Asymptotic Behavior of the Navier-Stokes Equations with Nonzero Far-Field Velocity
Concerning the nonstationary Navier-Stokes flow with a nonzero constant velocity at infinity, the temporal stability has been studied by Heywood (1970, 1972) and Masuda (1975) in 𝐿2 space and by Shibata (1999) and Enomoto-Shibata (2005) in 𝐿𝑝 spaces for 𝑝≥3. However, their results did not include en...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2011-01-01
|
| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/369571 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849305882900299776 |
|---|---|
| author | Jaiok Roh |
| author_facet | Jaiok Roh |
| author_sort | Jaiok Roh |
| collection | DOAJ |
| description | Concerning the nonstationary Navier-Stokes flow with a nonzero constant velocity at infinity, the temporal stability has been studied by Heywood (1970, 1972) and Masuda (1975) in 𝐿2 space and by Shibata (1999) and Enomoto-Shibata (2005) in 𝐿𝑝 spaces for 𝑝≥3. However, their results did not include enough information to find the spatial decay. So, Bae-Roh (2010) improved Enomoto-Shibata's results in some sense and estimated the spatial decay even though their results are limited. In this paper, we will prove temporal decay with a weighted function by using 𝐿𝑟−𝐿𝑝 decay estimates obtained by Roh (2011). Bae-Roh (2010) proved the temporal rate becomes slower by (1+𝜎)/2 if a weighted function is |𝑥|𝜎 for 0<𝜎<1/2. In this paper, we prove that the temporal decay becomes slower by 𝜎, where 0<𝜎<3/2 if a weighted function is |𝑥|𝜎. For the proof, we deduce an integral representation of the solution and then establish the temporal decay estimates of weighted 𝐿𝑝-norm of solutions. This method was first initiated by He and Xin (2000) and developed by Bae and Jin (2006, 2007, 2008). |
| format | Article |
| id | doaj-art-2f9769a74a504d3a95c01ef47da3bb9f |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-2f9769a74a504d3a95c01ef47da3bb9f2025-08-20T03:55:16ZengWileyAbstract and Applied Analysis1085-33751687-04092011-01-01201110.1155/2011/369571369571Asymptotic Behavior of the Navier-Stokes Equations with Nonzero Far-Field VelocityJaiok Roh0Department of Mathematics, Hallym University, Chuncheon 200-702, Republic of KoreaConcerning the nonstationary Navier-Stokes flow with a nonzero constant velocity at infinity, the temporal stability has been studied by Heywood (1970, 1972) and Masuda (1975) in 𝐿2 space and by Shibata (1999) and Enomoto-Shibata (2005) in 𝐿𝑝 spaces for 𝑝≥3. However, their results did not include enough information to find the spatial decay. So, Bae-Roh (2010) improved Enomoto-Shibata's results in some sense and estimated the spatial decay even though their results are limited. In this paper, we will prove temporal decay with a weighted function by using 𝐿𝑟−𝐿𝑝 decay estimates obtained by Roh (2011). Bae-Roh (2010) proved the temporal rate becomes slower by (1+𝜎)/2 if a weighted function is |𝑥|𝜎 for 0<𝜎<1/2. In this paper, we prove that the temporal decay becomes slower by 𝜎, where 0<𝜎<3/2 if a weighted function is |𝑥|𝜎. For the proof, we deduce an integral representation of the solution and then establish the temporal decay estimates of weighted 𝐿𝑝-norm of solutions. This method was first initiated by He and Xin (2000) and developed by Bae and Jin (2006, 2007, 2008).http://dx.doi.org/10.1155/2011/369571 |
| spellingShingle | Jaiok Roh Asymptotic Behavior of the Navier-Stokes Equations with Nonzero Far-Field Velocity Abstract and Applied Analysis |
| title | Asymptotic Behavior of the Navier-Stokes Equations with Nonzero Far-Field Velocity |
| title_full | Asymptotic Behavior of the Navier-Stokes Equations with Nonzero Far-Field Velocity |
| title_fullStr | Asymptotic Behavior of the Navier-Stokes Equations with Nonzero Far-Field Velocity |
| title_full_unstemmed | Asymptotic Behavior of the Navier-Stokes Equations with Nonzero Far-Field Velocity |
| title_short | Asymptotic Behavior of the Navier-Stokes Equations with Nonzero Far-Field Velocity |
| title_sort | asymptotic behavior of the navier stokes equations with nonzero far field velocity |
| url | http://dx.doi.org/10.1155/2011/369571 |
| work_keys_str_mv | AT jaiokroh asymptoticbehaviorofthenavierstokesequationswithnonzerofarfieldvelocity |