Braz e Silva, Pablo The resolvent technique for investigating the stability of plane Couette flow: bounds for the pressure terms. (English) Zbl 1254.76081 Nonlinear Anal., Real World Appl. 13, No. 6, 2680-2691 (2012). Summary: We derive bounds for the pressure terms of the operator used to prove nonlinear stability of plane Couette flow through the resolvent technique. MSC: 76E05 Parallel shear flows in hydrodynamic stability 35Q35 PDEs in connection with fluid mechanics 35B35 Stability in context of PDEs Keywords:Couette flow; resolvent estimates; nonlinear stability PDFBibTeX XMLCite \textit{P. Braz e Silva}, Nonlinear Anal., Real World Appl. 13, No. 6, 2680--2691 (2012; Zbl 1254.76081) Full Text: DOI arXiv References: [1] Kreiss, H.-O.; Lorenz, J., (Stability for Time Dependent Differential Equations. Stability for Time Dependent Differential Equations, Acta Numer., vol. 7 (1998), Cambridge University Press: Cambridge University Press Cambridge), 203-285 · Zbl 0909.35017 [2] Kreiss, H.-O.; Lorenz, J., Resolvent estimates and quantification of nonlinear stability, Acta Math. Sin. (Engl. Ser.), 1, 16, 1-20 (2000) · Zbl 0962.35021 [3] Kreiss, G.; Lundbladh, A.; Henningson, D. S., Bounds for threshold amplitudes in subcritical shear flows, J. Fluid Mech., 270, 175-198 (1994) · Zbl 0813.76024 [4] Åsén, P.-O.; Kreiss, G., On a rigorous resolvent estimate for plane Couette flow, J. Math. Fluid Mech., 9, 2, 153-180 (2007) · Zbl 1118.76031 [5] Braz e. Silva, P., Resolvent estimates for plane Couette flow, SIAM J. Appl. Math., 65, 2, 667-683 (2005) · Zbl 1078.76028 [6] Reddy, S. C.; Henningson, D. S., Energy growth in viscous channel flows, J. Fluid Mech., 252, 209-238 (1993) · Zbl 0789.76026 [7] Trefethen, L. N.; Trefethen, A. E.; Reddy, S. C.; Driscoll, T. A., Hydrodynamic stability without eigenvalues, Science, 261, 578-584 (1993) · Zbl 1226.76013 [8] Liefvendahl, M.; Kreiss, G., Bounds for the threshold amplitude for plane Couette flow, J. Nonlinear Math. Phys., 3, 9, 311-324 (2002) · Zbl 1009.35069 [9] Liefvendahl, M.; Kreiss, G., Analytical and numerical investigation of the resolvent for plane Couette flow, SIAM J. Appl. Math., 3, 63, 801-817 (2003) · Zbl 1033.76013 [10] Braz e. Silva, P., Nonlinear stability for 2 dimensional plane Couette flow, Revista Integración, 22, 67-81 (2004) [11] Braz e. Silva, P., Resolvent estimates for 2 dimensional perturbations of plane Couette flow, Electron. J. Diff. Eqns., 92, 2002, 1-15 (2002) · Zbl 1035.76014 [12] Bulícek, M.; Majdoub, M.; Málek, J., Unsteady flows of fluids with pressure dependent viscosity in unbounded domains, Nonlinear Anal. Real World Appl., 11, 5, 3968-3983 (2010) · Zbl 1201.35156 [13] Hron, J.; Málek, J.; Prusa, V.; Rajagopal, K. R., Further remarks on simple flows of fluids with pressure-dependent viscosities, Nonlinear Anal. Real World Appl., 12, 1, 394-402 (2011) · Zbl 1206.35207 [14] Huang, H.; Yang, X.; Krafczyk, M.; Lu, X.-Y., Rotation of spheroidal particles in Couette flows, J. Fluid Mech., 692, 369-394 (2012) · Zbl 1250.76172 [15] Heck, H.; Kim, H.; Kozono, H., Stability of plane Couette flows with respect to small periodic perturbations, Nonlinear Anal., 71, 9, 3739-3758 (2009) · Zbl 1173.35627 [16] Kagei, Y., Asymptotic behavior of solutions of the compressible Navier-Stokes equation around the plane Couette flow, J. Math. Fluid Mech., 13, 1, 1-31 (2011) · Zbl 1270.35348 [17] Kagei, Y., Global existence of solutions to the compressible Navier-Stokes equation around parallel flows, J. Differential Equations, 251, 11, 3248-3295 (2011) · Zbl 1232.35117 [18] Ko, J.; Lucor, D.; Sagaut, P., Effects of base flow uncertainty on Couette flow stability, Comput. & Fluids, 43, 82-89 (2011) · Zbl 1452.76067 [19] Li, C.; Zheng, L.; Zhang, X.; Niu, J., Couette flow of a generalized Oldroyd-B fluid due to constantly accelerated shear stress coupled with the pressure gradient, Advanced Materials Research, 354-355, 179-182 (2012) [20] Li, Y. C.; Lin, Z., A resolution of the Sommerfeld paradox, SIAM J. Math. Anal, 43, 4, 1923-1954 (2011) · Zbl 1383.76140 [21] Liu, Y.; Zheng, L.; Zhang, X., Unsteady MHD Couette flow of a generalized Oldroyd-B fluid with fractional derivative, Comput. Math. Appl., 61, 2, 443-450 (2011) · Zbl 1211.76152 [22] Zheng, L.; Liu, Y.; Zhang, X., Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative, Nonlinear Anal. Real World Appl., 13, 513-523 (2012) · Zbl 1238.76056 [23] Romanov, V. A., Stability of plane-parallel Couette flow, Funct. Anal. Appl., 7, 137-146 (1973) · Zbl 0287.76037 [24] Kreiss, H.-O.; Lorenz, J., (Initial-Boundary Value Problems and the Navier-Stokes Equations. Initial-Boundary Value Problems and the Navier-Stokes Equations, Pure and Applied Mathematics, vol. 136 (1989), Academic Press) · Zbl 0689.35001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.