An elementary proof of the continuity from L 2 0(Ω) to H 1 0 (Ω) n of bogovskii's right inverse of the divergence

The existence of right inverses of the divergence as an operator from H 1 0 (Ω) n to L 2 0(Ω) is a problem that has been widely studied because of its importance in the analysis of the classic equations of fluid dynamics. When Ω is a bounded domain which is star-shaped with respect to a ball B, a ri...

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Autor principal: Duran, Ricardo Guillermo
Publicado: 2012
Materias:
Divergence operator
Singular integrals
Stokes equations
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00416932_v53_n2_p59_Duran
http://hdl.handle.net/20.500.12110/paper_00416932_v53_n2_p59_Duran
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_00416932_v53_n2_p59_Duran
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spelling paper:paper_00416932_v53_n2_p59_Duran2023-06-08T15:04:47Z An elementary proof of the continuity from L 2 0(Ω) to H 1 0 (Ω) n of bogovskii's right inverse of the divergence Duran, Ricardo Guillermo Divergence operator Singular integrals Stokes equations The existence of right inverses of the divergence as an operator from H 1 0 (Ω) n to L 2 0(Ω) is a problem that has been widely studied because of its importance in the analysis of the classic equations of fluid dynamics. When Ω is a bounded domain which is star-shaped with respect to a ball B, a right inverse given by an integral operator was introduced by Bogovskii, who also proved its continuity using the Calderón-Zygmund theory of singular integrals. In this paper we give an alternative elementary proof of the continuity using the Fourier transform. As a consequence, we obtain estimates for the constant in the continuity in terms of the ratio between the diameter of Ω and that of B. Moreover, using the relation between the existence of right inverses of the divergence with the Korn and improved Poincaré inequalities, we obtain estimates for the constants in these two inequalities. We also show that one can proceed in the opposite way, that is, the existence of a continuous right inverse of the divergence, as well as estimates for the constant in that continuity, can be obtained from the improved Poincaré inequality. We give an interesting example of this situation in the case of convex domains. Fil:Durán, R.G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2012 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00416932_v53_n2_p59_Duran http://hdl.handle.net/20.500.12110/paper_00416932_v53_n2_p59_Duran
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Divergence operator
Singular integrals
Stokes equations
spellingShingle Divergence operator
Singular integrals
Stokes equations
Duran, Ricardo Guillermo
An elementary proof of the continuity from L 2 0(Ω) to H 1 0 (Ω) n of bogovskii's right inverse of the divergence
topic_facet Divergence operator
Singular integrals
Stokes equations
description The existence of right inverses of the divergence as an operator from H 1 0 (Ω) n to L 2 0(Ω) is a problem that has been widely studied because of its importance in the analysis of the classic equations of fluid dynamics. When Ω is a bounded domain which is star-shaped with respect to a ball B, a right inverse given by an integral operator was introduced by Bogovskii, who also proved its continuity using the Calderón-Zygmund theory of singular integrals. In this paper we give an alternative elementary proof of the continuity using the Fourier transform. As a consequence, we obtain estimates for the constant in the continuity in terms of the ratio between the diameter of Ω and that of B. Moreover, using the relation between the existence of right inverses of the divergence with the Korn and improved Poincaré inequalities, we obtain estimates for the constants in these two inequalities. We also show that one can proceed in the opposite way, that is, the existence of a continuous right inverse of the divergence, as well as estimates for the constant in that continuity, can be obtained from the improved Poincaré inequality. We give an interesting example of this situation in the case of convex domains.
author Duran, Ricardo Guillermo
author_facet Duran, Ricardo Guillermo
author_sort Duran, Ricardo Guillermo
title An elementary proof of the continuity from L 2 0(Ω) to H 1 0 (Ω) n of bogovskii's right inverse of the divergence
title_short An elementary proof of the continuity from L 2 0(Ω) to H 1 0 (Ω) n of bogovskii's right inverse of the divergence
title_full An elementary proof of the continuity from L 2 0(Ω) to H 1 0 (Ω) n of bogovskii's right inverse of the divergence
title_fullStr An elementary proof of the continuity from L 2 0(Ω) to H 1 0 (Ω) n of bogovskii's right inverse of the divergence
title_full_unstemmed An elementary proof of the continuity from L 2 0(Ω) to H 1 0 (Ω) n of bogovskii's right inverse of the divergence
title_sort elementary proof of the continuity from l 2 0(ω) to h 1 0 (ω) n of bogovskii's right inverse of the divergence
publishDate 2012
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00416932_v53_n2_p59_Duran
http://hdl.handle.net/20.500.12110/paper_00416932_v53_n2_p59_Duran
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AT duranricardoguillermo elementaryproofofthecontinuityfroml20ōtoh10ōnofbogovskiisrightinverseofthedivergence
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