强退化抛物聚集方程
A strongly degenerate parabolic aggregation equation
AI总结：
本文研究一维空间中强退化对流扩散方程，其对流通量依赖于给定位置一侧总质量的非线性函数，通过有限差分格式证明其收敛到唯一熵解，并数值模拟聚集现象。
摘要：
本文研究一维空间中强退化对流扩散方程，其对流通量涉及给定位置一侧总质量的非线性函数。该方程可理解为种群个体聚集的模型，解代表局部密度。聚集机制由退化扩散项平衡，后者代表扩散。在强退化情况下，非局部问题的解通常是不连续的，需要定义为满足熵条件的弱解。本文提出了非局部问题的有限差分格式，并证明了其收敛到唯一熵解。该格式通过对原函数的局部PDE的单调格式进行差商得到。数值例子说明了非局部问题熵解的行为，特别是聚集现象。
英文摘要：
This paper is concerned with a strongly degenerate convection-diffusion equation in one space dimension whose convective flux involves a non-linear function of the total mass to one side of the given position. This equation can be understood as a model of aggregation of the individuals of a population with the solution representing their local density. The aggregation mechanism is balanced by a degenerate diffusion term accounting for dispersal. In the strongly degenerate case, solutions of the non-local problem are usually discontinuous and need to be defined as weak solutions satisfying an entropy condition. A finite difference scheme for the non-local problem is formulated and its convergence to the unique entropy solution is proved. The scheme emerges from taking divided differences of a monotone scheme for the local PDE for the primitive. Numerical examples illustrate the behaviour of entropy solutions of the non-local problem, in particular the aggregation phenomenon.
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