Global solutions for 3D nonlocal Gross-Pitaevskii equations with rough data

Abstract

We study the Cauchy problem for the Gross-Pitaevskii equation with a nonlocal interaction potential of Hartree type in three space dimensions. If the potential is even and positive definite or a positive function and its Fourier transform decays sufficiently rapidly the problem is shown to be globally well-posed for large rough data which not necessarily have finite energy and also in a situation where the energy functional is not positive definite. The proof uses a suitable modification of the I-method.

Keywords

• Gross-Pitaevskii equation
• global well-posedness
• Fourier restriction norm method