On the ‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer
In this paper, we prove the ‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter , where , and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize th...
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| Published in | Mathematical methods in the applied sciences Vol. 38; no. 9; pp. 1888 - 1925 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English Japanese |
| Published |
01.06.2015
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 0170-4214 1099-1476 |
| DOI | 10.1002/mma.3201 |
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| Abstract | In this paper, we prove the
‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter
, where
, and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize that we can choose 0 <
ϵ
<
π
∕ 2 and
γ
0
> 0 arbitrarily, although usual parabolic theorem tells us that we must choose a large
γ
0
> 0 for given 0 <
ϵ
<
π
∕ 2. We also prove the maximal
L
p
−
L
q
regularity theorem of the nonstationary Stokes problem as an application of the
‐boundedness. The key of our approach is to apply several technical lemmas to the exact solution formulas of a resolvent problem. The formulas are obtained through the solutions of the ODEs, in the Fourier space, driven by the partial Fourier transform with respect to tangential space variable
. Copyright © 2014 John Wiley & Sons, Ltd. |
|---|---|
| AbstractList | In this paper, we prove the [Formulaomitted]-boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter [Formulaomitted], where [Formulaomitted], and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize that we can choose 0<< pi 2 and gamma sub(0)>0 arbitrarily, although usual parabolic theorem tells us that we must choose a large gamma sub(0)>0 for given 0<< pi 2. We also prove the maximal L sub()p L sub()qregularity theorem of the nonstationary Stokes problem as an application of the [Formulaomitted]-boundedness. The key of our approach is to apply several technical lemmas to the exact solution formulas of a resolvent problem. The formulas are obtained through the solutions of the ODEs, in the Fourier space, driven by the partial Fourier transform with respect to tangential space variable [Formulaomitted]. In this paper, we prove the ‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter , where , and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize that we can choose 0 < ϵ < π ∕ 2 and γ 0 > 0 arbitrarily, although usual parabolic theorem tells us that we must choose a large γ 0 > 0 for given 0 < ϵ < π ∕ 2. We also prove the maximal L p − L q regularity theorem of the nonstationary Stokes problem as an application of the ‐boundedness. The key of our approach is to apply several technical lemmas to the exact solution formulas of a resolvent problem. The formulas are obtained through the solutions of the ODEs, in the Fourier space, driven by the partial Fourier transform with respect to tangential space variable . Copyright © 2014 John Wiley & Sons, Ltd. |
| Author | Saito, Hirokazu |
| Author_xml | – sequence: 1 givenname: Hirokazu surname: Saito fullname: Saito, Hirokazu organization: Department of Pure and Applied Mathematics, Graduate School of Fundamental Science and Engineering Waseda University Okubo 3‐4‐1 Shinjuku‐ku, Tokyo 169‐8555 Japan |
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| CitedBy_id | crossref_primary_10_1619_fesi_62_337 crossref_primary_10_1016_j_jde_2017_09_045 crossref_primary_10_1007_s00021_022_00680_9 crossref_primary_10_1007_s00021_024_00914_y crossref_primary_10_1002_mma_6999 crossref_primary_10_1016_j_jde_2019_09_040 crossref_primary_10_1007_s00028_021_00674_6 crossref_primary_10_1016_j_jde_2023_11_020 |
