×

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)

Citations:

Zbl 0803.35095
PDFBibTeX XMLCite
Full Text: DOI

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 · doi:10.1007/BF01896020
[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 · doi:10.1007/BF01895688
[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 · doi:10.1215/S0012-7094-93-07101-3
[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 · doi:10.1002/cpa.3160460902
[5] H. Lindblad, Counterexamples to local existence for quasilinear wave equations , · Zbl 0932.35149 · doi:10.4310/MRL.1998.v5.n5.a5
[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 · doi:10.1080/03605309308820925
[7] Y. Zhou, private communication.
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.