Existence, uniqueness and regularity results on nonlocal balance laws
Existence, uniqueness and regularity results on nonlocal balance laws
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DOI：
10.1016/j.jde.2017.05.015
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发表时间：
2017-10
影响因子：
2.4
通讯作者：
Alexander Keimer;L. Pflug
中科院分区：
文献类型：
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作者：
Alexander Keimer;L. Pflug
本文考虑一类非局部平衡律作为有限时间水平上的初值问题，证明了相应弱解的存在唯一性。描述“非局部”是指平衡定律的速度取决于在任何给定时间在空间中的区域上的加权积分。利用特征线方法证明了初始数据和右端数据在L1 <$L∞、L∞和特殊情况下在L1中弱解的存在性，从而得到了非局部项的不动点问题.在通量函数和非局部项的假设相对较弱的情况下，建立了弱解的唯一性，使得唯一性结果不需要众所周知的“Kružkov”熵条件，因为它是（局部）平衡律的典型条件，并且到目前为止在现有文献中也用于非局部平衡律。
We consider a class of nonlocal balance laws as initial value problems on a finite time horizon and show existence and uniqueness of the corresponding weak solutions. The description “nonlocal” refers to the velocity of the balance law that depends on the weighted integral over an area in space at any given time. Existence of a weak solution for initial data and right hand side data in L 1∩ L∞, in L∞ and in special cases in L 1 is shown via the method of characteristics, resulting in a fixed-point problem in the nonlocal term. The uniqueness of a weak solution with relatively weak assumptions on the flux function and the nonlocal term is established, so that the uniqueness result does not require the well-known “Kružkov” entropy condition as it is typical for (local) balance laws and was up to now used in the available literature also for nonlocal balance laws.