Low regularity well-posedness of Hartree type Dirac equations in 2, 3-dimensions

Abstract

<p style='text-indent:20px;'>In this paper, we consider the Cauchy problem of <inline-formula><tex-math id="M1">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimension Hartree type Dirac equation with nonlinearity <inline-formula><tex-math id="M2">\begin{document}$ c|x|^{-\gamma} * \langle \psi, \beta \psi\rangle $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ c\in \mathbb R\setminus\{0\} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ 0 < \gamma < d $\end{document}</tex-math></inline-formula>(<inline-formula><tex-math id="M5">\begin{document}$ d = 2,3 $\end{document}</tex-math></inline-formula>). Our aim is to show the local well-posedness in <inline-formula><tex-math id="M6">\begin{document}$ H^s $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M7">\begin{document}$ s > \frac{\gamma-1}2 $\end{document}</tex-math></inline-formula> with mass-supercritical cases(<inline-formula><tex-math id="M8">\begin{document}$ 1 < \gamma<d $\end{document}</tex-math></inline-formula>) and mass-critical case(<inline-formula><tex-math id="M9">\begin{document}$ {\gamma} = 1 $\end{document}</tex-math></inline-formula>) via bilinear estimates and angular decomposition for which we use the null structure of nonlinear term effectively. We also provide the flow of Dirac equations cannot be <inline-formula><tex-math id="M10">\begin{document}$ C^3 $\end{document}</tex-math></inline-formula> at the origin for <inline-formula><tex-math id="M11">\begin{document}$ H^s $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M12">\begin{document}$ s < \frac{\gamma-1}2 $\end{document}</tex-math></inline-formula>.</p>

Local well-posedness of Dirac equations with nonlinearity derived from honeycomb structure in 2 dimensions

“…As known result for the equation (1.4), Bejenaru and Herr([3]) showed the GWP in H 1 2 (R 2 ). And the known results for the equation (1.5) are in [8,20]. In [20], An author of this paper revealed the LWP in H s (R 2 ) with s > γ−1 2 and 1 ≤ γ < 2.…”

“…And the known results for the equation (1.5) are in [8,20]. In [20], An author of this paper revealed the LWP in H s (R 2 ) with s > γ−1 2 and 1 ≤ γ < 2. It was studied in [8] that global well-posedness and small data scattering holds in H s (R 2 ) for s > γ − 1 and 1 < γ < 2.…”

“…This section aims to show Theorem 1.2. It adopts the argument [22,15,20] to prove of smoothness failure of flows of (1.1) with cubic and Hartree type nonlinearity. As in proof of [22,15], if the flow map ψ → u is C 3 at the origin from H s to C([0, T ) ; H s ), we have (6.4).…”

“…For this expectation we introduce the following theorem which denotes the smooth failure of our main equation (1.1) for s < s(ℓ). The smoothness failure of some equations was studied for many authors in [22,4,15,20]. Molinet, Saut, and Tzvetkov( [22]), Bejenaru and Tao([4]), and Herr and Lenzmann( [15]) have proved the ill-posedness results similar to Theorem (1.2) for Benjamin-Ono equations, 1-d Schrödinger equations and semi-relativistic equations, respectively.…”

“…Molinet, Saut, and Tzvetkov( [22]), Bejenaru and Tao([4]), and Herr and Lenzmann( [15]) have proved the ill-posedness results similar to Theorem (1.2) for Benjamin-Ono equations, 1-d Schrödinger equations and semi-relativistic equations, respectively. For Dirac equation, ill-posedness results have been shown in [20].…”