Markus Keel

Multilinear estimates for periodic KdV equations, and applications

We prove an endpoint multilinear estimate for the Xs,b spaces associated to the periodic Airy equation. As a consequence we obtain sharp local well-posedness results for periodic generalized KdV equations, as well as some global well-posedness results below the energy norm.

Almost global existence for some semilinear wave equations

We prove almost global existence for semilinear wave equations outside of nontrapping obstacles. We use the vector field method, but only use the generators of translations and Euclidean rotations. Our method exploits 1/r decay of wave equations, as opposed to the much harder to prove 1/t decay.

Almost global existence for quasilinear wave equations in three space dimensions

This article studies almost global existence for solutions of quadratically quasi linear systems of wave equations in three space dimensions. The approach here uses only the classical invariance of the wave operator under translations, spatial rotations, and scaling. Using these techniques we can handle wave equations in Minkowski space or Dirichlet-wave equations in the exterior of a smooth, star shaped obstacle. We can also apply our methods to systems of quasilinear wave equations having different wave speeds. This extends our work [11] for the semilinear case. Previous almost global ex istence theorems for quasilinear equations in three space dimensions were for the non-obstacle case. In [9], John and Klainerman proved almost global existence on Minkowski space for quadratic, quasilinear equations using the Lorentz invariance of the wave operator in addition to the symmetries listed above. Subsequently, in [14], Klainerman and Sideris obtained the same result for a class of quadratic, divergence-form nonlinearities without relying on Lorentz invariance. This line of thought was refined and applied to prove global-in-time results for null-form equa tions related to the theory of elasticity in Sideris [22], [23], and for multiple-speed systems of null-form quasilinear equations in Sideris and Tu [24], and Yokoyama [29]. The main difference between our approach and the earlier ones is that we ex ploit the 0(|x|-1) decay of solutions of wave equations with sufficiently decaying initial data as much as we involve the stronger 0(t~l) decay. Here, of course, x = (x\,X2,x

On global existence for nonlinear wave equations outside of convex obstacles

) is the spatial component, and t the time component, of a space time vector (t, x) G M+ x E3. Establishing 0(|x|_1) decay is considerably easier and can be achieved using only the invariance with respect to translations and spatial rotation. A weighted L2 space-time estimate for inhomogeneous wave equations (Proposition 3.1 below, from [11]) is important in making the spatial decay useful for the long-time existence argument. For semilinear systems, one can show almost global existence from small data using only this spatial decay [11]. For quasilinear systems, however, we also have to show that both first and second derivatives of u decay like 1/t. Fortunately, we can do this using a variant of some L1 ?> L?? estimates of John, H?rmander,

Endpoint Strichartz estimates

The authors prove global existence of small solutions to a semilinear wave equation outside of convex obstacles. This extends results of Christodoulou and Klainerman who handled the Minkowski space version. The proof is a compromise of the methods of Christodoulou and Klainerman. It relies on local estimates proved earlier by Smith and Sogge together with classical energy decay estimates for the wave equation of Morawetz, Lax and Phillips.