September 17 - 21, 2012
PDEs as conservative or dissipative systems. Stability and instability of solutions.
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 in Euclidean and Banach spaces. Solution operators as groups or semigroups. Nonlinear functional analysis. Infinitesimal generators.
Local invariant manifolds for dynamical systems in Banach space.
In the first lecture, we study the Lyapunov exponents and their associated invariant
subspaces for random dynamical systems, which are generated by, for example, stochastic or random differential equations. We present a multiplicative ergodic theorem for infinite dimensional random dynamical systems..
In the second lecture, we present several theorems on smooth conjugacy for random dynamical systems based on their Lyapunov exponents. We also present a stable and unstable manifold theorem with tempered estimates which are used in the construction of conjugacy.
In the third lecture, we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a Brownian motion. We show that, for almost all sample paths of the Brownian motion in the classical Wiener space, the forced equation admits a topological horseshoe of infinitely many branches. This result is then applied to the randomly forced Duffing equation and the pendulum equation.
The theory of dynamical systems is a well established theory dealing with the qualitative properties of difference equations or differential equations. Objectives of this theory are for instance the stability of
equilibrium points, bifurcations, existence of invariant manifolds, attractors, etc. The theory of random
dynamical systems studies the qualitative behavior of systems under the influence of noise. A noise is an
ergodic stochastic process. The main topic of these three lectures is to introduce the foundation of this theory.
In particular we give a mathematical description of several types of noise. In addition, we present the definition
of a random dynamical system as a measurable cocycle. Then we give several examples of random (partial)
differential equations and stochastic (partial) differential equations generating a random dynamical system.
Lecture 1: Basics on random dynamical systems.
We introduce a metric dynamical system as a model of a noise. As an example for such a metric dynamical
system we introduce the (fractional) Brownian motion. We then are able to introduce the ergodic theorem and
tempered random variables. Based on these foundations we can define a random dynamical system. We then
describe how to generate these systems by simple random differential equations. For these systems random
fixed points and describe their bifurcation are studied, [3], [1].
Lecture 2: Partial differential equations and random dynamical systems.
We consider several partial differential equations like 2D Navier-Stokes equations, reaction {diffusions equations, wave equations driven by a white or multiplicative noise. To generate a random dynamical system we
transform these equations by a random cohomology into partial differential equations with random coefficients.
These transformed equation then allows us to generate a random dynamical system. We also present some
ideas how to define random dynamical system for stochastic differential equations driven by a fractional Brownian motion where modern techniques of stochastic integration are used. These techniques allow to generate
random dynamical systems for stochastic partial differential equations with general coefficients in front of the
noise, [7], [5], [4].
Lecture 3: Random attractors.
We present the theory of pullback attractors. The existence of pullback attractors is based on two properties:
the existence of a pullback absorbing set or (asymptotical) compactness of the random dynamical systems. We
then explain methods how these properties can be found for particular equations from mathematical physics.
We will also discuss several questions related to the existence of random attractors, [2], [6].
References
[1] L. Arnold. Random dynamical systems. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998.
[2] L. Arnold and B. Schmalfuss. Fixed points and attractors for random dynamical systems. In Advances in nonlinear stochastic
mechanics (Trondheim, 1995), volume 47 of Solid Mech. Appl., pages 19{28. Kluwer Acad. Publ., Dordrecht, 1996.
[3] H. Bauer. Probability theory, volume 23 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1996.
Translated from the fourth (1991) German edition by Robert B. Burckel and revised by the author.
[4] M. Garrido-Atienza, K. Lu, and B. Schmalfuss. Unstable manifolds for a stochastic partial differential equation driven by a
ractional Brownian. J. Differential Equations, 248(7):1637{1667, 2010.
[5] B. Malowski and Nualart. D. Evolution equations driven by a fractional brownian motion. Journal of Functional Analysis,
202:277305, 2003.
[6] B. Schmalfuss. The random attractor of the stochastic Lorenz system. Z. Angew. Math. Phys., 48(6):951{975, 1997.
[7] B. Schmalfuss. Attractors for the non-autonomous dynamical systems. In International Conference on Differential Equations,
Vol. 1, 2 (Berlin, 1999), pages 684{689. World Sci. Publishing, River Edge, NJ, 2000.
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