Regularity theory for nonlocal kinetic equations
The central task of this master thesis is to investigate regularity results for kinetic equations with nonlocal integral diffusion operators. The manuscript is structured into three distinct parts. Initially, we introduce kinetic Hölder spaces, tailored specifically to the geometry and invariances i...
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| Tipo de recurso: | tesis de maestría |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/441561 |
| Acceso en línea: | https://hdl.handle.net/2117/441561 |
| Access Level: | acceso abierto |
| Palabra clave: | Differential equations, Partial Equacions diferencials en derivades parcials Classificació AMS::35 Partial differential equations::35Q Equations of mathematical physics and other areas of application Classificació AMS::35 Partial differential equations::35R Miscellaneous topics involving partial differential equations Classificació AMS::45 Integral equations::45K05 Integro-partial differential equations Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | The central task of this master thesis is to investigate regularity results for kinetic equations with nonlocal integral diffusion operators. The manuscript is structured into three distinct parts. Initially, we introduce kinetic Hölder spaces, tailored specifically to the geometry and invariances inherent to kinetic equations. Special attention is given to the definition of kinetic distances and appropriate scaling transformations that ensure consistent regularity frameworks. The second part addresses integro-differential operators of fractional order, examining their analytical properties, including convergence criteria and Hölder continuity estimates. Finally, we dedicate the third part to deriving Schauder-type estimates within the kinetic setting. Specifically, we establish C^{2s+\alpha} regularity for solutions to translation-invariant nonlocal kinetic equations by adapting blow-up and Liouville-type methods developed originally in elliptic contexts. |
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