# American Institute of Mathematical Sciences

## Journals

CPAA
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.
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CPAA
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].
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CPAA
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].
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CPAA
Communications on Pure & Applied Analysis 2019, 18(2): 663-688 doi: 10.3934/cpaa.2019033

We prove that the Yang-Mills equation in Lorenz gauge in the (3+1)-dimensional case is locally well-posed for data of the gauge potential in $H^s$ and the curvature in $H^r$, where $s >\frac{5}{7}$ and $r > -\frac{1}{7}$, respectively. This improves a result by Tesfahun [16]. The proof is based on the fundamental results of Klainerman-Selberg [6] and on the null structure of most of the nonlinear terms detected by Selberg-Tesfahun [14] and Tesfahun [16].

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