Fine structures in interpolation inequalities and application to parabolic problems

在偏微分方程的分析中，Gagliardo-Nirenberg型插值不等式在许多地方起着中心作用。在拟议的项目框架内，它旨在推导出这类函数不等式的经典版本的精化变体，而不是仅指普通的勒贝格范数，该范数适当地包括Orlicz空间中各个函数的大小。在这里，主要关注的是对数加权偏离相应的未受扰动的代数积分表达式的典型情况；然而，除此之外，还计划讨论推广的选择，例如，通过包括渐近为任意小尺寸的校正项。此后，将通过在抛物问题的示例性上下文中的应用来检验所获得的不等式的潜在相关性，最好是在由于某些临界星座的存在而使用标准的Gagliardo-Nirenberg插值法似乎不够充分的情况下。因此，它不仅旨在提供关于简单和基本扩散过程中的正则化的精确陈述，而且还旨在有效地解决一些复杂耦合抛物型系统，特别是交叉扩散类型的分析中的开放关键问题。

In the analysis of partial differential equations, interpolation inequalities of Gagliardo-Nirenberg type play a central role in numerous places. In the framework of the proposed project it is intended to derive refined variants of classical versions of such functional inequalities, which instead of exclusively referring to ordinary Lebesgue norms appropriately include the size of respectively functions in Orlicz spaces. Here, predominant attention isfirstly paid to prototypical cases of logarithmically weighted deviations from correspondingly unperturbed algebraic integral expressions; apart from that, however, it is planned to also discuss options to generalize, for instance, by including correction terms which asymptotically are of arbitrarily small size. The potential relevance of the obtained inequalities is thereafter to be examined by means of applications in exemplary contexts of parabolic problems, preferably in situations in which due to the presence of certain critical constellations, an employment of standard Gagliardo-Nirenberg interpolation seems insufficient. It is thereby intended not only to provide refined statements on regularization in simple and fundamental diffusion processes, but also to effectively address open key questions in the analysis of some intricately coupled parabolic systems, especially of cross-diffusive type.