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Global estimates and blow-up criteria for the generalized Hunter-Saxton system
Boundary layer separation of 2-D incompressible Dirichlet flows
1. | Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China |
2. | College of Mathematics and Software Science, Sichuan Normal University, Chengdu,Sichuan 610066, China, China |
References:
[1] |
P. Blanchonette and M. A. Page, Boundary-Layer separation in the two-layer flow past a cylinder in a rotating frame, Theoret. Comput. Fluid Dynamics, 11 (1998), 95-108. |
[2] |
Chorin, J. Alexandre and J. E. Marsden, A mathematical Introduction to Fluid Mechanics, Third edition. Texts in Applied Mathematics, 4. Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0883-9. |
[3] |
W. E and B. Engquist, Blow up of solutions of the unsteady Prandtl's equation, Commun. Pure Appl. Math, 50 (1997), 1287-1293.
doi: 10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4. |
[4] |
M. Ghil, T. Ma and S. Wang, Structural bifurcation of 2-D incompressible flows, Indiana Univ. Math. J, 50 (2001), 159-180.
doi: 10.1512/iumj.2001.50.2183. |
[5] |
M. Ghil, J. Liu, C. Wang and S. Wang, Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow, Physica D, 197 (2004), 149-173.
doi: 10.1016/j.physd.2004.06.012. |
[6] |
M. Ghil and T. Ma and S. Wang, Structural bifurcation of 2-D incompressible flows with Dirichlet boundary conditions: Applications to boundary-layer separation, SIAM J. Applied Math, 65 (2005), 1576-1596.
doi: 10.1137/S0036139903438818. |
[7] |
H. Luo, Q. Wang and T. Ma, A predicable condition for boundary layer separation of 2-D incompressible fluid flows, Nonlinear Analysis: Real World Applications, 22 (2015), 336-341.
doi: 10.1016/j.nonrwa.2014.09.007. |
[8] |
O. B. Larin and V. A. Levin, Boundary layer separation in a laminar supersonic flow with energy supply source, Technical Physics Letters, 34 (2008), 181-183.
doi: 10.1134/S1063785008030012. |
[9] |
O. B. Larin and V. A. Levin, Effect of energy supply to a gas on laminar boundary layer separation, Journal of Applied Mechanics and Technical Physics, 51 (2010), 11-15.
doi: 10.1007/s10808-010-0003-4. |
[10] |
J. Liu and Z. Xin, Boundary layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation, Arch. Rat. Mech. Anal, 135 (1996), 61-105.
doi: 10.1007/BF02198435. |
[11] |
T. Ma and S. Wang, Rigorous characterization of boundary layer separations, Proc. of the Second MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, (2003), 1008-1010.
doi: 10.1016/B978-008044046-0/50246-3. |
[12] |
T. Ma and S. Wang, Interior structural bifurcation and separation of $2-D$ incompressible flows, J. Math. Phy, 45 (2004), 1762-1776.
doi: 10.1063/1.1689005. |
[13] |
T. Ma and S. Wang, Asymptotic structure for solutions of the Navier-Stokes equations, Discrete and Continuous Dynamical Systems, 11 (2004), 189-204.
doi: 10.3934/dcds.2004.11.189. |
[14] |
T. Ma and S. Wang, Boundary Layer Separation and Structural Bifurcation for 2-D Incompressible Fluid Flows, Discrete and Continuous Dynamical Systems, 10 (2004), 459-472.
doi: 10.3934/dcds.2004.10.459. |
[15] |
T. Ma and S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, Mathematical Surveys and Monographs, 119. American Mathematical Society, Providence, RI, 2005. x+234 pp.
doi: 10.1090/surv/119. |
[16] |
T. Ma and S. Wang, Boundary layer and interior separations in the Taylor-Couette-Poiseuille flow, Journal of Mathematical Physics, 50 (2009), 1-29.
doi: 10.1063/1.3093268. |
[17] |
O. A. Oleinik, On the mathematical theory of boundary layer for unsteady flow of incompressible fluid, J. Appl. Math. Mech, 30 (1966), 951-974.
doi: 10.1016/0021-8928(66)90001-3. |
[18] |
O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Chapman and Hall/CRC, Boca Raton, FL, 1999. |
[19] |
L. Prandtl, On the motion of fluids with very little friction, in Verhandlungen des dritten internationalen Mathematiker-Konggresses, Heidelberg, 1904, Leipeizig, (1905), 484-491. |
[20] |
H. Schlichting, Boundary Layer Theory, eighth ed., Springer, Berlin-Heidelberg, 2000.
doi: 10.1007/978-3-642-85829-1. |
[21] |
F.T. Smith and S.N. Brown, Boundary-Layer Separation, Proceedings of the IUTAM Symposium London, 1986, Springer -Verlag, 1987.
doi: 10.1007/978-3-642-83000-6. |
[22] |
Quan. Wang, Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders, Discrete and Continuous Dynamical Systems-B, 19 (2014), 543-563.
