**Title:** Damped chaotic waves: exponential stabilization in absence of geometric control

**Speaker:** Stéphane Nonnenmacher

**Speaker Info:** CEA-Saclay

**Brief Description:**

**Special Note**:

**Abstract:**

We consider the (linear) damped wave equation on a Riemannian manifold with Anosov geodesic flow (e.g. a manifold of negative sectional curvature). We assume that damping function vanishes in some regions of the manifold. We want to quantitatively describe the stabilization (decay of the energy) of the waves in such a "chaotic" situation. This leads us to study the spectrum of the damped wave equation, and especially search for a spectrum-free strip below the real axis ("spectral gap"). The presence of a gap is well-known in the case of "geometric control" (all geodesics cross the damping region). In the alternative case, we show that the existence of such a gap relies on the structure of the set of undamped geodesics: roughly speaking, if this set is "thin enough", then there is a (finite) gap, and thereby exponential decay of smooth initial data.Our result uses a "new" type of hyperbolic dispersion estimate, as well as large deviation theory for Anosov flows. See also G.Rivière's talk on the same topic.

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