论文标题
建设性稳定性导致插值不平等和快速扩散方程的衰减速率的明确改善
Constructive stability results in interpolation inequalities and explicit improvements of decay rates of fast diffusion equations
论文作者
论文摘要
我们为Gagliardo-Nirenberg-Sobolev（GNS）不平等的家族提供了最新稳定性结果的方案，这相当于改善的熵生产 - 熵产生的不平等现象与适当的快速扩散方程（FDE）相关的不平等，以自相似变量书写。可以将该结果改写为（FDE）溶液的熵的提高衰减速率，以获得精心准备的初始数据。有一个非常相似的结构的Caffarelli-Kohn-Nirenberg（CKN）不平等的家族。当指数处于（CKN）径向对称的最佳函数的范围内时，我们研究了（GNS）的方法如何扩展到（CKN）。特别是，我们证明与（CKN）相关的演化方程的溶液在明显的延迟后还满足了熵的衰减率的提高。但是，提高的速率是在不假定初始数据准备充分的情况下获得的，这与（GNS）情况是主要区别。
We provide a scheme of a recent stability result for a family of Gagliardo-Nirenberg-Sobolev (GNS) inequalities, which is equivalent to an improved entropy - entropy production inequality associated with an appropriate fast diffusion equation (FDE) written in self-similar variables. This result can be rephrased as an improved decay rate of the entropy of the solution of (FDE) for well prepared initial data. There is a family of Caffarelli-Kohn-Nirenberg (CKN) inequalities which has a very similar structure. When the exponents are in a range for which the optimal functions for (CKN) are radially symmetric, we investigate how the methods for (GNS) can be extended to (CKN). In particular, we prove that the solutions of the evolution equation associated to (CKN) also satisfy an improved decay rate of the entropy, after an explicit delay. However, the improved rate is obtained without assuming that initial data are well prepared, which is a major difference with the (GNS) case.
