^K)= 0\quad (I= 1,\dots,N)\tag{1}$ with the inhomogeneous data $$\Phi(0, x)= f_0(x)$$, $$\partial_t\Phi(0, x)= f_1(x)$$ is well posed for $$f_0\in H^{3/2+s}$$, $$f_1\in H^{1/2+s}$$, where $$\Gamma^I_{J,K}(\Phi)$$ are real analytic functions in $$\Phi= (\Phi^1,\dots, \Phi^N)$$. The quoted result is sharp for the equation (1).

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] 1 J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations , Geom. Funct. Anal. 3 (1993), no. 2, 107-156. · Zbl 0787.35097 [2] 2 J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation , Geom. Funct. Anal. 3 (1993), no. 3, 209-262. · Zbl 0787.35098 [3] C. Kenig, G. Ponce, and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices , Duke Math. J. 71 (1993), no. 1, 1-21. · Zbl 0787.35090 [4] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem , Comm. Pure Appl. Math. 46 (1993), no. 9, 1221-1268. · Zbl 0803.35095 [5] H. Lindblad, Counterexamples to local existence for quasilinear wave equations , · Zbl 0932.35149 [6] G. Ponce and T. Sideris, Local regularity of nonlinear wave equations in three space dimensions , Comm. Partial Differential Equations 18 (1993), no. 1-2, 169-177. · Zbl 0803.35096 [7] Y. Zhou, private communication.
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