One dimensional Dirac equation with quadratic nonlinearities
The author considers the \(1\)-dimensional Cauchy problem of Dirac equations with quadratic nonlinear term: \[ (i\gamma^0 \partial_t +i \gamma^1 \partial_x) u+mu=F(u), \quad u(0,x)=\phi(x), \leqno (\text{NLD})_m \] where \(m\) is a mass and nonnegative, \(u(t,x)\), \(t,x \in \mathbb{R}\), is an unknown function with values in \(\mathbb{C}^2\), \(\gamma^0\) and \(\gamma^1\) are canonical Dirac matrices \[ \gamma^0= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad\gamma^1= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \] \(F(u)\) is of the form \[ F(u)= \binom{c_1u_1u_2}{c_2u_1u_2}, \] with some constants \(c_1,c_2 \in \mathbb{C}\). Then he proves the local \(H^s\) well-posedness of \((\text{NLD})_m\) for \(s>-1/8\). Namely, for any data \(\phi \in H^s\), there exists a time \(T=T(\| \phi \|_{H^s})\) so that \((\text{ NLD})_m\) has a unique solution \(u \in C([-T,T];H^s)\) with finite Fourier restriction norms satisfying \[ \| \psi(t/T)u_1 \|_{X^{s,b}_{+}} \equiv \| \langle \tau+\xi \rangle^b \langle \xi \rangle^s \mathcal{F}_t \mathcal{F}_x \psi(t/T)u_1 \|_{L^{2}_{\tau}L^{2}_{\xi}} <\infty, \] \[ \| \psi(t/T)u_2 \|_{X^{s,b}_{-}} \equiv \| \langle \tau-\xi \rangle^b \langle \xi \rangle^s \mathcal{F}_t \mathcal{F}_x \psi(t/T)u_2 \|_{L^{2}_{\tau}L^{2}_{\xi}} <\infty \] for some \(b \in (1/2,1)\), where \(\psi(t)\) is \(C^\infty\)-cut-off function so that \(0 \leq \psi(t) \leq 1\) for \(-\infty<t<+\infty\), \(\psi(t)=1\) for \(| t | \leq 1\) and \(=0\) for \(| t | \geq 2\). The proof is based on the contraction mapping principle combined with appropriate bilinear estimates of nonlinear term in \(X^{s,b}_\pm\). \textit{J.-P. Dias} and \textit{M. Figueira} [Ric. Mat. 35, 309--316 (1986; Zbl 0658.35076)] proved \(H^s\) well-posedness of 3-dimensional Cauchy problems \((\text{NLD})_m\) for \(s>1/2\) by using \(L^\infty\) time-decay estimates. As to the 1-dimensional case, such time-decay estimates can be unexpected. To overcome this difficulty, the present author notices the bilinear estimates of nonlinear term, and hence, the result in this paper contributes to the well-posedness in 1-dimensional Cauchy problem and weakens the regulariry \(s\) to \(s>-1/8\). The author also obtains local \(H^s\), \(s>1/4\), well-posedness to the massless Cauchy problem \((\text{NLD})_0\) with nonlinear term \(F(u)\) replaced by \[ F(u)= \binom{c_1u_1u_1}{c_2u_2u_2}. \] Finally, local \(L^2\) well-posedness of the Cauchy problem \((\text{NLD})_m\) is given by deriving appropriate bilinear estimates similar to \textit{N. Bournaveas} [J. Funct. Anal. 173, 203--213 (2000; Zbl 0953.35003)], and applying the iteration procedure to solution formulas of 1-dimensional wave equations, which provides an alternative proof of the first result in the case when \(s=0\).
scientific article