论文标题
偶极量子气体的质量能阈值动力学
Mass-Energy threshold dynamics for dipolar Quantum Gases
论文作者
论文摘要
我们考虑了一个毛状的pitaevskii方程，该方程式在偶极玻色 - 因斯坦冷凝物的描述中显示为模型，而没有限制外部捕获势。我们描述了在不同配置中相对于质量能量阈值的能量空间中相应考奇问题的溶液的渐近动力学，即对于上述和质量能量阈值的初始数据。我们首先为方程式建立一个散射标准，我们通过浓度/紧凑性和刚度方案来证明。该标准使我们能够以高于质量能阈值的数据显示解决方案的能量散射，仅知道爆炸。我们还证明了该方程的爆炸/成长标准，并在能量空间中具有一般数据。作为散射和爆破标准的副产品，以及最小化Gagliardo-Nirenberg的不平等的序列的紧凑性，我们研究了解决方案的长时间动力学，数据完全位于质量能量阈值。
We consider a Gross-Pitaevskii equation which appears as a model in the description of dipolar Bose-Einstein condensates, without a confining external trapping potential. We describe the asymptotic dynamics of solutions to the corresponding Cauchy problem in the energy space in different configurations with respect to the mass-energy threshold, namely for initial data above and at the mass-energy threshold. We first establish a scattering criterion for the equation that we prove by means of the concentration/compactness and rigidity scheme. This criterion enables us to show the energy scattering for solutions with data above the mass-energy threshold, for which only blow-up was known. We also prove a blow-up/grow-up criterion for the equation with general data in the energy space. As a byproduct of scattering and blow-up criteria, and the compactness of minimizing sequences for the Gagliardo-Nirenberg's inequality, we study long time dynamics of solutions with data lying exactly at the mass-energy threshold.
