- 19-6 Hongyu Cheng, Rafael de la Llave
- Stable Manifolds to Bounded Solutions in Possibly Ill-posed PDEs
Jan 23, 19
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Abstract. We prove several results establishing existence and regularity of
stable manifolds for different classes of special solutions
for evolution equations (these equations may be ill-posed):
a single specific solution, an invariant torus
filled with quasiperiodic orbits
or more general manifolds of solutions. In the later cases, which
include several orbits, we also establish the invariant manifolds
of an orbit depend smoothly on the orbit (analytically
in the case of quasi-periodic orbits and finitely differentiably
in the case of more general families).
We first establish a general abstract theorem which, under suitable
(spectral, non-degeneracy, analyticity) assumptions on the linearized equation,
establishes the existence of the desired manifold.
Then we present concrete applications of the abstract results
to the ill-posed Boussinesq equation for long wave approximation of water waves and complex Ginzburg-Landau equation.
Since the equations we consider may be ill-posed, part of the requirements
for the stable manifold is that one can define the (forward) dynamics on
them. Note also that the methods that are based in the
existence of dynamics (such as graph transform) do not apply to ill-posed
We use the methods based on integral equations (Perron method) associated with the partial dynamics,
but we need to take advantage of smoothing properties of
the partial dynamics.
Note that, even if the families of solutions we started with are finite
dimensional, the stable manifolds may be infinite dimensional.