Two Remarks on Solutions of Gross-Pitaevskii Equations on Zhidkov Spaces

Abstract

Abstract. We consider the so-called Gross-Pitaevskii equations supplemented with non-standard boundary conditions. We prove two mathematical results concerned with the initial value problem for these equations in Zhidkov spaces.

The Cauchy Problem for Defocusing Nonlinear Schrödinger Equations with Non-Vanishing Initial Data at Infinity

“…Using a Brezis-Gallouët method, Goubet (2007) proved the global well-posedness for the Gross-Pitaevskii equation in X 2 2 , if the initial data has finite energy. More recently, Gérard (2006) obtained a global well-posedness result for the Gross-Pitaevskii equation in dimension two and three in the energy space u ∈ H 1 loc u ∈ L 2 1 − u 2 ∈ L 2…”

The Cauchy problem for the Gross–Pitaevskii equation

“…Let us mention that local existence results in the spaces X k (R d ), k > d/2, were already proved by Gallo [6], and that a global existence result in X 2 (R 2 ) has been obtained recently by Goubet [9] using logarithmic estimates as in Brezis-Gallouët 's contribution to cubic nonlinear Schrödinger equation on plane domains [3].…”

On the Dirac delta as initial condition for nonlinear Schrödinger equations

“…if γ solves (13). Therefore for A + we get the Schödinger map onto the unit sphere S 2 , and for A − we have it onto 2d hyperbolic space:…”

“…In a class of larger spaces, the Zhidkov spaces X k (R d ), it has been solved in X 1 (R) and in X 2 (R 2 ) by Zhidkov, by Gallo and by Goubet ([23], [24], [8], [13]). Also, considered in the natural energy space {f ∈ H 1 loc , ∇f ∈ L 2 , |f | 2 − 1 ∈ L 2 }, it has been solved for d ∈ {2, 3} and for d = 4 with smallness assumption, by Gérard in [10].…”