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arXiv · 2302.10683

The mixed fractional Hartree equations in Fourier amalgam and modulation spaces

Divyang G. Bhimani·Hichem Hajaiej·Saikatul Haque
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Abstract

We prove local and global well-posedness for mixed fractional Hartree equation and with low regularity Cauchy data in Fourier amalgam $\F W(L^p,\ell^q)$ and modulation $M^{p,q}$ spaces. Similar results also hold for the Hartree equation with harmonic potential in some modulation spaces. Our approach also addresses Hartree-Fock equations of finitely many (but arbitrary large) particles. A key ingredient of our method is to establish trilinear estimates for Hartree non-linearity and the use of Strichartz estimates. As a consequence, we could gain $\F W(L^p,\ell^q)$ and $M^{p,q}-$regularity for all $p,q\in [1, \infty].$ In particular, we extend result of Bhimani-Grillakis-Okoudju \cite{bhimani2020hartree} in $M^{p,q}$ for all $p,q$ and complement known results in Sobolev spaces.

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Divyang G. Bhimani, Hichem Hajaiej, Saikatul Haque. 2023-02-21. The mixed fractional Hartree equations in Fourier amalgam and modulation spaces. https://arxiv.org/abs/2302.10683

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