A mathematical model for the use of energy resources: a singular parabolic equation
We consider a singular parabolic equation, for , arising in symmetric boundary layer flows. Here is a bounded domain with C2 boundary is bounded, and T > 0 is some fixed time. We establish sufficient conditions for the existence and uniqueness of a weak solution of this singular parabolic equatio...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Vilnius Gediminas Technical University
2020-01-01
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| Series: | Mathematical Modelling and Analysis |
| Subjects: | |
| Online Access: | https://journals.vgtu.lt/index.php/MMA/article/view/9792 |
| _version_ | 1818872868354654208 |
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| author | Daniel López-García Rosa Pardo |
| author_facet | Daniel López-García Rosa Pardo |
| author_sort | Daniel López-García |
| collection | DOAJ |
| description | We consider a singular parabolic equation, for , arising in symmetric boundary layer flows. Here is a bounded domain with C2 boundary is bounded, and T > 0 is some fixed time. We establish sufficient conditions for the existence and uniqueness of a weak solution of this singular parabolic equation with Dirichlet boundary conditions, and we investigate its regularity.
There are two different cases depending on β. If β < 1, for any initial data, there exists a unique weak solution, which in fact is a strong solution. The singularity is removable when β < 1. While if β = 1, there exists a unique solution of the singular parabolic problem The initial data cannot be arbitrarily chosen. In fact, if f is continuous and , as t → 0, then, this solution converges, as t → 0, to the solution of the elliptic problem, for , with Dirichlet boundary conditions. Hence, no initial data can be prescribed when β = 1, and the singularity in that case is strong. |
| first_indexed | 2024-12-19T12:45:39Z |
| format | Article |
| id | doaj.art-74e74d46e9324c9dabcd43b2f14463b1 |
| institution | Directory Open Access Journal |
| issn | 1392-6292 1648-3510 |
| language | English |
| last_indexed | 2024-12-19T12:45:39Z |
| publishDate | 2020-01-01 |
| publisher | Vilnius Gediminas Technical University |
| record_format | Article |
| series | Mathematical Modelling and Analysis |
| spelling | doaj.art-74e74d46e9324c9dabcd43b2f14463b12022-12-21T20:20:48ZengVilnius Gediminas Technical UniversityMathematical Modelling and Analysis1392-62921648-35102020-01-0125110.3846/mma.2020.9792A mathematical model for the use of energy resources: a singular parabolic equationDaniel López-García0Rosa Pardo1Departamento de Análisis Matemático y Matemática Aplicada, Universidad, Complutense de Madrid, 28040, Madrid, SpainDepartamento de Análisis Matemático y Matemática Aplicada, Universidad, Complutense de Madrid, 28040, Madrid, SpainWe consider a singular parabolic equation, for , arising in symmetric boundary layer flows. Here is a bounded domain with C2 boundary is bounded, and T > 0 is some fixed time. We establish sufficient conditions for the existence and uniqueness of a weak solution of this singular parabolic equation with Dirichlet boundary conditions, and we investigate its regularity. There are two different cases depending on β. If β < 1, for any initial data, there exists a unique weak solution, which in fact is a strong solution. The singularity is removable when β < 1. While if β = 1, there exists a unique solution of the singular parabolic problem The initial data cannot be arbitrarily chosen. In fact, if f is continuous and , as t → 0, then, this solution converges, as t → 0, to the solution of the elliptic problem, for , with Dirichlet boundary conditions. Hence, no initial data can be prescribed when β = 1, and the singularity in that case is strong.https://journals.vgtu.lt/index.php/MMA/article/view/9792singular parabolic equationdegenerate parabolic equationsexistenceuniquenesssymmetric boundary layerregularity |
| spellingShingle | Daniel López-García Rosa Pardo A mathematical model for the use of energy resources: a singular parabolic equation Mathematical Modelling and Analysis singular parabolic equation degenerate parabolic equations existence uniqueness symmetric boundary layer regularity |
| title | A mathematical model for the use of energy resources: a singular parabolic equation |
| title_full | A mathematical model for the use of energy resources: a singular parabolic equation |
| title_fullStr | A mathematical model for the use of energy resources: a singular parabolic equation |
| title_full_unstemmed | A mathematical model for the use of energy resources: a singular parabolic equation |
| title_short | A mathematical model for the use of energy resources: a singular parabolic equation |
| title_sort | mathematical model for the use of energy resources a singular parabolic equation |
| topic | singular parabolic equation degenerate parabolic equations existence uniqueness symmetric boundary layer regularity |
| url | https://journals.vgtu.lt/index.php/MMA/article/view/9792 |
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