论文标题
快速扩散导致凯勒 - 塞格型固定溶液中的部分质量浓度
Fast Diffusion leads to partial mass concentration in Keller-Segel type stationary solutions
论文作者
论文摘要
我们表明，对于在快速扩散范围内具有均匀有吸引力的核的聚合扩散方程的固定溶液可能会发生部分质量浓度。更确切地说，我们证明，自由能在一组概率度量中接受了径向全局最小化器，该概率度量可能会在给定点将其质量集中在Dirac Delta中。在四分之一的相互作用电位的情况下，我们找到了扩散指数的确切范围，其中浓度发生在空间尺寸$ n \ geq6 $中。然后，我们提供数值计算，这些计算表明在所有维度$ n \ geq3 $中出现质量浓度，对于具有较高功率的均质相互作用势。
We show that partial mass concentration can happen for stationary solutions of aggregation-diffusion equations with homogeneous attractive kernels in the fast diffusion range. More precisely, we prove that the free energy admits a radial global minimizer in the set of probability measures which may have part of its mass concentrated in a Dirac delta at a given point. In the case of the quartic interaction potential, we find the exact range of the diffusion exponent where concentration occurs in space dimensions $N\geq6$. We then provide numerical computations which suggest the occurrence of mass concentration in all dimensions $N\geq3$, for homogeneous interaction potentials with higher power.
