Multilinear Space-Time Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations

We prove a quadrilinear integral estimate in space-time for solutions of the homogeneous wave equation on R. This estimate is a generalization of a previously known bilinear L estimate, and it arises naturally in the study of the local regularity properties of a hyperbolic model equation connected with wave maps from Minkowski space R into a sphere. The scale invariant data space for this equation is L(R), and we prove local well-posedness for data in H(R) for all s > 1/4. In space dimension three and higher, the same equation has previously been studied by Klainerman and Machedon. Using a recently proved Lt (L ∞ x ) bilinear estimate for solutions of the homogeneous wave equation, we obtain a simpler proof of their result, and we also extend it to the full system from which the model equation was derived. The main new idea introduced by Klainerman and Machedon in their work on the aforementioned model equation was to estimate a Picard iterate using information not just from the preceding iterate, but from two previous iterates. This procedure leads to integrals of quadrilinear expressions involving functions in certain “hyperbolic” Sobolev spaces which are adapted to the wave operator. Klainerman and Machedon estimated these expressions by reducing them to trilinear and bilinear L estimates in space-time for solutions of the homogeneous wave equation. Here we show that this reduction is impossible in the two-dimensional case, so the problem is of a genuinely quadrilinear nature. A general framework for proving local well-posedness for nonlinear wave equations based on estimates in space-time Sobolev norms is developed, refining and unifying earlier results of this type.