The Hartree and Hartree–Fock Equations in Lebesgue Lp and Fourier–Lebesgue Lˆp Spaces

Abstract

We establish some local and global well-posedness for Hartree–Fock equations of N particles (HFP) with Cauchy data in Lebesgue spaces Lp∩L2 for 1≤p≤∞. Similar results are proven for fractional HFP in Fourier–Lebesgue spaces Lˆp∩L2 (1≤p≤∞). On the other hand, we show that the Cauchy problem for HFP is mildly ill-posed if we simply work in Lˆp (2<p≤∞). 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 Lp and Lˆp extension of classical well-posedness theories of Hartree and Hartree–Fock equations with Cauchy data in just L2−based Sobolev spaces.