Hongyu Cheng, Rafael de la Llave
Stable Manifolds to Bounded Solutions in Possibly Ill-posed PDEs
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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 
equation. 
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.