Abstract
We study the formation of singularities for cylindrical symmetric solutions to the Gross–Pitaevskii equation describing a dipolar Bose–Einstein condensate. We prove that solutions arising from initial data with energy below the energy of the Ground State and that do not scatter collapse in finite time. The main tools to prove our result are the variational characterization of the Ground State energy, suitable localized virial identities for cylindrical symmetric functions, and general integral and pointwise estimates for operators involving powers of the Riesz transform.
Highlights
Since the first experimental observation in 1995 of a quantum state of matter at very low temperature called Bose–Einstein condensate (BEC), see e.g. [1,7,11], the study of the asymptotic dynamics of nonlinear equations describing this phenomena rapidly increased, both numerically and theoretically
Forcella see e.g. [3,4,27,29,31], and their peculiarity is given by the long-range anisotropic interaction between particles, in contrast with the short-range, isotropic character of the contact interaction of BEC
As the nonlinearity were defocusing. This notation is incorrect in the context of the Gross–Pitaevskii equation (GPE), as we will emphasize in some remark later on in the paper, after we introduce some basic notation
Summary
Since the first experimental observation in 1995 of a quantum state of matter at very low temperature called Bose–Einstein condensate (BEC), see e.g. [1,7,11], the study of the asymptotic dynamics of nonlinear equations describing this phenomena rapidly increased, both numerically and theoretically. The dipolar kernel K (x) is a Calderón–Zigmund operator of the form |x|−3O(x) where O is a zero-order function having zero average on the sphere This implies that the restriction to radial symmetric solutions makes disappear the effect of the nonlocal term in (1.2), the equation reduces to the classical NLS equation in the radial framework, see [8]. Remark 1.8 From the identity (1.11) and the fact that there exists a positive constant δ > 0 such that G(u(t)) ≤ −δ for any t ∈ (−Tmin, Tmax ) (see Lemma 3.5 below), it is straightforward to see that the assumption G(u0) < 0 implies that P(u(t)) < 0 for any time in the maximal interval of existence of the solution to (1.2). Which in turn implies the finite time blow-up via a convexity argument, provided R 1
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