On the Existence of Leray-Hopf Weak Solutions to the Navier-Stokes Equations
We give a rather short and self-contained presentation of the global existence for Leray-Hopf weak solutions to the three dimensional incompressible Navier-Stokes equations, with constant density. We give a unified treatment in terms of the domains and the relative boundary conditions and in terms o...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-01-01
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| Series: | Fluids |
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| Online Access: | https://www.mdpi.com/2311-5521/6/1/42 |
| _version_ | 1797412122499481600 |
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| author | Luigi C. Berselli Stefano Spirito |
| author_facet | Luigi C. Berselli Stefano Spirito |
| author_sort | Luigi C. Berselli |
| collection | DOAJ |
| description | We give a rather short and self-contained presentation of the global existence for Leray-Hopf weak solutions to the three dimensional incompressible Navier-Stokes equations, with constant density. We give a unified treatment in terms of the domains and the relative boundary conditions and in terms of the approximation methods. More precisely, we consider the case of the whole space, the flat torus, and the case of a general bounded domain with a smooth boundary (the latter supplemented with homogeneous Dirichlet conditions). We consider as approximation schemes the Leray approximation method, the Faedo-Galerkin method, the semi-discretization in time and the approximation by adding a Smagorinsky-Ladyžhenskaya term. We mainly focus on developing a unified treatment especially in the compactness argument needed to show that approximations converge to the weak solutions. |
| first_indexed | 2024-03-09T04:57:32Z |
| format | Article |
| id | doaj.art-c91bdaa3748341c6954eae590ff124f5 |
| institution | Directory Open Access Journal |
| issn | 2311-5521 |
| language | English |
| last_indexed | 2024-03-09T04:57:32Z |
| publishDate | 2021-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fluids |
| spelling | doaj.art-c91bdaa3748341c6954eae590ff124f52023-12-03T13:04:36ZengMDPI AGFluids2311-55212021-01-01614210.3390/fluids6010042On the Existence of Leray-Hopf Weak Solutions to the Navier-Stokes EquationsLuigi C. Berselli0Stefano Spirito1Dipartimento di Matematica, Università di Pisa, Via F. Buonarroti 1/c, I-56127 Pisa, ItalyDISIM—Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Via Vetoio, I-67100 L’Aquila, ItalyWe give a rather short and self-contained presentation of the global existence for Leray-Hopf weak solutions to the three dimensional incompressible Navier-Stokes equations, with constant density. We give a unified treatment in terms of the domains and the relative boundary conditions and in terms of the approximation methods. More precisely, we consider the case of the whole space, the flat torus, and the case of a general bounded domain with a smooth boundary (the latter supplemented with homogeneous Dirichlet conditions). We consider as approximation schemes the Leray approximation method, the Faedo-Galerkin method, the semi-discretization in time and the approximation by adding a Smagorinsky-Ladyžhenskaya term. We mainly focus on developing a unified treatment especially in the compactness argument needed to show that approximations converge to the weak solutions.https://www.mdpi.com/2311-5521/6/1/42Navier-Stokes equationsLeray-Hopf weak solutionsexistence |
| spellingShingle | Luigi C. Berselli Stefano Spirito On the Existence of Leray-Hopf Weak Solutions to the Navier-Stokes Equations Fluids Navier-Stokes equations Leray-Hopf weak solutions existence |
| title | On the Existence of Leray-Hopf Weak Solutions to the Navier-Stokes Equations |
| title_full | On the Existence of Leray-Hopf Weak Solutions to the Navier-Stokes Equations |
| title_fullStr | On the Existence of Leray-Hopf Weak Solutions to the Navier-Stokes Equations |
| title_full_unstemmed | On the Existence of Leray-Hopf Weak Solutions to the Navier-Stokes Equations |
| title_short | On the Existence of Leray-Hopf Weak Solutions to the Navier-Stokes Equations |
| title_sort | on the existence of leray hopf weak solutions to the navier stokes equations |
| topic | Navier-Stokes equations Leray-Hopf weak solutions existence |
| url | https://www.mdpi.com/2311-5521/6/1/42 |
| work_keys_str_mv | AT luigicberselli ontheexistenceoflerayhopfweaksolutionstothenavierstokesequations AT stefanospirito ontheexistenceoflerayhopfweaksolutionstothenavierstokesequations |