Stable manifolds to bounded solutions in possibly ill-posed PDEs

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:268

Article

Cheng, Hongyu^1,2,3  de la Llave, Rafael³ 

[1] Nankai Univ, Chern Inst Math, Tianjin 300071, Peoples R China

[2] Nankai Univ, LPMC, Tianjin 300071, Peoples R China

[3] Georgia Inst Technol, Sch Math, 686 Cherry St, Atlanta, GA 30332 USA

关键词: Stable manifold;    Partial differential equation;    Boussinesq equation;    Complex Ginzburg-Landau equation;    Mean field games;   

DOI  :  10.1016/j.jde.2019.10.042

来源: Elsevier

【 摘 要 】

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. Related results appear in the literature, but our results allow that the nonlinearity is unbounded and we obtain smoothness of the invariant manifolds. This makes the results in this paper applicable to some several models of current interest that could not be treated otherwise. We discuss in detail the Boussinesq equation of water waves (similar phenomena happen in other long wave approximations) and complex Ginzburg-Landau equation. More recently, we observed [11] that our results also apply to Mean Field Games. 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. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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