doi: 10.3934/dcdsb.2014.19.543. |
show all references
References:
[1] |
P. Blanchonette and M. A. Page, Boundary-Layer separation in the two-layer flow past a cylinder in a rotating frame, Theoret. Comput. Fluid Dynamics, 11 (1998), 95-108. |
[2] |
Chorin, J. Alexandre and J. E. Marsden, A mathematical Introduction to Fluid Mechanics, Third edition. Texts in Applied Mathematics, 4. Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0883-9. |
[3] |
W. E and B. Engquist, Blow up of solutions of the unsteady Prandtl's equation, Commun. Pure Appl. Math, 50 (1997), 1287-1293.
doi: 10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4. |
[4] |
M. Ghil, T. Ma and S. Wang, Structural bifurcation of 2-D incompressible flows, Indiana Univ. Math. J, 50 (2001), 159-180.
doi: 10.1512/iumj.2001.50.2183. |
[5] |
M. Ghil, J. Liu, C. Wang and S. Wang, Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow, Physica D, 197 (2004), 149-173.
doi: 10.1016/j.physd.2004.06.012. |
[6] |
M. Ghil and T. Ma and S. Wang, Structural bifurcation of 2-D incompressible flows with Dirichlet boundary conditions: Applications to boundary-layer separation, SIAM J. Applied Math, 65 (2005), 1576-1596.
doi: 10.1137/S0036139903438818. |
[7] |
H. Luo, Q. Wang and T. Ma, A predicable condition for boundary layer separation of 2-D incompressible fluid flows, Nonlinear Analysis: Real World Applications, 22 (2015), 336-341.
doi: 10.1016/j.nonrwa.2014.09.007. |
[8] |
O. B. Larin and V. A. Levin, Boundary layer separation in a laminar supersonic flow with energy supply source, Technical Physics Letters, 34 (2008), 181-183.
doi: 10.1134/S1063785008030012. |
[9] |
O. B. Larin and V. A. Levin, Effect of energy supply to a gas on laminar boundary layer separation, Journal of Applied Mechanics and Technical Physics, 51 (2010), 11-15.
doi: 10.1007/s10808-010-0003-4. |
[10] |
J. Liu and Z. Xin, Boundary layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation, Arch. Rat. Mech. Anal, 135 (1996), 61-105.
doi: 10.1007/BF02198435. |
[11] |
T. Ma and S. Wang, Rigorous characterization of boundary layer separations, Proc. of the Second MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, (2003), 1008-1010.
doi: 10.1016/B978-008044046-0/50246-3. |
[12] |
T. Ma and S. Wang, Interior structural bifurcation and separation of $2-D$ incompressible flows, J. Math. Phy, 45 (2004), 1762-1776.
doi: 10.1063/1.1689005. |
[13] |
T. Ma and S. Wang, Asymptotic structure for solutions of the Navier-Stokes equations, Discrete and Continuous Dynamical Systems, 11 (2004), 189-204.
doi: 10.3934/dcds.2004.11.189. |
[14] |
T. Ma and S. Wang, Boundary Layer Separation and Structural Bifurcation for 2-D Incompressible Fluid Flows, Discrete and Continuous Dynamical Systems, 10 (2004), 459-472.
doi: 10.3934/dcds.2004.10.459. |
[15] |
T. Ma and S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, Mathematical Surveys and Monographs, 119. American Mathematical Society, Providence, RI, 2005. x+234 pp.
doi: 10.1090/surv/119. |
[16] |
T. Ma and S. Wang, Boundary layer and interior separations in the Taylor-Couette-Poiseuille flow, Journal of Mathematical Physics, 50 (2009), 1-29.
doi: 10.1063/1.3093268. |
[17] |
O. A. Oleinik, On the mathematical theory of boundary layer for unsteady flow of incompressible fluid, J. Appl. Math. Mech, 30 (1966), 951-974.
doi: 10.1016/0021-8928(66)90001-3. |
[18] |
O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Chapman and Hall/CRC, Boca Raton, FL, 1999. |
[19] |
L. Prandtl, On the motion of fluids with very little friction, in Verhandlungen des dritten internationalen Mathematiker-Konggresses, Heidelberg, 1904, Leipeizig, (1905), 484-491. |
[20] |
H. Schlichting, Boundary Layer Theory, eighth ed., Springer, Berlin-Heidelberg, 2000.
doi: 10.1007/978-3-642-85829-1. |
[21] |
F.T. Smith and S.N. Brown, Boundary-Layer Separation, Proceedings of the IUTAM Symposium London, 1986, Springer -Verlag, 1987.
doi: 10.1007/978-3-642-83000-6. |
[22] |
Quan. Wang, Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders, Discrete and Continuous Dynamical Systems-B, 19 (2014), 543-563.
doi: 10.3934/dcdsb.2014.19.543. |
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