# Invariant manifolds of PDEs and applications.

Wednesday, September 19, 2012 - 1:00pm - 2:30pm

Keller 3-180

Chongchun Zeng (Georgia Institute of Technology)

Invariant manifolds and foliations have become very useful tools in

dynamical systems. For infinite dimensional systems generated by

evolutionary PDEs, the mere existence of these structures is

non-trivial compared to those of ODEs due to issues such as the

non-existence of backward (in time) solutions of some PDEs or

nonlinear terms causing derivative losses. In addition to systematic

generalization of the standard theory, often specific treatment has to

be adopted based on the structure of the PDEs under consideration. We

will briefly go through the general invariant manifold theory,

followed by a few concrete PDEs. Also, applications to singular

perturbations and homoclinic orbits for PDEs will be discussed.

REFERENCES:

1. Dan Henry; Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes 840. 2. Chapters 1 and 6 at least.

1. A. Pazy; Semigroups of Linear Operators and Applications to PDEs, Springer Applied Math 44, 2. Chapters 1, 2, and 4.

3. R. Temam; Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer Applied Math 68, Chapters 1-3.

4. P. Bates and C. Jones; Invariant Manifolds for Semilinear PDEs, Dynamics Reported V2, 1989.

5. K, Nakanishi and W. Schlag, Invariant Manifolds and Dispersive Hamiltonian Evolution Eqts, European Math Soc. 2011.

6. Unstable manifolds of Euler equations, Z. Lin and C. Zeng, â€¨avalaible at arxiv.org

7. Inviscid dynamical structures near Couette flow, Z. Lin and C. â€¨Zeng, ARMA and also avalaible at arxiv.org

8. Existence and persistence of invariant manifolds for semiflows in â€¨Banach space, P. Bates, K. Lu, and C. Zeng, Memoirs of AMS

9. Approximately invariant manifolds and global dynamics of spike â€¨states, P. Bates, K. Lu, and C. Zeng, Inventiones mathematicae

10. Invariant manifolds around soliton manifolds for the nonlinear ¨Klein-Gordon equation, Kenji Nakanishi, Wilhelm Schla, SIAM Math. Anal. and also avalaible at arxiv.org

dynamical systems. For infinite dimensional systems generated by

evolutionary PDEs, the mere existence of these structures is

non-trivial compared to those of ODEs due to issues such as the

non-existence of backward (in time) solutions of some PDEs or

nonlinear terms causing derivative losses. In addition to systematic

generalization of the standard theory, often specific treatment has to

be adopted based on the structure of the PDEs under consideration. We

will briefly go through the general invariant manifold theory,

followed by a few concrete PDEs. Also, applications to singular

perturbations and homoclinic orbits for PDEs will be discussed.

REFERENCES:

1. Dan Henry; Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes 840. 2. Chapters 1 and 6 at least.

1. A. Pazy; Semigroups of Linear Operators and Applications to PDEs, Springer Applied Math 44, 2. Chapters 1, 2, and 4.

3. R. Temam; Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer Applied Math 68, Chapters 1-3.

4. P. Bates and C. Jones; Invariant Manifolds for Semilinear PDEs, Dynamics Reported V2, 1989.

5. K, Nakanishi and W. Schlag, Invariant Manifolds and Dispersive Hamiltonian Evolution Eqts, European Math Soc. 2011.

6. Unstable manifolds of Euler equations, Z. Lin and C. Zeng, â€¨avalaible at arxiv.org

7. Inviscid dynamical structures near Couette flow, Z. Lin and C. â€¨Zeng, ARMA and also avalaible at arxiv.org

8. Existence and persistence of invariant manifolds for semiflows in â€¨Banach space, P. Bates, K. Lu, and C. Zeng, Memoirs of AMS

9. Approximately invariant manifolds and global dynamics of spike â€¨states, P. Bates, K. Lu, and C. Zeng, Inventiones mathematicae

10. Invariant manifolds around soliton manifolds for the nonlinear ¨Klein-Gordon equation, Kenji Nakanishi, Wilhelm Schla, SIAM Math. Anal. and also avalaible at arxiv.org

MSC Code:

35Q90

Keywords: