The Transition to a Giant Vortex Phase in a Fast Rotating Bose-Einstein Condensate

We study the Gross-Pitaevskii (GP) energy functional for a fast rotating Bose-Einstein condensate on the unit disc in two dimensions. Writing the coupling parameter as 1 / \eps^2 we consider the asymptotic regime \eps \to 0 with the angular velocity $\Omega$ proportional to (\eps^2|\log\eps|)^{-1} . We prove that if \Omega = \Omega_0 (\eps^2|\log\eps|)^{-1} and $\Omega_0 > 2(3\pi)^{-1}$ then a minimizer of the GP energy functional has no zeros in an annulus at the boundary of the disc that contains the bulk of the mass. The vorticity resides in a complementary `hole' around the center where the density is vanishingly small. Moreover, we prove a lower bound to the ground state energy that matches, up to small errors, the upper bound obtained from an optimal giant vortex trial function, and also that the winding number of a GP minimizer around the disc is in accord with the phase of this trial function.Comment: 52 pages, PDFLaTex. Minor corrections, sign convention modified. To be published in Commun. Math. Phy

Existence of continuous eigenvalues for a class of parametric problems involving the (p,2)-laplacian operator

We discuss a parametric eigenvalue problem, where the differential operator is of (p,2)-Laplacian type. We show that, when p≠2, the spectrum of the operator is a half line, with the end point formulated in terms of the parameter and the principal eigenvalue of the Laplacian with zero Dirichlet boundary conditions. Two cases are considered corresponding to p>2 and p2, and to infinity in the case of p<2

On global solutions and blow-up for Kuramoto\u2013Sivashinsky-type models, and well-posed Burnett equations

The initial-boundary-value problem (IBVP) and the Cauchy problem for the Kuramoto\u2013Sivashinsky equation v_t + v_xxxx + v_xx = 1/2 (v^2 )_x and other related 2mth-order semilinear parabolic partial differential equations in one dimension and in R^N are considered. iewed by using: (i) classic tools of interpolation theory and Galerkin methods, (ii) eigenfunction and nonlinear capacity methods, (iii) Henry\u2019s version of weighted Gronwall\u2019s inequalities, (iv) two types of scaling (blow-up) arguments. For the IBVPs, existence of global solutions is proved for both Dirichlet and \u2018\u2018Navier\u2019\u2019 boundary conditions. For some related 2mth-order PDEs in RN 7 R+ , uniform boundedness of global solutions of the Cauchy problem are established. As another related application, the well-posed Burnett-type equations vt +(v\ub7 07)v= 12 07p 12( 12 06)mv, divv=0 inRN 7R+, m 651, are considered. For m = 1 these are the classic Navier\u2013Stokes equations. As a simple illustration, it is shown that a uniform Lp(RN)-bound on locally sufficiently smooth v(x, t ) for p > N implies a uniform L 1e (RN )-bound, hence the solutions do not 2m 121 Crown Copyright \ua9 2008 Published by Elsevier Ltd. All rights reserved. \ufffc\ufffc\ufffc\ufffcblow-up. For m = 1 and N = 3, this gives p > 3, which reflects the famous Leray\u2013Prodi\u2013Serrin\u2013Ladyzhenskaya regularity results (Lp,q criteria), and re-derives Kato\u2019s class of unique mild solutions in RN . Truly bounded classic L2 -solutions are shown to exist in dimensions N < 2 (2m 12 1)