标题： 具有测度初值条件的均匀玻尔兹曼方程的熵有限性 显示英文标题

标题： Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition

摘要： 我们研究了空间均匀的$3D$型玻尔兹曼方程，适用于（真实的）硬势和适度软势。 我们假设初始条件是一个具有有限能量的概率测度，并且不是狄拉克测度。 对于硬势情形，我们证明任意合理的弱解会立即属于某个Besov空间。 对于适度软势情形，我们额外假设初始条件具有足够高阶的矩（$8$即可），并证明存在一个解，该解会立即属于某个Besov空间。 因此所考虑的解会瞬时成为具有有限熵的函数。 我们还证明，在任何情况下，任意弱解都会立即支持${\mathbb {R}}^3$。 显示英文摘要

摘要： We consider the $3D$ spatially homogeneous Boltzmann equation for (true) hard and moderately soft potentials. We assume that the initial condition is a probability measure with finite energy and is not a Dirac mass. For hard potentials, we prove that any reasonable weak solution immediately belongs to some Besov space. For moderately soft potentials, we assume additionally that the initial condition has a moment of sufficiently high order ($8$ is enough) and prove the existence of a solution that immediately belongs to some Besov space. The considered solutions thus instantaneously become functions with a finite entropy. We also prove that in any case, any weak solution is immediately supported by ${\mathbb {R}}^3$.