Global existence for nonlinear wave equations with multiple speeds
| dc.creator | Sogge, Christopher D. | |
| dc.date | 2002-02-04 | |
| dc.date | 2002-10-10 | |
| dc.date.accessioned | 2026-07-07T04:46:17Z | |
| dc.date.available | 2026-07-07T04:46:17Z | |
| dc.description | We shall be concerned with the Cauchy problem for quasilinear systems in three space dimensions of the form \label{i.1} \partial^2_tu^I-c^2_IΔu^I = C^{IJK}_{abc}\partial_c u^J\partial_a\partial_b u^K + B^{IJK}_{ab}\partial_a u^J\partial_b u^K, \quad I=1,..., D. Here we are using the convention of summing repeated indices, and $\partial u$ denotes the space-time gradient, $\partial u=(\partial_0 u, \partial_1 u, \partial_2 u, \partial_3u)$, with $\partial_0=\partial_t$, and $\partial_j=\partial_{x_j}$, $j=1,2,3$. We shall be in the nonrelativistic case where we assume that the wave speeds $c_k$ are all positive but not necessarily equal. Using a new pointwise estimate of the M. Keel, H. Smith and the author we shall prove global existence of small amplitude solutions for such equations satisfying a null condition. This generalizes the earlier result of Christodoulou and Klainerman where all the wave speeds are the same. Our approach is related to that of Klainerman; however, since we are in the non-relativistic case we cannot use the Lorentz boost vector fields or the Morawetz vector fields. Instead we exploit both the 1/t decay of linear solutions as well as the much easier to prove 1/|x| decay. | |
| dc.description | 14 pages, to appear in Proceedings of the 2001 Mount Holyoke Conference on Harmonic Analysis. Corrected a couple of typos | |
| dc.identifier | https://arxiv.org/abs/math/0202031 | |
| dc.identifier | http://arxiv.org/abs/math/0202031 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/63273 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35L70; 42B99 | |
| dc.title | Global existence for nonlinear wave equations with multiple speeds | |
| dc.type | text |