| Cites_doi | 10.1007/s00021-012-0130-1 10.57262/die/1356060651 10.4064/sm-93-3-201-222 10.1515/CRELLE.2008.013 10.1007/BF00250586 10.1002/mma.483 10.57262/ade/1355867895 10.1007/BF01442184 10.2969/jmsj/04640607 10.1002/mana.200310365 10.2969/jmsj/06420561 10.1007/s00209-007-0120-9 10.1002/cpa.3160340305 10.1007/BF00375146 10.1016/j.na.2009.05.061 10.1007/s00021-004-0116-8 10.1007/s00021-004-0117-7 10.1007/PL00004457 10.1007/s00021-003-0075-5 10.1090/memo/0788 10.1016/S0304-0208(08)72355-7 10.1007/BF00375142 10.2969/jmsj/1191419127 10.1007/978-3-319-02000-6 10.1007/PL00012599 10.1007/PL00000970 |
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| References_xml | – ident: e_1_2_10_21_1 doi: 10.1007/s00021-012-0130-1 – volume: 16 start-page: 385 issue: 4 year: 2003 ident: e_1_2_10_31_1 article-title: On a resolvent estimate for the Stokes system with Neumann boundary condition publication-title: Differential Integral Equations doi: 10.57262/die/1356060651 – volume: 3 start-page: 201 year: 1989 ident: e_1_2_10_27_1 article-title: On vector‐valued Fourier multiplier theorems publication-title: Studia Mathematica doi: 10.4064/sm-93-3-201-222 – ident: e_1_2_10_20_1 doi: 10.1515/CRELLE.2008.013 – volume: 84 start-page: 307 issue: 4 year: 1983 ident: e_1_2_10_12_1 article-title: Large‐time regularity of viscous surface waves publication-title: Archive for Rational Mechanics and Analysis doi: 10.1007/BF00250586 – volume: 49 start-page: 691 issue: 4 year: 2009 ident: e_1_2_10_16_1 article-title: Decaying solution of a Navier–Stokes flow without surface tension publication-title: Journal of Mathematics of Kyoto University – ident: e_1_2_10_6_1 doi: 10.1002/mma.483 – volume: 10 start-page: 45 issue: 1 year: 2005 ident: e_1_2_10_9_1 article-title: The initial‐value problem for the Navier–Stokes equations with a free surface in L q ‐Sobolev spaces publication-title: Advances in Differential Equations doi: 10.57262/ade/1355867895 – ident: e_1_2_10_13_1 doi: 10.1007/BF01442184 – ident: e_1_2_10_22_1 doi: 10.2969/jmsj/04640607 – ident: e_1_2_10_7_1 doi: 10.1002/mana.200310365 – ident: e_1_2_10_19_1 doi: 10.2969/jmsj/06420561 – ident: e_1_2_10_26_1 doi: 10.1007/s00209-007-0120-9 – ident: e_1_2_10_11_1 doi: 10.1002/cpa.3160340305 – ident: e_1_2_10_14_1 doi: 10.1007/BF00375146 – ident: e_1_2_10_18_1 doi: 10.1016/j.na.2009.05.061 – ident: e_1_2_10_10_1 doi: 10.1007/s00021-004-0116-8 – ident: e_1_2_10_23_1 – ident: e_1_2_10_8_1 doi: 10.1007/s00021-004-0117-7 – ident: e_1_2_10_24_1 doi: 10.1007/PL00004457 – ident: e_1_2_10_29_1 – volume: 5 start-page: 245 issue: 3 year: 2003 ident: e_1_2_10_2_1 article-title: On a resolvent estimate of the Stokes equation on an infinite layer part 2, λ=0 case publication-title: Journal of Mathematical Fluid Mechanics doi: 10.1007/s00021-003-0075-5 – ident: e_1_2_10_28_1 doi: 10.1090/memo/0788 – ident: e_1_2_10_17_1 doi: 10.1016/S0304-0208(08)72355-7 – ident: e_1_2_10_15_1 doi: 10.1007/BF00375142 – volume: 10 start-page: 1081 year: 2005 ident: e_1_2_10_4_1 article-title: Resolvent estimates for the Stokes operator on an infinite layer publication-title: Differential Integral Equations – ident: e_1_2_10_3_1 doi: 10.2969/jmsj/1191419127 – ident: e_1_2_10_25_1 doi: 10.1007/978-3-319-02000-6 – ident: e_1_2_10_5_1 doi: 10.1007/PL00012599 – volume-title: Multidimensional Singular Integrals and Integral equations year: 1965 ident: e_1_2_10_30_1 – ident: e_1_2_10_32_1 doi: 10.1007/PL00000970 |
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| Snippet | In this paper, we prove the
‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent... In this paper, we prove the [Formulaomitted]-boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with... |
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| SubjectTerms | Boundaries Dirichlet problem Fluid flow Fourier analysis Fourier transforms Mathematical analysis Operators Theorems |
| Title | On the ‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer |
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