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Hartmut Pecher

Publications and source records attributed to Hartmut Pecher.

At least 37 records · Page 2Linked to original sources
arXiv2016

Low regularity solutions for the (2+1) - dimensional Maxwell-Klein-Gordon equations in temporal gauge

The Maxwell-Klein-Gordon equations in 2+1 dimensions in temporal gauge are locally well-posed for low regularity data even below energy level. The corresponding (3+1)-dimensional case was considered by Yuan. Fundamental for the proof is a partial null structure in the nonlinearity which allows to rely on bilinear estimates in wave-Sobolev spaces by d'Ancona, Foschi and Selberg, on an $(L^p_x L^q_t)$ - estimate for the solution of the wave equation, and on the proof of a related result for the Yang-Mills equations by Tao.

math.AP↗
arXiv2015

Global well-posedness in energy space 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 globally well-posed in energy space improving a result of Huh. The proof uses the bilinear space-time estimates in wave-Sobolev spaces by d'Ancona, Foschi and Selberg, an $L^6_x L^2_t$-estimate for solutions of the wave equation, and also takes advantage of a null condition.

math.AP↗
arXiv2014

A remark on low regularity solutions of the Chern-Simons-Dirac system

An alternative proof of low regularity well-posedness for the Chern-Simons-Dirac system in Coulomb gauge is given which completely avoids the use of any null structure similarly to a recent result of Bournaveas-Candy-Machihara. An unconditional uniqueness result is also given.

math.AP↗
arXiv2014

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.

math.AP↗
arXiv2013

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.

math.AP↗
arXiv2012

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.

math.AP↗
arXiv2012

Global solutions for 3D nonlocal Gross-Pitaevskii equations with rough data

We study the Cauchy problem for the Gross-Pitaevskii equation with a nonlocal interaction potential of Hartree type in three space dimensions. If the potential is even and positive definite or a positive function and its Fourier transform decays sufficiently rapidly the problem is shown to be globally well-posed for large rough data which not necessarily have finite energy and also in a situation where the energy functional is not positive definite. The proof uses a suitable modification of the I-method.

math.AP↗
arXiv2012

Global rough solutions for the Zakharov system in two spatial dimensions

We show an improved global well-posedness result for the Zakharov system in two space dimensions with minimal regularity assumptions for the data. Especially we are able to allow Schroedinger and wave data, which do not belong to H^1 and L^2, respectively, thus with infinite energy. The proof uses a refined I-method originally initiated by Colliander, Keel, Staffilani, Takaoka and Tao and bilinear estimates by Bejenaru, Herr, Holmer and Tataru. A polynomial growth bound for the solution is also given.

math.AP↗
arXiv2011

Some new well-posedness results for the Klein-Gordon-Schrödinger system

We consider the Cauchy problem for the 2D and 3D Klein-Gordon-Schrödinger system. In 2D we show local well-posedness for Schrödinger data in H^s and wave data in H^σ x H^{σ-1} for s=-1/4 + and σ= -1/2, whereas ill-posedness holds for s<- 1/4 or σ<-1/2, and global well-posedness for s\ge 0 and s- 1/2 \le σ< s+ 3/2. In 3D we show global well-posedness for s \ge 0, s - 1/2 < σ\le s+1. Fundamental for our results are the studies by Bejenaru, Herr, Holmer and Tataru, and Bejenaru and Herr for the Zakharov system, and also the global well-posedness results for the Zakharov and Klein-Gordon-Schrödinger system by Colliander, Holmer and Tzirakis.

math.AP↗
arXiv2011

Low regularity well-posedness for the 3D Klein-Gordon-Schrödinger system

The Klein-Gordon-Schrödinger system in 3D is shown to be locally well-posed for Schrödinger data in H^s and wave data in H^σ \times H^{σ-1}, if s > - 1/4, σ> - 1/2, σ-2s > 3/2 and σ-2 < s < σ+1 . This result is optimal up to the endpoints in the sense that the local flow map is not C^2 otherwise. It is also shown that (unconditional) uniqueness holds for s=σ=0 in the natural solution space C^0([0,T],L^2) \times C^0([0,T],L^2) \times C^0([0,T],H^{-1/2}) . This solution exists even globally by Colliander, Holmer and Tzirakis. The proofs are based on new well-posedness results for the Zakharov system by Bejenaru, Herr, Holmer and Tataru, and Bejenaru and Herr.

math.AP↗
arXiv2011

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.

math.AP↗
arXiv2009

Global solutions for the Dirac-Klein-Gordon system in two space dimensions

The Cauchy problem for the classical Dirac-Klein-Gordon system in two space dimensions is globally well-posed for L^2 Schoedinger data and wave data in H^{1/2} \times H^{-1/2}. In the case of smooth data there exists a global smooth (classical) solution. The proof uses function spaces of Bourgain type based on Besov spaces - previously applied by Colliander, Kenig and Staffilani for generalized Benjamin-Ono equations and also by Bejenaru, Herr, Holmer and Tataru for the 2D Zakharov system - and the null structure of the system detected by d'Ancona, Foschi and Selberg, and a refined bilinear Strichartz estimate due to Selberg. The global existence proof uses an idea of Colliander, Holmer and Tzirakis for the 1D Zakharov system.

math.AP↗
arXiv2009

Low regularity global well-posedness for the two-dimensional Zakharov system

The two-dimensional Zakharov system is shown to have a unique global solution for data without finite energy if the L^2 - norm of the Schrödinger part is small enough. The proof uses a refined I-method originally initiated by Colliander, Keel, Staffilani, Takaoka and Tao. A polynomial growth bound for the solution is also given.

math.AP↗
arXiv2008

An improved local well-posedness result for the one-dimensional Zakharov system

The 1D Cauchy problem for the Zakharov system is shown to be locally well-posed for low regularity Schrödinger data u_0 \in \hat{H^{k,p}} and wave data (n_0,n_1) \in \hat{H^{l,p}} \times \hat{H^{l-1,p}} under certain assumptions on the parameters k,l and 1 ^k \hat{u_0}\|_{L^{p'}}, generalizing the results for p=2 by Ginibre, Tsutsumi, and Velo. Especially we are able to improve the results from the scaling point of view, and also allow suitable k<0, l<-1/2, i.e. data u_0 \not\in L^2 and (n_0,n_1)\not\in H^{-1/2}\times H^{-3/2}, which was excluded in the case p=2.

math.AP↗