Understanding the Landau equation as a gradient flow
<p>This thesis provides and investigates the rigorous gradient flow viewpoint of the spatially homogeneous Landau equation for soft potentials. This extends the similar perspective recently developed for the Boltzmann equation with Maxwellian potentials by Erbar. Taking advantage of the H-theo...
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| Format: | Thesis |
| Language: | English |
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2022
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| _version_ | 1797107303316455424 |
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| author | Wu, JSH |
| author2 | Carrillo De La Plata, J |
| author_facet | Carrillo De La Plata, J Wu, JSH |
| author_sort | Wu, JSH |
| collection | OXFORD |
| description | <p>This thesis provides and investigates the rigorous gradient flow viewpoint of the spatially homogeneous Landau equation for soft potentials. This extends the similar perspective recently developed for the Boltzmann equation with Maxwellian potentials by Erbar. Taking advantage of the H-theorem for the Landau equation, we construct a metric from a dynamic optimal transportation viewpoint (Benamou-Brenier and Dolbeault-Nazaret-Savaré) for which the Landau equation can be viewed as the gradient flow of the Boltzmann entropy with respect to this metric. The gradient flow description is further reinforced by recovering the grazing collision limit from the Boltzmann equation to the Landau equation with simpler arguments and intuition in the spirit of Gamma-convergence (Sandier-Serfaty). Finally, a robust and efficient particle approximation is numerically analysed for a regularised version of the Landau equation which preserves the gradient flow structure.</p>
<p>This thesis invites and advertises the use of gradient flow techniques to pursue some of the future research directions discussed here.</p> |
| first_indexed | 2024-03-07T07:14:10Z |
| format | Thesis |
| id | oxford-uuid:a4613fd6-781a-43fc-bfe1-9ec3223f4675 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:14:10Z |
| publishDate | 2022 |
| record_format | dspace |
| spelling | oxford-uuid:a4613fd6-781a-43fc-bfe1-9ec3223f46752022-07-21T14:30:52ZUnderstanding the Landau equation as a gradient flowThesishttp://purl.org/coar/resource_type/c_db06uuid:a4613fd6-781a-43fc-bfe1-9ec3223f4675Kinetic theory of gasesDifferential equations, PartialEnglishHyrax Deposit2022Wu, JSHCarrillo De La Plata, J<p>This thesis provides and investigates the rigorous gradient flow viewpoint of the spatially homogeneous Landau equation for soft potentials. This extends the similar perspective recently developed for the Boltzmann equation with Maxwellian potentials by Erbar. Taking advantage of the H-theorem for the Landau equation, we construct a metric from a dynamic optimal transportation viewpoint (Benamou-Brenier and Dolbeault-Nazaret-Savaré) for which the Landau equation can be viewed as the gradient flow of the Boltzmann entropy with respect to this metric. The gradient flow description is further reinforced by recovering the grazing collision limit from the Boltzmann equation to the Landau equation with simpler arguments and intuition in the spirit of Gamma-convergence (Sandier-Serfaty). Finally, a robust and efficient particle approximation is numerically analysed for a regularised version of the Landau equation which preserves the gradient flow structure.</p> <p>This thesis invites and advertises the use of gradient flow techniques to pursue some of the future research directions discussed here.</p> |
| spellingShingle | Kinetic theory of gases Differential equations, Partial Wu, JSH Understanding the Landau equation as a gradient flow |
| title | Understanding the Landau equation as a gradient flow |
| title_full | Understanding the Landau equation as a gradient flow |
| title_fullStr | Understanding the Landau equation as a gradient flow |
| title_full_unstemmed | Understanding the Landau equation as a gradient flow |
| title_short | Understanding the Landau equation as a gradient flow |
| title_sort | understanding the landau equation as a gradient flow |
| topic | Kinetic theory of gases Differential equations, Partial |
| work_keys_str_mv | AT wujsh understandingthelandauequationasagradientflow |