On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type
Abstract We study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) + α ( x , t ) ∇ A ( u ) ) + f ( u , x , t ) . $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2021-01-01
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| Series: | Journal of Inequalities and Applications |
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| Online Access: | https://doi.org/10.1186/s13660-021-02550-w |
| _version_ | 1818854937314983936 |
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| author | Sujun Weng |
| author_facet | Sujun Weng |
| author_sort | Sujun Weng |
| collection | DOAJ |
| description | Abstract We study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) + α ( x , t ) ∇ A ( u ) ) + f ( u , x , t ) . $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). $$ We assume that A ′ ( s ) = a ( s ) ≥ 0 $A'(s)=a(s)\geq 0$ , A ( s ) $A(s)$ is a strictly increasing function, A ( 0 ) = 0 $A(0)=0$ , b ( x , t ) ≥ 0 $b(x,t)\geq 0$ , and α ( x , t ) ≥ 0 $\alpha (x,t)\geq 0$ . If b ( x , t ) = α ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ 0 , T ] , $$ b(x,t)=\alpha (x,t)=0,\quad (x,t)\in \partial \Omega \times [0,T], $$ then we prove the stability of weak solutions without the boundary value condition. |
| first_indexed | 2024-12-19T08:00:38Z |
| format | Article |
| id | doaj.art-08769eb987954a29b3dfe7728d7b20a8 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-19T08:00:38Z |
| publishDate | 2021-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-08769eb987954a29b3dfe7728d7b20a82022-12-21T20:29:53ZengSpringerOpenJournal of Inequalities and Applications1029-242X2021-01-012021111910.1186/s13660-021-02550-wOn a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed typeSujun Weng0Chengyi University College, Jimei UniversityAbstract We study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) + α ( x , t ) ∇ A ( u ) ) + f ( u , x , t ) . $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). $$ We assume that A ′ ( s ) = a ( s ) ≥ 0 $A'(s)=a(s)\geq 0$ , A ( s ) $A(s)$ is a strictly increasing function, A ( 0 ) = 0 $A(0)=0$ , b ( x , t ) ≥ 0 $b(x,t)\geq 0$ , and α ( x , t ) ≥ 0 $\alpha (x,t)\geq 0$ . If b ( x , t ) = α ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ 0 , T ] , $$ b(x,t)=\alpha (x,t)=0,\quad (x,t)\in \partial \Omega \times [0,T], $$ then we prove the stability of weak solutions without the boundary value condition.https://doi.org/10.1186/s13660-021-02550-wNewtonian fluid∼non-Newtonian fluid mixed-type equationThe existenceStabilityBoundary value condition |
| spellingShingle | Sujun Weng On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type Journal of Inequalities and Applications Newtonian fluid∼non-Newtonian fluid mixed-type equation The existence Stability Boundary value condition |
| title | On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type |
| title_full | On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type |
| title_fullStr | On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type |
| title_full_unstemmed | On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type |
| title_short | On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type |
| title_sort | on a degenerate parabolic equation with newtonian fluid∼non newtonian fluid mixed type |
| topic | Newtonian fluid∼non-Newtonian fluid mixed-type equation The existence Stability Boundary value condition |
| url | https://doi.org/10.1186/s13660-021-02550-w |
| work_keys_str_mv | AT sujunweng onadegenerateparabolicequationwithnewtonianfluidnonnewtonianfluidmixedtype |