Global existence for nonlinear wave equations with multiple speeds

By Christopher D. Sogge. 2002

53 citations recorded by the source corpus.

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_I\Delta u^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.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/90892f2bb5dc4e0c60f97351c75368531353b767

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Hyperbolic trapped rays and global existence of quasilinear wave equations (2003, background): “Hidano [4], Sideris and Tu [24], Sogge [ 27 ], and Yokoyama [28]).”
Global existence of null-form wave equations in exterior domains (2006, methodology): “In the multiple speed boundaryless case, related results were established, e.g., by Sideris and Tu [26], Sogge [ 28 ], Agemi and Yokoyama [1], Kubota and Yokoyama [16], Hidano [4], Yokoyama [31], and Katayama [7–9].”