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Oberseminar AG Mathematische Modellierung
Montag, den 03. Juni 2024 um 17.00 Uhr in B7- 210 - Seminarraum A
Vortrag von Anatole Gaudin (Marseille) zum Thema: "Homogeneous Function Spaces for Global-in-Time Well-Posedness"
Montag, den 03. Juni 2024 um 17.00 Uhr in B7- 210 - Seminarraum A
Vortrag von Anatole Gaudin (Marseille) zum Thema: "Homogeneous Function Spaces for Global-in-Time Well-Posedness"
Abstract:
In the last decades, function spaces that control specific amounts of localization and regularity appearing in a PDE have become central to the global-in-time well-posedness of evolution PDEs: those spaces are called homogeneous Sobolev and Besov spaces.
This presentation will focus on the realization of homogeneous function spaces on some half-spaces, extending certain approaches established on the whole space and the flat half-space. The construction we focus on is particularly well-suited for addressing nonlinear partial differential equations with possibly boundary conditions. We will demonstrate how these spaces appear naturally when studying some standard PDEs, and why their usual construction does not fit the goal of studying nonlinear and boundary value problems.
We will specifically discuss the interpolation of these spaces, trace results, and adapted operator theory. This will lead to some global-in-time $\mathrm{L}^q$-maximal regularity results, as well as numerous other variants in this framework, offering a natural extension of some results recently obtained by Danchin, Hieber, Mucha, and Tolksdorf. When considering the flat half-space, it is possible to obtain a Hodge/Helmholtz decomposition (or equivalently the boundedness of the Leray projector) for homogeneous Besov spaces with "sufficiently high" regularity indices. It allows us to also recover various global-in-time $\mathrm{L}^q$-maximal regularity results, such as, for example, an $\mathrm{L}^1_t(\dot{\mathrm{B}}^{s}_{p,1})$ result, which can be of central interest in viscous fluid mechanics.