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The Hartree and Hartree–Fock Equations in Lebesgue \(L^p\) and Fourier–Lebesgue \(\widehat{L}^p\) Spaces

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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 \(1\le p \le \infty \). Similar results are proven for fractional HFP in Fourier–Lebesgue spaces \( \widehat{L}^p \cap L^2 \ (1\le p \le \infty ).\) On the other hand, we show that the Cauchy problem for HFP is mildly ill-posed if we simply work in \(\widehat{L}^p \ (2<p\le \infty )\). 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 \(\widehat{L}^p\) extension of classical well-posedness theories of Hartree and Hartree–Fock equations with Cauchy data in just \(L^2-\)based Sobolev spaces.

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Notes

  1. In [6], Bourgain introduced this approach to establish failure of \(C^3-\)smoothness for the solution map of KdV and mKdv, see also [32]. Since then, many authors have used this approach, see for example, [14, Proposition 4.1] for cubic nonlinear half-wave equation.

  2. i.e. repeated indexed in a product are summed up: for example \(a_lb_lc_k\) stands for \(c_k\sum _{l=1}^Na_lb_l\)

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Acknowledgements

Both authors are thankful to Prof. K. Sandeep for the encouragement during this project and his thoughtful suggestions on the preliminary draft of this paper. D.G. B is thankful to DST-INSPIRE (DST/INSPIRE/04/2016/001507) for the financial support. Both authors are thankful to TIFR CAM for the excellent research facilities. Both authors are also thankful to Prof. Rémi Carles for his thoughtful suggestions and for bringing to our notice the reference [18]. Both authors are thankful to Prof. H. Hajaiej for the encouragement and thoughtful suggestions. We are grateful to an anonymous referee for pointing out the error in the previous version of this paper. This has helped us to improve the presentation of Sect. 4.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune, 411008, India

    Divyang G. Bhimani

  2. Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Prayagraj, Allahabad, 211019, India

    Saikatul Haque

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  1. Divyang G. Bhimani
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Correspondence to Saikatul Haque.

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Bhimani, D.G., Haque, S. The Hartree and Hartree–Fock Equations in Lebesgue \(L^p\) and Fourier–Lebesgue \(\widehat{L}^p\) Spaces. Ann. Henri Poincaré 24, 1005–1049 (2023). https://doi.org/10.1007/s00023-022-01234-5

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