Spectral theory and nonlinear partial differential equations: A survey
Wilhelm Schlag
Discrete & Continuous Dynamical Systems - A 2006, 15(3): 703-723 doi: 10.3934/dcds.2006.15.703
This paper reviews some of the recent work of the author on stable manifolds for unstable evolution equations. In particular, we discuss such a result, jointly with Joachim Krieger, for the critical focusing nonlinear wave equation in three dimensions.
keywords: orbital and asymptotic stability of stationary waves Focusing NLS and NLW spectral theory. criticality
Regularity and convergence rates for the Lyapunov exponents of linear cocycles
Wilhelm Schlag
Journal of Modern Dynamics 2013, 7(4): 619-637 doi: 10.3934/jmd.2013.7.619
We consider cocycles $\tilde A: \mathbb{T}\times K^d \ni (x,v)\mapsto ( x+\omega, A(x,E)v)$ with $\omega$ Diophantine, $K=\mathbb{R}$ or $K=\mathbb{C}$. We assume that $A: \mathbb{T}\times \mathfrak{E} \to GL(d,K)$ is continuous, depends analytically on $x\in\mathbb{T}$ and is Hölder in $E\in \mathfrak{E} $, where $\mathfrak{E}$ is a compact metric space. It is shown that if all Lyapunov exponents are distinct at one point $E_{0}\in\mathfrak{E}$, then they remain distinct near $E$. Moreover, they depend in a Hölder fashion on $E\in B$ for any ball $B\subset \mathfrak{E}$ where they are distinct. Similar results, with a weaker modulus of continuity, hold for higher-dimensional tori $\mathbb{T}^\nu$ with a Diophantine shift. We also derive optimal statements about the rate of convergence of the finite-scale Lyapunov exponents to their infinite-scale counterparts. A key ingredient in our arguments is the Avalanche Principle, a deterministic statement about long finite products of invertible matrices, which goes back to work of Michael Goldstein and the author. We also discuss applications of our techniques to products of random matrices.
keywords: Lyapunov exponents. shift dynamics Multiplicative Ergodic Theorem
Global dynamics of the nonradial energy-critical wave equation above the ground state energy
Joachim Krieger Kenji Nakanishi Wilhelm Schlag
Discrete & Continuous Dynamical Systems - A 2013, 33(6): 2423-2450 doi: 10.3934/dcds.2013.33.2423
In this paper we establish the existence of certain classes of solutions to the energy critical nonlinear wave equation in dimensions $3$ and $5$ assuming that the energy exceeds the ground state energy only by a small amount. No radial assumption is made. We find that there exist four sets in $\dot H^{1}\times L^{2}$ with nonempty interiors which correspond to all possible combinations of finite-time blowup on the one hand, and global existence and scattering to a free wave, on the other hand, as $t → ±∞$.
keywords: blowup stability invariant manifold. scattering Critical wave equation

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