The Boltzmann collision operator for long-range interactions is usually employed in its “weak form” in the literature. However the weak form utilizes the symmetry property of the spherical integral and thus should be understood more or less in the principle value sense especially for strong angular singularity. To study the integral in the Lebesgue sense, it is natural to define the collision operator via the cutoff approximation. In this way, we give a rigorous proof to the local well-posedness of the Boltzmann equation with the long-range interactions. The result has the following main features and innovations: (1). The initial data is not necessarily a small perturbation around equilibrium but satisfies compatible conditions. (2). A quasi-linear method instead of the standard linearization method is used to prove existence and non-negativity of the solution in a suitably designed energy space depending heavily on the initial data. In such space, we derive the first uniqueness result for the equation in particular for hard potential case.

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具有长程相互作用的玻尔兹曼方程的截止逼近

用于长程相互作用的 Boltzmann 碰撞算子在文献中通常以其“弱形式”使用。然而，弱形式利用了球积分的对称性，因此应该或多或少地在主值意义上理解，特别是对于强角奇异性。为了研究 Lebesgue 意义上的积分，通过截止近似来定义碰撞算子是很自然的。通过这种方式，我们对具有长程相互作用的玻尔兹曼方程的局部适定性给出了严格的证明。该成果有以下主要特点和创新点： (1)。初始数据不一定是平衡附近的小扰动，但满足兼容条件。(2). 使用拟线性方法代替标准线性化方法来证明在很大程度上依赖于初始数据的适当设计的能量空间中解的存在性和非负性。在这样的空间中，我们推导出方程的第一个唯一性结果，特别是对于硬潜在情况。