Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow
We establish the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. For the Navier-Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the unifo...
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| Format: | Journal article |
| Language: | English |
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2010
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| author | Chen, G Perepelitsa, M |
| author_facet | Chen, G Perepelitsa, M |
| author_sort | Chen, G |
| collection | OXFORD |
| description | We establish the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. For the Navier-Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup-norm of solutions with respect to the physical viscosity coefficient may not be directly controllable. Furthermore, convex entropy-entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy-type estimates with respect to the viscosity coefficient for solutions of the Navier-Stokes equations and establish the existence of measure-valued solutions of the isentropic Euler equations generated by the Navier-Stokes equations. Based on the uniform energy-type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier-Stokes equations for weak entropy-entropy flux pairs, generated by compactly supported C2 test functions, are confined in a compact set in H-1, which leads to the existence of measure-valued solutions that are confined by the Tartar-Murat commutator relation. A careful characterization of the unbounded support of the measure-valued solution confined by the commutator relation yields the reduction of themeasurevalued solution to a Dirac mass, which leads to the convergence of solutions of the Navier-Stokes equations to a finite-energy entropy solution of the isentropic Euler equations with finite-energy initial data, relative to the different end-states at infinity. © 2010 Wiley Periodicals, Inc. |
| first_indexed | 2024-03-07T03:57:48Z |
| format | Journal article |
| id | oxford-uuid:c378f207-367f-4ff5-9de2-adf03161dca9 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T03:57:48Z |
| publishDate | 2010 |
| record_format | dspace |
| spelling | oxford-uuid:c378f207-367f-4ff5-9de2-adf03161dca92022-03-27T06:16:39ZVanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid FlowJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:c378f207-367f-4ff5-9de2-adf03161dca9EnglishSymplectic Elements at Oxford2010Chen, GPerepelitsa, MWe establish the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. For the Navier-Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup-norm of solutions with respect to the physical viscosity coefficient may not be directly controllable. Furthermore, convex entropy-entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy-type estimates with respect to the viscosity coefficient for solutions of the Navier-Stokes equations and establish the existence of measure-valued solutions of the isentropic Euler equations generated by the Navier-Stokes equations. Based on the uniform energy-type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier-Stokes equations for weak entropy-entropy flux pairs, generated by compactly supported C2 test functions, are confined in a compact set in H-1, which leads to the existence of measure-valued solutions that are confined by the Tartar-Murat commutator relation. A careful characterization of the unbounded support of the measure-valued solution confined by the commutator relation yields the reduction of themeasurevalued solution to a Dirac mass, which leads to the convergence of solutions of the Navier-Stokes equations to a finite-energy entropy solution of the isentropic Euler equations with finite-energy initial data, relative to the different end-states at infinity. © 2010 Wiley Periodicals, Inc. |
| spellingShingle | Chen, G Perepelitsa, M Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow |
| title | Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow |
| title_full | Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow |
| title_fullStr | Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow |
| title_full_unstemmed | Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow |
| title_short | Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow |
| title_sort | vanishing viscosity limit of the navier stokes equations to the euler equations for compressible fluid flow |
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