543 research outputs found
The Cauchy Problem for Wave Maps on a Curved Background
We consider the Cauchy problem for wave maps u: \R times M \to N for
Riemannian manifolds, (M, g) and (N, h). We prove global existence and
uniqueness for initial data that is small in the critical Sobolev norm in the
case (M, g) = (\R^4, g), where g is a small perturbation of the Euclidean
metric. The proof follows the method introduced by Statah and Struwe for
proving global existence and uniqueness of small data wave maps u : \R \times
\R^d \to N in the critical norm, for d at least 4. In our argument we employ
the Strichartz estimates for variable coefficient wave equations established by
Metcalfe and Tataru.Comment: Fixed minor typos in previous version. To appear in Calculus of
Variations and Partial Differential Equation
Scattering for radial, semi-linear, super-critical wave equations with bounded critical norm
In this paper we study the focusing cubic wave equation in 1+5 dimensions
with radial initial data as well as the one-equivariant wave maps equation in
1+3 dimensions with the model target manifolds and
. In both cases the scaling for the equation leaves the
-norm of the solution
invariant, which means that the equation is super-critical with respect to the
conserved energy. Here we prove a conditional scattering result: If the
critical norm of the solution stays bounded on its maximal time of existence,
then the solution is global in time and scatters to free waves both forwards
and backwards in infinite time. The methods in this paper also apply to all
supercritical power-type nonlinearities for both the focusing and defocusing
radial semi-linear equation in 1+5 dimensions, yielding analogous results.Comment: 59 pages, minor typos have been correcte
Relaxation of wave maps exterior to a ball to harmonic maps for all data
In this paper we study 1-equivariant wave maps of finite energy from
1+3-dimensional Minkowski space exterior to the unit ball at the origin into
the 3-sphere. We impose a Dirichlet boundary condition at r=1, meaning that the
unit sphere in R^3 gets mapped to the north pole. Finite energy implies that
spacial infinity gets mapped to either the north or south pole. In particular,
each such equivariant wave map has a well-defined topological degree which is
an integer. We establish relaxation of such a map of arbitrary energy and
degree to the unique stationary harmonic map in its degree class. This settles
a recent conjecture of Bizon, Chmaj, Maliborski who observed this asymptotic
behavior numerically.Comment: keywords: equivariant wave maps, concentration compactness, profile
decomposition, soliton resolution conjecture. Fixed minor typos. To appear in
GAF
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