A solution formula and the R-boundedness for the generalized Stokes resolvent problem in an infinite layer with Neumann boundary condition
We consider the generalized Stokes resolvent problem in an infinite layer with Neumann boundary conditions. This problem arises from a free boundary problem describing the motion of incompressible viscous one-phase fluid flow without surface tension in an infinite layer bounded both from above and f...
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| description | We consider the generalized Stokes resolvent problem in an infinite layer with Neumann boundary conditions. This problem arises from a free boundary problem describing the motion of incompressible viscous one-phase fluid flow without surface tension in an infinite layer bounded both from above and from below by free surfaces. We derive a new exact solution formula to the generalized Stokes resolvent problem and prove the \(\mathscr{R}\)-boundedness of the solution operator families with resolvent parameter \(\lambda\) varying in a sector \(\Sigma_{\varepsilon,\gamma_0}\) for any \(\gamma_0>0\) and \(00\) arbitrarily, while in general domains we only know the \(\mathscr{R}\)-boundedness for \(\gamma_0\gg1\) from the result by Shibata. As compared with the case of Neumann-Dirichlet boundary condition studied by Saito, analysis is even harder on account of higher singularity of the symbols in the solution formula. |
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This problem arises from a free boundary problem describing the motion of incompressible viscous one-phase fluid flow without surface tension in an infinite layer bounded both from above and from below by free surfaces. We derive a new exact solution formula to the generalized Stokes resolvent problem and prove the \(\mathscr{R}\)-boundedness of the solution operator families with resolvent parameter \(\lambda\) varying in a sector \(\Sigma_{\varepsilon,\gamma_0}\) for any \(\gamma_0>0\) and \(0<\varepsilon<\pi/2\), where \(\Sigma_{\varepsilon,\gamma_0} =\{ \lambda\in\mathbb{C}\setminus\{0\} \mid |\arg\lambda|\leq\pi-\varepsilon, \ |\lambda|>\gamma_0 \}\). As applications, we obtain the maximal \(L_p\)-\(L_q\) regularity for the nonstationary Stokes problem and then establish the well-posedness locally in time of the nonlinear free boundary problem mentioned above in \(L_p\)-\(L_q\) setting. We make full use of the solution formula to take \(\gamma_0>0\) arbitrarily, while in general domains we only know the \(\mathscr{R}\)-boundedness for \(\gamma_0\gg1\) from the result by Shibata. As compared with the case of Neumann-Dirichlet boundary condition studied by Saito, analysis is even harder on account of higher singularity of the symbols in the solution formula.</description><identifier>EISSN: 2331-8422</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Boundary conditions ; Computational fluid dynamics ; Dirichlet problem ; Exact solutions ; Fluid flow ; Free boundaries ; Free surfaces ; Incompressible flow ; Surface tension ; Well posed problems</subject><ispartof>arXiv.org, 2020-10</ispartof><rights>2020. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). 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We derive a new exact solution formula to the generalized Stokes resolvent problem and prove the \(\mathscr{R}\)-boundedness of the solution operator families with resolvent parameter \(\lambda\) varying in a sector \(\Sigma_{\varepsilon,\gamma_0}\) for any \(\gamma_0>0\) and \(0<\varepsilon<\pi/2\), where \(\Sigma_{\varepsilon,\gamma_0} =\{ \lambda\in\mathbb{C}\setminus\{0\} \mid |\arg\lambda|\leq\pi-\varepsilon, \ |\lambda|>\gamma_0 \}\). As applications, we obtain the maximal \(L_p\)-\(L_q\) regularity for the nonstationary Stokes problem and then establish the well-posedness locally in time of the nonlinear free boundary problem mentioned above in \(L_p\)-\(L_q\) setting. We make full use of the solution formula to take \(\gamma_0>0\) arbitrarily, while in general domains we only know the \(\mathscr{R}\)-boundedness for \(\gamma_0\gg1\) from the result by Shibata. As compared with the case of Neumann-Dirichlet boundary condition studied by Saito, analysis is even harder on account of higher singularity of the symbols in the solution formula.