Unique solvability of the free-boundary Navier-Stokes equations with surface tension
We prove the existence and uniqueness of solutions to the time-dependent incompressible Navier-Stokes equations with a free-boundary governed by surface tension. The solution is found using a topological fixed-point theorem for a nonlinear iteration scheme, requiring at each step, the solution of a...
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| Format: | Journal article |
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2002
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| _version_ | 1797092868955832320 |
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| author | Coutand, D Shkoller, S |
| author_facet | Coutand, D Shkoller, S |
| author_sort | Coutand, D |
| collection | OXFORD |
| description | We prove the existence and uniqueness of solutions to the time-dependent incompressible Navier-Stokes equations with a free-boundary governed by surface tension. The solution is found using a topological fixed-point theorem for a nonlinear iteration scheme, requiring at each step, the solution of a model linear problem consisting of the time-dependent Stokes equation with linearized mean-curvature forcing on the boundary. We use energy methods to establish new types of spacetime inequalities that allow us to find a unique weak solution to this problem. We then prove regularity of the weak solution, and establish the a priori estimates required by the nonlinear iteration process. |
| first_indexed | 2024-03-07T03:52:10Z |
| format | Journal article |
| id | oxford-uuid:c1a6ce77-e116-42db-8d6a-cf8de78693ce |
| institution | University of Oxford |
| last_indexed | 2024-03-07T03:52:10Z |
| publishDate | 2002 |
| record_format | dspace |
| spelling | oxford-uuid:c1a6ce77-e116-42db-8d6a-cf8de78693ce2022-03-27T06:03:07ZUnique solvability of the free-boundary Navier-Stokes equations with surface tensionJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:c1a6ce77-e116-42db-8d6a-cf8de78693ceSymplectic Elements at Oxford2002Coutand, DShkoller, SWe prove the existence and uniqueness of solutions to the time-dependent incompressible Navier-Stokes equations with a free-boundary governed by surface tension. The solution is found using a topological fixed-point theorem for a nonlinear iteration scheme, requiring at each step, the solution of a model linear problem consisting of the time-dependent Stokes equation with linearized mean-curvature forcing on the boundary. We use energy methods to establish new types of spacetime inequalities that allow us to find a unique weak solution to this problem. We then prove regularity of the weak solution, and establish the a priori estimates required by the nonlinear iteration process. |
| spellingShingle | Coutand, D Shkoller, S Unique solvability of the free-boundary Navier-Stokes equations with surface tension |
| title | Unique solvability of the free-boundary Navier-Stokes equations with
surface tension |
| title_full | Unique solvability of the free-boundary Navier-Stokes equations with
surface tension |
| title_fullStr | Unique solvability of the free-boundary Navier-Stokes equations with
surface tension |
| title_full_unstemmed | Unique solvability of the free-boundary Navier-Stokes equations with
surface tension |
| title_short | Unique solvability of the free-boundary Navier-Stokes equations with
surface tension |
| title_sort | unique solvability of the free boundary navier stokes equations with surface tension |
| work_keys_str_mv | AT coutandd uniquesolvabilityofthefreeboundarynavierstokesequationswithsurfacetension AT shkollers uniquesolvabilityofthefreeboundarynavierstokesequationswithsurfacetension |