On the cauchy problem for equivariant wave maps

Abstract

IntroductionWave maps are maps U from a Lorentzian manifold M"" (a manifold with a metric g of index 1) into a Riemannian manifold N" (one with a positive definite metric h ) that are the critical points (with respect to compactly supported variations) of the Lagrangian If M and N are rotationally symmetric, i.e., SO(n) and SO(K) act on M and N as isometries, then we can define equivariant maps by requiring that the orbit of any point in M under spatial rotations maps into the orbit of the image point in N . When this mapping of the orbits has degree 1, we call the map corotatzonaf.In a previous paper (see [ 171) we have shown that in the case M = [W2,I, the 2 + 1 -dimensional Minkowski space-time, and N , a rotationally symmetric surface diffeomorphic to I F! ', the Cauchy problem with smooth data of any size for the corotational wave map from M into N has globally smooth solutions, provided that the target N is geodesically convex. This condition was later laxed by Grillakis (see [9]), to include a certain class of nonconvex targets. Our proof of regularity was also simplified later by Shatah and Struwe; see [16]. In this paper we obtain several results for equivariant wave maps in the areas of existence and uniqueness, regularity, asymptotic behavior, development of singularities, and weak solutions:1. Existence and Uniqueness. We obtain a local-in-time existence and uniqueness theorem for equivariant wave maps that requires a minimal amount of

Almost optimal local well-posedness for the (3+1)-dimensional Maxwell–Klein–Gordon equations

“…2 The local and global existence theory for the nonlinear wave equations of mathematical physics has seen exciting and fast-paced progress recently for a particular set of equations, those of the wave-maps from Minkowski space into a complete Riemannian manifold. See for instance [32] and [34], [30]- [31], [19], [25], [10], [11], and finally [35]. While the small data regularity problem for these equations is now largely complete, the corresponding problem for gauge field equations seems to be much more difficult and is still far from understood.…”

The Cauchy problem for wave maps on a curved background

“…The argument given for uniqueness in [31] adapts perfectly to our case and we reproduce it below for completeness.…”

“…In [31], Shatah and Struwe consider the Cauchy problem for wave maps u : R 1+d → N with initial data (u 0 , u 1 …”

“…In what follows, we restrict our attention to the case that the background manifold (M, g) is (R 4 , g), with g a small perturbation of the Euclidean metric, as in this case we have suitable replacements for the technical tools used in [31]. Here we view the equations for the components of connection form, A, as a system of variable coefficient elliptic equations and prove elliptic estimates via a perturbative argument, see Proposition 3.2.…”

“…We will assume that (N, h) is a smooth complete Riemannian manifold, without boundary that is isometrically embedded into R m . Following [31], we also assume that N has bounded geometry in the sense that the curvature tensor, R, and the second fundamental form, S, of the embedding are bounded and all of their derivatives are bounded.…”