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