On the regularity of spherically symmetric wave maps

Abstract

Wave maps are critical points U : M -N of the Lagrangian 2 [ U l = s , 11dLI1I2, where M is an Einsteinian manifold and N a Riemannian one. For the case M = R2.' and LI a spherically symmetric map, it is shown that the solution to the Cauchy problem for LI with smooth initial data of arbitrary size is smooth for all time, provided the target manifold N satisfies the two conditions that: (I) it is either compact or there exists an orthonormal frame of smooth vectorfields on N whose structure functions are bounded and (2) there are two constants c and C such that the smallest eigenvalue X and the largest eigenvalue A of the second fundamental form km of any geodesic sphere X(p, s) of radius s centered at p E N satisfy sh 2 c and sA 5 C( 1 + s). This is proved by first analyzing the energy-momentum tensor and using the second condition to show that near the first possible singularity, the energy of the solution cannot concentrate, and hence is small. One then proves that for targets satisfying the first condition, initial data of small energy imply global regularity of the solution. 01993 John Wiley & Sons, Inc. T [ U ] = (dU, dU) = -/ gaPh,bd,U"dpUb dvM MIf the metric tensor g is positive definite, the resulting Euler-Lagrange equations for the critical point U of 2 will be elliptic, and U is called a harmonic map. If, however, g is Einsteinian, i.e., of index 1, then the equations will be hyperbolic, and the solutions are called wave maps. We study the case M = R2*', the 2 + 1dimensional Minkowski space-time, with the metric (gap) = diag(-1,1,1). Thus, in polar coordinates (t, r, 0) E R x W + x S', the line element of M isThe target N is a smooth, complete and connected Riemannian manifold of dimension n and positive definite metric h. In local coordinates, the equation for U is (1 .2) nu" + ~,(u)apuba~u' = 0,

Geometric renormalization of large energy wave maps

“…for any one-form P α smooth on K [t 2 ,t 1 ] , where P L (t, x) := P 0 (t, x) − x j |x| P j (t, x) the energy from dispersing into multiple points of concentration, or disappearing into the light cone; the basic problem is that control of the ψ S component of the energy density does not seem to directly control the other components without further structural control on φ, even near the light cone (the presence of angular components in the stress energy seems to prevent the Gronwall inequality approach in [7], [50], etc. from being effective, and naive attempts to exploit the negative curvature of the target N seem to require more boundedness control on φ than is currently available.…”

Polyhomogeneous solutions of wave equations in the radiation regime

“…On the other hand, Theorem 3.2 below gives actually r^ = z/2 for wave-maps on 2 + 1 dimensional Minkowski space-time. We note that the 2 + 1 dimensional wave map equation is related to the vacuum Einstein equations with cylindrical symmetry {cf eg [8,19,20]). …”

Conservative Solutions to a Nonlinear Variational Wave Equation

“…In the special case α = β = γ, this variational principle (1.9) yields the equation for harmonic wave maps from (1+3)-dimensional Minkowski space into the two sphere, see [8,31,32] for example. For planar deformations depending on a single space variable x, the director field has the special form n = cos u(x, t)e x + sin u(x, t)e y , where the dependent variable u ∈ R 1 measures the angle of the director field to the x-direction, and e x and e y are the coordinate vectors in the x and y directions, respectively.…”