38 research outputs found
Small data scattering for a cubic Dirac equation with Hartree type nonlinearity in
We prove that the initial value problem for the Dirac equation
is globally well-posed and the solution scatters to free waves asymptotically
as , if we start with initial data that is small in
for . This is an almost critical well-posedness result in the sense
that is the critical space for the equation. The main ingredients in the
proof are Strichartz estimates, space-time bilinear null-form estimates for
free waves in , and an application of the and -function spaces.Comment: 34 pages: To appear in SIAM J. Math. Analysi
Null structure and local well-posedness in the energy class for the Yang-Mills equations in Lorenz gauge
We demonstrate null structure in the Yang-Mills equations in Lorenz gauge.
Such structure was found in Coulomb gauge by Klainerman and Machedon, who used
it to prove global well-posedness for finite-energy data. Compared with Coulomb
gauge, Lorenz gauge has the advantage---shared with the temporal gauge---that
it can be imposed globally in space even for large solutions. Using the null
structure and bilinear space-time estimates, we also prove local-in-time
well-posedness of the equations in Lorenz gauge, for data with finite energy.
The time of existence depends on the initial energy and on the -norm of the initial potential, for some .Comment: Minor typos corrected, references update