2,202 research outputs found
Low regularity local well-posedness for the Chern-Simons-Higgs system in temporal gauge
The Cauchy problem for the Chern-Simons-Higgs system in the (2+1)-dimensional
Minkowski space in temporal gauge is locally well-posed for low regularity
initial data improving a result of Huh. The proof uses the bilinear space-time
estimates in wave-Sobolev spaces by d'Ancona, Foschi and Selberg and takes
advantage of a null condition.Comment: 11pages, slight modification of version
Unconditional global well-posedness for the 3D Gross-Pitaevskii equation for data without finite energy
The Cauchy problem for the Gross-Pitaevskii equation in three space
dimensions is shown to have an unconditionally unique global solution for data
of the form 1 + H^s for 5/6 < s < 1, which do not have necessarily finite
energy. The proof uses the I-method which is complicated by the fact that no
L^2 -conservation law holds. This improves former results of Bethuel-Saut and
Gerard.Comment: 23 pages. Final version to appear in Nonlinear Differential Equations
and Application
Unconditional well-posedness for the Dirac - Klein - Gordon system in two space dimensions
The solution of the Dirac - Klein - Gordon system in two space dimensions
with Dirac data in H^s and wave data in H^{s+1/2} x H^{s-1/2} is uniquely
determined in the natural solution space C^0([0,T],H^s) x
C^0([0,T],H^{s+\frac1/2}), provided s > 1/30 . This improves the uniqueness
part of the global well-posedness result by A. Gruenrock and the author, where
uniqueness was proven in (smaller) spaces of Bourgain type. Local
well-posedness is also proven for Dirac data in L^2 and wave data in H^{3/5}+}
x H^{-2/5+} in the solution space C^0([0,T],L^2) x C^0([0,T],H^{3/5+}) and also
for more regular data.Comment: 6 page
Well-posedness for a modified Zakharov system
The Cauchy problem for a modified Zakharov system is proven to be locally
well-posed for rough data in two and three space dimensions. In the three
dimensional case the problem is globally well-posed for data with small energy.
Under this assumption there also exists a global classical solution for
sufficiently smooth data.Comment: 30 pages. Final version to appear in Hokkaido Mathematical Journa
Local well-posedness for the nonlinear Dirac equation in two space dimensions
The Cauchy problem for the cubic nonlinear Dirac equation in two space
dimensions is locally well-posed for data in H^s for s > 1/2. The proof given
in spaces of Bourgain-Klainerman-Machedon type relies on the null structure of
the nonlinearity as used by d'Ancona-Foschi-Selberg for the Dirac-Klein-Gordon
system before and bilinear Strichartz type estimates for the wave equation by
Selberg and Foschi-Klainerman.Comment: 21 pages. This is (almost) identical with version 1. The versions 2-4
contain an error in the proof of Proposition 2.
Global well-posedness below energy space for the 1D Zakharov system
The Cauchy problem for the 1-dimensional Zakharov system is shown to be
globally well-posed for large data which not necessarily have finite energy.
The proof combines the local well-posedness result of Ginibre, Tsutsumi, Velo
and a general method introduced by Bourgain to prove a similar result for
nonlinear Schr\"odinger equations
Local well-posedness for the Klein-Gordon-Zakharov system in 3D
We study the Cauchy problem for the Klein-Gordon-Zakharov system in 3D with
low regularity data. We lower down the regularity to the critical value with
respect to scaling up to the endpoint. The decisive bilinear estimates are
proved by means of methods developed by Bejenaru-Herr for the Zakharov system
and already applied by Kinoshita to the Klein-Gordon-Zakharov system in 2D.Comment: 28 page
Unconditional global well-posedness in energy space for the Maxwell-Klein-Gordon system in temporal gauge
The Maxwell-Klein-Gordon system in temporal gauge is unconditionally globally
well-posed in energy space, especially uniqueness holds in the natural solution
space. This improves earlier results where uniqueness was only shown in a
suitable subspace. It is also locally well-posed for large data below energy
space.Comment: Minor changes in the proof of Proposition 2.2. (17 pages
The Cauchy problem for a Schroedinger - Korteweg - de Vries system with rough data
The Cauchy problem for a coupled system of the Schroedinger and the KdV
equation is shown to be globally well-posed for data with infinite energy. The
proof uses refined bilinear Strichartz estimates and the I-method introduced by
Colliander, Keel, Staffilani, Takaoka, and Tao.Comment: 25 pages. Minor corrections have been mad
Low regularity local well-posedness for the Maxwell-Klein-Gordon equations in Lorenz gauge
The Cauchy problem for the Maxwell-Klein-Gordon equations in Lorenz gauge in
two and three space dimensions is locally well-posed for low regularity data
without finite energy. The result relies on the null structure for the main
bilinear terms which was shown to be not only present in Coulomb gauge but also
in Lorenz gauge by Selberg and Tesfahun, who proved global well-posedness for
finite energy data in three space dimensions. This null structure is combined
with product estimates for wave-Sobolev spaces given systematically by
d'Ancona, Foschi and Selberg.Comment: 23 page
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