4,142 research outputs found
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
Global bifurcation for monotone fronts of elliptic equations
In this paper, we present two results on global continuation of monotone
front-type solutions to elliptic PDEs posed on infinite cylinders. This is done
under quite general assumptions, and in particular applies even to fully
nonlinear equations as well as quasilinear problems with transmission boundary
conditions. Our approach is rooted in the analytic global bifurcation theory of
Dancer and Buffoni--Toland, but extending it to unbounded domains requires
contending with new potential limiting behavior relating to loss of
compactness. We obtain an exhaustive set of alternatives for the global
behavior of the solution curve that is sharp, with each possibility having a
direct analogue in the bifurcation theory of second-order ODEs.
As a major application of the general theory, we construct global families of
internal hydrodynamic bores. These are traveling front solutions of the full
two-phase Euler equation in two dimensions. The fluids are confined to a
channel that is bounded above and below by rigid walls, with incompressible and
irrotational flow in each layer. Small-amplitude fronts for this system have
been obtained by several authors. We give the first large-amplitude result in
the form of continuous curves of elevation and depression bores. Following the
elevation curve to its extreme, we find waves whose interfaces either overturn
(develop a vertical tangent) or become exceptionally singular in that the flow
in both layers degenerates at a single point on the boundary. For the curve of
depression waves, we prove that either the interface overturns or it comes into
contact with the upper wall.Comment: 60 pages, 6 figure
- β¦