The well-posedness problem of an anisotropic porous medium equation with a convection term
Abstract The initial boundary value problem of an anisotropic porous medium equation is considered in this paper. The existence of a weak solution is proved by the monotone convergent method. By showing that ∇ u ∈ L ∞ ( 0 , T ; L loc 2 ( Ω ) ) $\nabla u\in L^{\infty}(0,T; L^{2}_{\mathrm{loc}}(\Omega...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2022-08-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13660-022-02847-4 |
| _version_ | 1818084564451459072 |
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| author | Yuan Zhi Huashui Zhan |
| author_facet | Yuan Zhi Huashui Zhan |
| author_sort | Yuan Zhi |
| collection | DOAJ |
| description | Abstract The initial boundary value problem of an anisotropic porous medium equation is considered in this paper. The existence of a weak solution is proved by the monotone convergent method. By showing that ∇ u ∈ L ∞ ( 0 , T ; L loc 2 ( Ω ) ) $\nabla u\in L^{\infty}(0,T; L^{2}_{\mathrm{loc}}(\Omega ))$ , according to different boundary value conditions, some stability theorems of weak solutions are obtained. The unusual thing is that the partial boundary value condition is based on a submanifold Σ of ∂ Ω × ( 0 , T ) $\partial \Omega \times (0,T)$ and, in some special cases, Σ = { ( x , t ) ∈ ∂ Ω × ( 0 , T ) : ∏ a i ( x , t ) > 0 } $\Sigma = \{(x,t)\in \partial \Omega \times (0,T): \prod a_{i}(x,t)>0 \}$ . |
| first_indexed | 2024-12-10T19:55:54Z |
| format | Article |
| id | doaj.art-e0add740646c40038728afdbd4514359 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-10T19:55:54Z |
| publishDate | 2022-08-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-e0add740646c40038728afdbd45143592022-12-22T01:35:39ZengSpringerOpenJournal of Inequalities and Applications1029-242X2022-08-012022112210.1186/s13660-022-02847-4The well-posedness problem of an anisotropic porous medium equation with a convection termYuan Zhi0Huashui Zhan1School of Sciences, Jimei UniversitySchool of Applied Mathematics, Xiamen University of TechnologyAbstract The initial boundary value problem of an anisotropic porous medium equation is considered in this paper. The existence of a weak solution is proved by the monotone convergent method. By showing that ∇ u ∈ L ∞ ( 0 , T ; L loc 2 ( Ω ) ) $\nabla u\in L^{\infty}(0,T; L^{2}_{\mathrm{loc}}(\Omega ))$ , according to different boundary value conditions, some stability theorems of weak solutions are obtained. The unusual thing is that the partial boundary value condition is based on a submanifold Σ of ∂ Ω × ( 0 , T ) $\partial \Omega \times (0,T)$ and, in some special cases, Σ = { ( x , t ) ∈ ∂ Ω × ( 0 , T ) : ∏ a i ( x , t ) > 0 } $\Sigma = \{(x,t)\in \partial \Omega \times (0,T): \prod a_{i}(x,t)>0 \}$ .https://doi.org/10.1186/s13660-022-02847-4Anisotropic porous medium equationStability theoremPartial boundary conditionSubmanifold |
| spellingShingle | Yuan Zhi Huashui Zhan The well-posedness problem of an anisotropic porous medium equation with a convection term Journal of Inequalities and Applications Anisotropic porous medium equation Stability theorem Partial boundary condition Submanifold |
| title | The well-posedness problem of an anisotropic porous medium equation with a convection term |
| title_full | The well-posedness problem of an anisotropic porous medium equation with a convection term |
| title_fullStr | The well-posedness problem of an anisotropic porous medium equation with a convection term |
| title_full_unstemmed | The well-posedness problem of an anisotropic porous medium equation with a convection term |
| title_short | The well-posedness problem of an anisotropic porous medium equation with a convection term |
| title_sort | well posedness problem of an anisotropic porous medium equation with a convection term |
| topic | Anisotropic porous medium equation Stability theorem Partial boundary condition Submanifold |
| url | https://doi.org/10.1186/s13660-022-02847-4 |
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