42 research outputs found
On the regularity of the composition of diffeomorphisms
For being a closed manifold or the Euclidean space we present a detailed
proof of regularity properties of the composition of -regular
diffeomorphisms of for
Hierarchy of integrable geodesic flows
A family of integrable geodesic flows is obtained. Any such a family corresponds to a pair of geodesically equivalent metrics
The Miura Map on the Line
The Miura map (introduced by Miura) is a nonlinear map between function
spaces which transforms smooth solutions of the modified Korteweg - de Vries
equation (mKdV) to solutions of the Korteweg - de Vries equation (KdV). In this
paper we study relations between the Miura map and Schroedinger operators with
real-valued distributional potentials (possibly not decaying at infinity) from
various spaces. We also investigate mapping properties of the Miura map in
these spaces. As an application we prove existence of solutions of the Korteweg
- de Vries equation in the negative Sobolev space H^{-1} for the initial data
in the range of the Miura map.Comment: 33 page
Spatially quasi-periodic solutions of the Euler equation
We develop a framework for studying quasi-periodic maps and diffeomorphisms
on . As an application, we prove that the Euler equation is
locally well posed in a space of quasi-periodic vector fields on
. In particular, the equation preserves the spatial
quasi-periodicity of the initial data. Several results on the analytic
dependence of solutions on the time and the initial data are proved
On the low regularity phase space of the Benjamin-Ono equation
In this paper we prove that the Benjamin-Ono equation is globally in time
-well-posed in the Hilbert space
of periodic distributions in
with -weights. The space
can thus be considered as a
maximal low regularity phase space for the Benjamin-Ono equation corresponding
to the scale ,