Quasigeostrophic Equations for Fractional Powers of Infinitesimal Generators
In this paper we treat the following partial differential equation, the quasigeostrophic equation: ∂/∂t+u·∇f=-σ-Aαf, 0≤α≤1, where (A,D(A)) is the infinitesimal generator of a convolution C0-semigroup of positive kernel on Lp(Rn), with 1≤p<∞. Firstly, we give remarkable pointwise and integral ine...
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| Format: | Article |
| Language: | English |
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Wiley
2019-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2019/4763450 |
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| author | Luciano Abadias Pedro J. Miana |
| author_facet | Luciano Abadias Pedro J. Miana |
| author_sort | Luciano Abadias |
| collection | DOAJ |
| description | In this paper we treat the following partial differential equation, the quasigeostrophic equation: ∂/∂t+u·∇f=-σ-Aαf, 0≤α≤1, where (A,D(A)) is the infinitesimal generator of a convolution C0-semigroup of positive kernel on Lp(Rn), with 1≤p<∞. Firstly, we give remarkable pointwise and integral inequalities involving the fractional powers (-A)α for 0≤α≤1. We use these estimates to obtain Lp-decayment of solutions of the above quasigeostrophic equation. These results extend the case of fractional derivatives (taking A=Δ, the Laplacian), which has been studied in the literature. |
| format | Article |
| id | doaj-art-4b0d0091867046aaafc7775d2170d705 |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2019-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-4b0d0091867046aaafc7775d2170d7052025-02-03T01:12:11ZengWileyJournal of Function Spaces2314-88962314-88882019-01-01201910.1155/2019/47634504763450Quasigeostrophic Equations for Fractional Powers of Infinitesimal GeneratorsLuciano Abadias0Pedro J. Miana1Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, SpainDepartamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, SpainIn this paper we treat the following partial differential equation, the quasigeostrophic equation: ∂/∂t+u·∇f=-σ-Aαf, 0≤α≤1, where (A,D(A)) is the infinitesimal generator of a convolution C0-semigroup of positive kernel on Lp(Rn), with 1≤p<∞. Firstly, we give remarkable pointwise and integral inequalities involving the fractional powers (-A)α for 0≤α≤1. We use these estimates to obtain Lp-decayment of solutions of the above quasigeostrophic equation. These results extend the case of fractional derivatives (taking A=Δ, the Laplacian), which has been studied in the literature.http://dx.doi.org/10.1155/2019/4763450 |
| spellingShingle | Luciano Abadias Pedro J. Miana Quasigeostrophic Equations for Fractional Powers of Infinitesimal Generators Journal of Function Spaces |
| title | Quasigeostrophic Equations for Fractional Powers of Infinitesimal Generators |
| title_full | Quasigeostrophic Equations for Fractional Powers of Infinitesimal Generators |
| title_fullStr | Quasigeostrophic Equations for Fractional Powers of Infinitesimal Generators |
| title_full_unstemmed | Quasigeostrophic Equations for Fractional Powers of Infinitesimal Generators |
| title_short | Quasigeostrophic Equations for Fractional Powers of Infinitesimal Generators |
| title_sort | quasigeostrophic equations for fractional powers of infinitesimal generators |
| url | http://dx.doi.org/10.1155/2019/4763450 |
| work_keys_str_mv | AT lucianoabadias quasigeostrophicequationsforfractionalpowersofinfinitesimalgenerators AT pedrojmiana quasigeostrophicequationsforfractionalpowersofinfinitesimalgenerators |