CPAA
Local well-posedness for the nonlinear Dirac equation in two space dimensions
Hartmut Pecher
Communications on Pure & Applied Analysis 2014, 13(2): 673-685 doi: 10.3934/cpaa.2014.13.673
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.
keywords: Dirac equation well-posedness Fourier restriction norm method.
CPAA
Corrigendum of "Local well-posedness for the nonlinear Dirac equation in two space dimensions"
Hartmut Pecher
Communications on Pure & Applied Analysis 2015, 14(2): 737-742 doi: 10.3934/cpaa.2015.14.737
The stated theorems in [1] remain completely unchanged. However, the proof of Proposition 2.1 has to be modified, because in several places Cor. 1.1 was used for $\beta_- < \frac{1}{4}$, which is not admissible. Instead we use that the nonlinearity satisfies two null conditions, namely $\langle \beta \psi,\psi \rangle$ on one hand and the factor $\beta \psi$ produces a second null condition by duality on the other hand. The latter property was not used before and gives an additional regularizing factor which allows to use Cor. 1.1 correctly. Here and in the following we use the numbering and notation of [1].
keywords: well-posedness Fourier restriction norm method Dirac equation
CPAA
Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system
Hartmut Pecher
Communications on Pure & Applied Analysis 2012, 11(3): 1081-1096 doi: 10.3934/cpaa.2012.11.1081
The Klein-Gordon-Schrödinger system in 3D is shown to be locally well-posed for Schrödinger data in $H^s$ and wave data in $H^{\sigma}\times H^{\sigma -1}$ , if $ s > - \frac{1}{4},$ $\sigma > - \frac{1}{2}$ , $\sigma -2s > \frac{3}{2} $ and $\sigma -2 < s < \sigma +1$ . 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 $s = \sigma = 0$ in the natural solution space $C^0([0,T],L^2) \times C^0([0,T],L^2) \times C^0([0,T],H^{-\frac{1}{2}}).$ This solution exists even globally by Colliander, Holmer and Tzirakis [6]. The proofs are based on new well-posedness results for the Zakharov system by Bejenaru, Herr, Holmer and Tataru [3], and Bejenaru and Herr [4].
keywords: Klein - Gordon - Schrödinger system well-posedness Fourier restriction norm method.
CPAA
Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge
Hartmut Pecher
Communications on Pure & Applied Analysis 2016, 15(6): 2203-2219 doi: 10.3934/cpaa.2016034
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.
keywords: local well-posedness Maxwell-Klein-Gordon temporal gauge.
DCDS
Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge
Hartmut Pecher
Discrete & Continuous Dynamical Systems - A 2016, 36(4): 2193-2204 doi: 10.3934/dcds.2016.36.2193
We consider the Maxwell-Chern-Simons-Higgs system in Lorenz gauge and use a null condition to show local well-psoedness for low regularity data. This improves a recent result of J. Yuan.
keywords: Maxwell-Chern-Simons-Higgs Lorenz gauge. local well-posedness

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