</description><subject>Boundary conditions</subject><subject>Computational fluid dynamics</subject><subject>Dirichlet problem</subject><subject>Exact solutions</subject><subject>Fluid flow</subject><subject>Free boundaries</subject><subject>Free surfaces</subject><subject>Incompressible flow</subject><subject>Surface tension</subject><subject>Well posed problems</subject><issn>2331-8422</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><recordid>eNqNjU0OAUEUhDsSCcEdXmI9yUzPMLYixMoCe-nRb2h6XtM_hCM4tSYOYFOVVFW-arEuz_MsmRScd9jAuVOapnxc8tEo77LXFJzRwStDUBvbBC1AkAR_RFgnlQkkURI692m_6QEJrdDqiRI23pzRgcXIuCF5uFhTaWxAUaRErRUpj6DFAy3clT_CCkMjiOCLFvYBe0NSff77rF0L7XDw8x4bLubb2TKJ0GtA53cnEyzFaseLrCxTnvEi_2_1BvbuVRw</recordid><startdate>20201020</startdate><enddate>20201020</enddate><creator>Oishi, Kenta</creator><general>Cornell University Library, arXiv.org</general><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>HCIFZ</scope><scope>L6V</scope><scope>M7S</scope><scope>PIMPY</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope></search><sort><creationdate>20201020</creationdate><title>A solution formula and the R-boundedness for the generalized Stokes resolvent problem in an infinite layer with Neumann boundary condition</title><author>Oishi, Kenta</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-proquest_journals_24177021243</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Boundary conditions</topic><topic>Computational fluid dynamics</topic><topic>Dirichlet problem</topic><topic>Exact solutions</topic><topic>Fluid flow</topic><topic>Free boundaries</topic><topic>Free surfaces</topic><topic>Incompressible flow</topic><topic>Surface tension</topic><topic>Well posed problems</topic><toplevel>online_resources</toplevel><creatorcontrib>Oishi, Kenta</creatorcontrib><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Engineering Collection</collection><collection>Engineering Database</collection><collection>Access via ProQuest (Open Access)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Oishi, Kenta</au><format>book</format><genre>document</genre><ristype>GEN</ristype><atitle>A solution formula and the R-boundedness for the generalized Stokes resolvent problem in an infinite layer with Neumann boundary condition</atitle><jtitle>arXiv.org</jtitle><date>2020-10-20</date><risdate>2020</risdate><eissn>2331-8422</eissn><abstract>We consider the generalized Stokes resolvent problem in an infinite layer with Neumann boundary conditions. This problem arises from a free boundary problem describing the motion of incompressible viscous one-phase fluid flow without surface tension in an infinite layer bounded both from above and from below by free surfaces. We derive a new exact solution formula to the generalized Stokes resolvent problem and prove the \(\mathscr{R}\)-boundedness of the solution operator families with resolvent parameter \(\lambda\) varying in a sector \(\Sigma_{\varepsilon,\gamma_0}\) for any \(\gamma_0>0\) and \(0<\varepsilon<\pi/2\), where \(\Sigma_{\varepsilon,\gamma_0} =\{ \lambda\in\mathbb{C}\setminus\{0\} \mid |\arg\lambda|\leq\pi-\varepsilon, \ |\lambda|>\gamma_0 \}\). As applications, we obtain the maximal \(L_p\)-\(L_q\) regularity for the nonstationary Stokes problem and then establish the well-posedness locally in time of the nonlinear free boundary problem mentioned above in \(L_p\)-\(L_q\) setting. We make full use of the solution formula to take \(\gamma_0>0\) arbitrarily, while in general domains we only know the \(\mathscr{R}\)-boundedness for \(\gamma_0\gg1\) from the result by Shibata. As compared with the case of Neumann-Dirichlet boundary condition studied by Saito, analysis is even harder on account of higher singularity of the symbols in the solution formula.</abstract><cop>Ithaca</cop><pub>Cornell University Library, arXiv.org</pub><oa>free_for_read</oa></addata></record> |
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| subjects | Boundary conditions Computational fluid dynamics Dirichlet problem Exact solutions Fluid flow Free boundaries Free surfaces Incompressible flow Surface tension Well posed problems |
| title | A solution formula and the R-boundedness for the generalized Stokes resolvent problem in an infinite layer with Neumann boundary condition |
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