Artículo Original
Publicado 2004-09-16
Palabras clave
- Couette flow,
- resolvent estimates,
- nonlinear stability
Cómo citar
e Silva, P. B. (2004). Nonlinear stability for 2 dimensional plane Couette flow. Revista Integración, Temas De matemáticas, 22(1 y 2), 67–81. Recuperado a partir de https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/513
Resumen
In this expository article, we discuss the application of the resolvent technique to prove nonlinear stability of 2 dimensional plane Couette flow. Using this technique, we show how one can derive a threshold amplitude for perturbations that can lead to turbulence in terms of the parameter called Reynolds number. Our objective is to present this argument in details, trying to be accessible to a wide class of readers, and hopefully catching their attention to the beautiful subject of stability questions in fluid mechanics.
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Referencias
[1]Braz e Silva P.“Resolvent Estimates for 2 Dimensional Perturbations of PlaneCouette Flow”.Electron. J. Diff. Eqns.,92(2002), 1–15.
[2]Kreiss H.-O.andLorenz J.“Initial-Boundary Value Problems and the Navier-Stokes Equations”.Pure and Applied Mathematics,136, Academic Press, 1989.
[3]Kreiss H.-O.andLorenz J.“Stability for Time Dependent Differential Equa-tions”.Acta Numer.,7(1998), 203–285, Cambridge University Press, Cambridge.
[4]Kreiss H.-O.andLorenz J.“Resolvent Estimates and Quantification of Non-linear Stability”.Acta Math. Sin.(Engl. Ser.),16(2000), No1, 1–20.
[5]Kreiss G., Lundbladh A.andHenningson D. S.“Bounds for ThresholdAmplitudes in Subcritical Shear Flows”.J. Fluid Mech.,270(1994), 175–198.
[6]Liefvendahl M.andKreiss G.“Bounds for the Threshold Amplitude forPlane Couette Flow”.J. Nonlinear Math. Phys.,9(2002), No3, 311–324.
[7]Liefvendahl M.andKreiss G.“Analytical and Numerical Investigation ofthe Resolvent for Plane Couette Flow”.SIAM J. Appl. Math.,63(2003), No3,801–817.
8]Reddy S. C.andHenningson D. S.“Energy Growth in Viscous ChannelFlows”.J. Fluid Mech.,252(1993), 209–238.
[9]Romanov V. A.“Stability of Plane-Parallel Couette Flow”.Functional Anal.Applics.,7(1973), 137–146.
[10]Trefethen L. N.,Trefethen A. E.,Reddy S. C.andDriscoll T. A.“Hy-drodynamic Stability Without Eigenvalues”.Science,261(1993), 578-584.
[11]Schiff J. L.The Laplace Transform: Theory and Applications, UndergraduateTexts in Mathematics, Springer-Verlag, 1999.
[2]Kreiss H.-O.andLorenz J.“Initial-Boundary Value Problems and the Navier-Stokes Equations”.Pure and Applied Mathematics,136, Academic Press, 1989.
[3]Kreiss H.-O.andLorenz J.“Stability for Time Dependent Differential Equa-tions”.Acta Numer.,7(1998), 203–285, Cambridge University Press, Cambridge.
[4]Kreiss H.-O.andLorenz J.“Resolvent Estimates and Quantification of Non-linear Stability”.Acta Math. Sin.(Engl. Ser.),16(2000), No1, 1–20.
[5]Kreiss G., Lundbladh A.andHenningson D. S.“Bounds for ThresholdAmplitudes in Subcritical Shear Flows”.J. Fluid Mech.,270(1994), 175–198.
[6]Liefvendahl M.andKreiss G.“Bounds for the Threshold Amplitude forPlane Couette Flow”.J. Nonlinear Math. Phys.,9(2002), No3, 311–324.
[7]Liefvendahl M.andKreiss G.“Analytical and Numerical Investigation ofthe Resolvent for Plane Couette Flow”.SIAM J. Appl. Math.,63(2003), No3,801–817.
8]Reddy S. C.andHenningson D. S.“Energy Growth in Viscous ChannelFlows”.J. Fluid Mech.,252(1993), 209–238.
[9]Romanov V. A.“Stability of Plane-Parallel Couette Flow”.Functional Anal.Applics.,7(1973), 137–146.
[10]Trefethen L. N.,Trefethen A. E.,Reddy S. C.andDriscoll T. A.“Hy-drodynamic Stability Without Eigenvalues”.Science,261(1993), 578-584.
[11]Schiff J. L.The Laplace Transform: Theory and Applications, UndergraduateTexts in Mathematics, Springer-Verlag, 1999.