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Derivation of Hartree’s theory for mean-field Bose gases

Mathieu Lewin (2013)

Journées Équations aux dérivées partielles

This article is a review of recent results with Phan Thành Nam, Nicolas Rougerie, Sylvia Serfaty and Jan Philip Solovej. We consider a system of N bosons with an interaction of intensity 1 / N (mean-field regime). In the limit N → ∞ , we prove that the first order in the expansion of the eigenvalues of the many-particle Hamiltonian is given by the nonlinear Hartree theory, whereas the next order is predicted by the Bogoliubov Hamiltonian. We also discuss the occurrence of Bose-Einstein condensation in these...

Divergent solutions to the 5D Hartree equations

Daomin Cao, Qing Guo (2011)

Colloquium Mathematicae

We consider the Cauchy problem for the focusing Hartree equation i u t + Δ u + ( | · | - 3 ∗ | u | ² ) u = 0 in ℝ⁵ with initial data in H¹, and study the divergence property of infinite-variance and nonradial solutions. For the ground state solution of - Q + Δ Q + ( | · | - 3 ∗ | Q | ² ) Q = 0 in ℝ⁵, we prove that if u₀ ∈ H¹ satisfies M(u₀)E(u₀) < M(Q)E(Q) and ||∇u₀||₂||u₀||₂ > ||∇Q||₂||Q||₂, then the corresponding solution u(t) either blows up in finite forward time, or exists globally for positive time and there exists a time sequence tₙ → ∞ such that ||∇u(tₙ)||₂ → ∞....

Dynamique des points vortex dans une équation de Ginzburg-Landau complexe

Evelyne Miot (2009/2010)

Séminaire Équations aux dérivées partielles

On considère une équation de Ginzburg-Landau complexe dans le plan. On étudie un régime asymptotique à petit paramètre dans lequel les solutions comportent des singularités ponctuelles, appelées points vortex, et on détermine un système d’équations différentielles ordinaires du premier ordre décrivant la dynamique de ces points jusqu’au premier temps de collision.

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