玻色子Boltzmann-Nordheim方程中三波碰撞算子线性化的光滑性质
Smoothing Properties of a Linearization of the Three Waves Collision Operator in the bosonic Boltzmann-Nordheim Equation
AI总结：
本文研究玻色子Boltzmann-Nordheim方程在含凝聚体平衡态附近线性化后三波碰撞算子的光滑性质，证明在满足Hölder条件的Banach空间中，线性化演化至少有限时间内能提高Hölder正则性。
摘要：
我们考虑三维弱相互作用玻色子流体处于非平衡态时的动理学理论，该非平衡态同时包含正常流体和凝聚体。更准确地说，我们研究此前已提出的描述此类系统的非线性Boltzmann-Nordheim方程，在空间均匀且具有各向同性动量分布的状态下，将方程在具有凝聚体的平衡态附近进行线性化。我们研究由超临界初始数据的三波碰撞算子所产生的线性化算子中最奇异的部分。该算子具有两类奇异性，其中一类类似于由符号$\ln(1+p^2)$定义的边际光滑算子。在此背景下，我们的主要结果是：对于满足Hölder型条件的某个Banach函数空间中的初始数据，至少在某个有限时间内，由线性化算子决定的演化能够提高Hölder正则性。该问题的主要困难来自线性算子中同时存在的点奇异性和线奇异性的组合，我们必须使用某些精细调制的函数空间来完成分析。
英文摘要：
We consider the kinetic theory of a three-dimensional fluid of weakly interacting bosons in a non-equilibrium state which includes both normal fluid and a condensate. More precisely, we look at the previously postulated nonlinear Boltzmann-Nordheim equations for such systems, in a spatially homogeneous state which has an isotropic momentum distribution, and we linearize the equation around an equilibrium state which has a condensate. We study the most singular part of the linearized operator coming from the three waves collision operator for supercritical initial data. The operator has two types of singularities, one of which is similar to the marginally smoothing operator defined by the symbol $\ln(1+p^2)$. Our main result in this context is that for initial data in a certain Banach space of functions satisfying a H\"{older} type condition, at least for some finite time, evolution determined by the linearized operator improves the Hölder regularity. The main difficulty in this problem arises from the combination of a point singularity and a line singularity present in the linear operator, and we have to use certain fine-tuned function spaces in order to carry out our analysis.
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