Summer School

Hamiltonian PDE's and Variational Methods

CAPRI - Villa Orlandi 8-12 september 2008

Titles of lectures and talks

• Walter Craig (McMaster University, Hamilton, Canada): Hamiltonian Partial Differential Equations
• Michael Struwe (ETH Zürich, Switzerland): Equivariant Wave Maps
• John F. Toland (University of Bath, United Kingdom): Lectures on Real-Analytic Operator Equations
• Pietro Baldi: Periodic travelling water waves under nonlinear elastic membranes
• Massimiliano Berti: Un teorema di funzione implicita alla Nash-Moser
• Luca Biasco: Periodic orbits close to invariant manifolds for infinite dimensional hamiltonian systems
• Michela Procesi: Nonlinear wave equations on compact Lie groups

Timetable

Abstracts

Many partial differential equations of physical relevance to nonlinear wave evolution can be formulated as Hamiltonian dynamical systems with infinitely many degrees of freedom. Examples include the Euler equations for surface and interfacial waves in incompressible fluids, the nonlinear Schrödinger equations of modulation and many-body quantum theory, the Gross-Pitaevski equation of Bose-Einstein condensates, and the Fermi-Pasta-Ulam system.

The fact that a system can be written in Hamiltonian form has certain implications for its conserved integrals of motion, and for the structure of solutions in its phase space. In these lectures we will

1. introduce the general picture of the phase space, the flow for PDEs, and some of their basic conserved quantities.
2. We will go on to study invariant tori constructed through infinite dimensional versions of KAM theory,
3. normal forms transformations and the concept of Nekhoroshev stability,
4. and finally we will give several cases in which slowly diffusing orbits can be constructed, which significantly increase its higher Sobolev norms, and which may be considered a version of Arnol'd difusion.

Notes of the course are also available: Lecture 1, Lecture 2, Lectures 3-4..

We introduce the concept of wave maps and show that the Cauchy problem is well-posed for small data in the critical norm.

We then focus on the co-rotational case in 2 space dimensions for which global results are available. Finally, we show that the Cauchy problem for radially symmetric data is globally well-posed in 2 space dimensions for arbitrary closed targets.

Notes for the course are also available.

Our presentation will be based on the following references:

• J. Shatah, M. Struwe: The Cauchy problem for wave maps, Int. Math. Res. Not. 2002, no. 11, 555-571
• M. Struwe: Equivariant wave maps in two space dimensions, Comm. Pure Appl. Math. 56 (2003), no. 7, 815-823.
• M. Struwe: Radially symmetric wave maps from (1+2)-dimensional Minkowski space to general targets, Calc. Var. Partial Differential Equations 16 (2003), no. 4, 431-437.

Please consult

• J. Shatah, M. Struwe: Geometric wave equations, Courant Lecture Notes 2, New York University (1998), 2nd edition: AMS (2000).

for background material and further references.

When X and Y are Banach spaces and U is open in X, a function ƒ : U → Y is real-analytic if it is C^∞ on U and, in a neighbourhood of every point of U, it is given by the sum of its Taylor series at that point. For example, the function ƒ(x, y) = xy is a polynomial, and hence real-analytic, from R² to R. This simple example shows that the zeros of a real-analytic function need not be isolated (very different from the well known complex-analytic case). However, if ƒ is real-analytic on a connected set U, and zero on an open subset of U, then ƒ must be identically zero on U. On the other hand, any closed set E in R² is the zero set of a C^∞ function¹. The contrast between these observations highlights the huge difference, in the theory of equations, between operators that are real-analytic and that those are merely C^∞.

The special nature of the solution set of ƒ(x) = 0 when ƒ is real-analytic is the main topic of these lectures. We will see that, after Lyapunov-Schmidt reduction, it suffices to study finite-dimensional equations of the form h(z) = 0 where h : Rⁿ → R^m is real-analytic. The solution set of equations like this are called real analytic varieties. The structural implications of that statement will explained in the lectures.

A corollary of the analysis will be a theory of unique global continuation of one-parameter families of solutions to equations of the form F (λ, x) = 0, where F : R × X → Y is real-analytic and λ ∈ R is a distinguished parameter. This includes bifurcation problems and one outcome is a very strong global version of the theorem on bifurcation from a simple eigenvalue.

Topological degree theory is not involved in the argument, which is due to Norman Dancer (Bifurcation theory for analytic operators, Proc. Lond. Math. Soc. XXVI (1973), 359-384, and Global structure of the solution set of non-linear real-analytic eigenvalue problems, Proc. Lond. Math. Soc. XXVI (1973), 359-384.)

The exposition will be along the lines of Analytic Theory of Global Bifurcation, PUP 2003, which was written in a collaboration with Boris Buffoni. Notes are also available.

The course will assume an acquaintance with

• analytic functions of one complex variable;
• linear operator theory in Banach spaces, projections, subspaces etc;
• calculus in Banach spaces, including the multi-linear representation of higher Fréchet derivatives, and C^k and C^∞ operators; implicit and inverse function theorems for C^k functions;
• Lyapunov-Schmidt reduction; bifurcation from a simple eigenvalue for C^k operator equations;

The main material on analytic operators will be reviewed in as much detail as time permits and applications will be mentioned.

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