Solutions of the divergence and analysis of the stokes equations in planar Hölder-α domains
If Ω ⊂ n is a bounded domain, the existence of solutions u∈ H10(Ω)n of div u = f for f ∈ L 2(Ω) with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular, it allows to show the existence of a solution (u,p)∈ H10(Ω)n× L2(Ω ), where u is the velocity and p the...
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todo:paper_02182025_v20_n1_p95_DurAn2023-10-03T15:10:49Z Solutions of the divergence and analysis of the stokes equations in planar Hölder-α domains DurÁn, R.G. López GarcÍa, F. Divergence operator Hölder-α domains Stokes equations Basic results Bounded domain Divergence operators Existence of Solutions Lipschitz domain Mean values Stokes equations Weighted Sobolev spaces Wellposedness Holmium Sobolev spaces Mathematical operators If Ω ⊂ n is a bounded domain, the existence of solutions u∈ H10(Ω)n of div u = f for f ∈ L 2(Ω) with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular, it allows to show the existence of a solution (u,p)∈ H10(Ω)n× L2(Ω ), where u is the velocity and p the pressure. It is known that the above-mentioned result holds when Ω is a Lipschitz domain and that it is not valid for arbitrary Hölder-α domains. In this paper we prove that if Ω is a planar simply connected Hölder-α domain, there exist solutions of div u = f in appropriate weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Moreover, we show that the powers of the distance in the results obtained are optimal. For some particular domains with an external cusp, we apply our results to show the well-posedness of the Stokes equations in appropriate weighted Sobolev spaces obtaining as a consequence the existence of a solution (u,p)∈ H10(Ω) n× Lr(Ω) for some r < 2 depending on the power of the cusp. © 2010 World Scientific Publishing Company. Fil:DurÁn, R.G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:López GarcÍa, F. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_02182025_v20_n1_p95_DurAn |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Divergence operator Hölder-α domains Stokes equations Basic results Bounded domain Divergence operators Existence of Solutions Lipschitz domain Mean values Stokes equations Weighted Sobolev spaces Wellposedness Holmium Sobolev spaces Mathematical operators |
| spellingShingle |
Divergence operator Hölder-α domains Stokes equations Basic results Bounded domain Divergence operators Existence of Solutions Lipschitz domain Mean values Stokes equations Weighted Sobolev spaces Wellposedness Holmium Sobolev spaces Mathematical operators DurÁn, R.G. López GarcÍa, F. Solutions of the divergence and analysis of the stokes equations in planar Hölder-α domains |
| topic_facet |
Divergence operator Hölder-α domains Stokes equations Basic results Bounded domain Divergence operators Existence of Solutions Lipschitz domain Mean values Stokes equations Weighted Sobolev spaces Wellposedness Holmium Sobolev spaces Mathematical operators |
| description |
If Ω ⊂ n is a bounded domain, the existence of solutions u∈ H10(Ω)n of div u = f for f ∈ L 2(Ω) with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular, it allows to show the existence of a solution (u,p)∈ H10(Ω)n× L2(Ω ), where u is the velocity and p the pressure. It is known that the above-mentioned result holds when Ω is a Lipschitz domain and that it is not valid for arbitrary Hölder-α domains. In this paper we prove that if Ω is a planar simply connected Hölder-α domain, there exist solutions of div u = f in appropriate weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Moreover, we show that the powers of the distance in the results obtained are optimal. For some particular domains with an external cusp, we apply our results to show the well-posedness of the Stokes equations in appropriate weighted Sobolev spaces obtaining as a consequence the existence of a solution (u,p)∈ H10(Ω) n× Lr(Ω) for some r < 2 depending on the power of the cusp. © 2010 World Scientific Publishing Company. |
| format |
JOUR |
| author |
DurÁn, R.G. López GarcÍa, F. |
| author_facet |
DurÁn, R.G. López GarcÍa, F. |
| author_sort |
DurÁn, R.G. |
| title |
Solutions of the divergence and analysis of the stokes equations in planar Hölder-α domains |
| title_short |
Solutions of the divergence and analysis of the stokes equations in planar Hölder-α domains |
| title_full |
Solutions of the divergence and analysis of the stokes equations in planar Hölder-α domains |
| title_fullStr |
Solutions of the divergence and analysis of the stokes equations in planar Hölder-α domains |
| title_full_unstemmed |
Solutions of the divergence and analysis of the stokes equations in planar Hölder-α domains |
| title_sort |
solutions of the divergence and analysis of the stokes equations in planar hölder-α domains |
| url |
http://hdl.handle.net/20.500.12110/paper_02182025_v20_n1_p95_DurAn |
| work_keys_str_mv |
AT duranrg solutionsofthedivergenceandanalysisofthestokesequationsinplanarholderadomains AT lopezgarciaf solutionsofthedivergenceandanalysisofthestokesequationsinplanarholderadomains |
| _version_ |
1782026501304090624 |