Fine structures in interpolation inequalities and application to parabolic problems

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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.