Local smoothing

Local smoothing refers to the phenomenon that highly dispersive equations (in particular, those with infinite speed of propagation) are often locally in higher regularity spaces than they are in globally, provided that one averages in time or assumes some decay condition on the initial data.

The local smoothing for the free Schrödinger equation

$i\partial _{t}u+\Delta u=\chi f,\quad u_{|t=0}=u_{0}\in L^{2}(\mathbb {R} ^{N}).$

writes $\|\chi u\|_{L^{2}(\mathbb {R} ,H^{1/2}(\mathbb {R} ^{N}))}\leq C{\Big (}\|u_{0}\|_{L^{2}}+\|f\|_{L^{2}(\mathbb {R} ^{N},H^{-1/2}(\mathbb {R} ^{N})}{\Big )}$ (Here $\chi$ is a smooth function decreasing fast enough at infinity) and was first observed independently by Constantin and Saut ConSau1988, Sjölin Sl1987, and Vega Ve1988 in the late 80's. The smoothing effect does not always persist when replacing $\mathbb {R} ^{N}$ by another non-compact manifold.

(A lot more references are needed in what follows.) Things are well-understood for metrics that are close to the Euclidian one at infinity, on $\mathbb {R} ^{N}$, or outside a compact obstacle. In this case, the smoothing effect is known to be equivalent to a non-trapping condition on the metric, namely that all rays of geometric optics escape at infinity . This condition is equivalent to the necessary and sufficient condition of local, uniform decay of the energy for the wave equation outside an obstacle, first studied in the pioneer works of Morawetz Mz1961 Mz1966, and Morawetz, Ralston and Strauss MzRalSr1977.