Small-data solutions of the semiclassical Hartree equation with long-range interaction satisfy the uniform-in-hbar optimal density decay ||rho(t)||_{L∞} ≲ <t>^{-3}.
Modified scattering for long-range Hartree equations of infinite rank near vacuum
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We establish the asymptotic behavior and decay of solutions near vacuum to the Hartree equation with the Coulomb interaction potential in three dimensions. Our approach is direct, which consists of independently deriving the sharp dispersive decay estimates for the density function and establishing the boundedness of energy norms. The global in time well-posedness is done without introducing the phase correction, while the phase modification is explicit in terms of the density function.
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Uniform dispersive estimates for the semi-classical Hartree equation with long-range interaction
Small-data solutions of the semiclassical Hartree equation with long-range interaction satisfy the uniform-in-hbar optimal density decay ||rho(t)||_{L∞} ≲ <t>^{-3}.