71 research outputs found
On local well-posedness of nonlinear dispersive equations with partially regular data
We revisit the local well-posedness theory of nonlinear Schr\"odinger and
wave equations in Sobolev spaces and , . The
theory has been well established over the past few decades under Sobolev
initial data regular with respect to all spatial variables. But here, we reveal
that the initial data do not need to have complete regularity like Sobolev
spaces, but only partially regularity with respect to some variables is
sufficient. To develop such a new theory, we suggest a refined Strichartz
estimate which has a different norm for each spatial variable. This makes it
possible to extract a different integrability/regularity of the data from each
variable.Comment: To appear in J. Differential Equations, 15 page
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