The Hartree and Hartree-Fock equations in Lebesgue $L^p$ and Fourier-Lebesgue $\hat{L}^p$ spaces

Abstract

We establish some local and global well-posedness for Hartree-Fock equations of $N$ particles (HFP) with Cauchy data in Lebesgue spaces $L^p \cap L^2 $ for $1\leq p \leq \infty$. Similar results are proven for fractional HFP in Fourier-Lebesgue spaces $ \hat{L}^p \cap L^2 \ (1\leq p \leq \infty).$ On the other hand, we show that the Cauchy problem for HFP is ill-posed if we simply work in $\hat{L}^p \ (2<p\leq \infty).$ Analogue results hold for reduced HFP. In the process, we prove the boundedeness of various trilinear estimates for Hartree type non linearity in these spaces which may be of independent interest. As a consequence, we get natural $L^p$ and $\hat{L}^p$ extension of classical well-posedness theories of Hartree and Hartree-Fock equations with Cauchy data in just $L^2-$based Sobolev spaces.