## Vanishing viscosity limit for incompressible flow inside a rotating circle.(English)Zbl 1143.76416

Summary: We consider circularly symmetric incompressible viscous flow in a disk. The boundary condition is no-slip with respect to a prescribed time-dependent rotation of the boundary about the center of the disk. We prove that, if the prescribed angular velocity of the boundary has finite total variation, then the Navier-Stokes solutions converge strongly in $$L^{2}$$ to the corresponding stationary solution of the Euler equations when viscosity vanishes. Our approach is based on a semigroup treatment of the symmetry-reduced scalar equation.

### MSC:

 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76U05 General theory of rotating fluids 35Q30 Navier-Stokes equations
Full Text:

### References:

 [1] K. Asano, Zero viscosity limit of incompressible Navier-Stokes equations I, II, unpublished preprints, 1988 [2] Bona, J.; Wu, Jiahong, The zero-viscosity limit of the 2D navier – stokes equations, Stud. appl. math., 109, 4, 265-278, (2002) · Zbl 1141.35431 [3] Clopeau, T.; Mikelic, A.; Robert, R., On the vanishing viscosity limit for the 2D incompressible navier – stokes equations with friction-type boundary conditions, Nonlinearity, 11, 1625-1636, (1998) · Zbl 0911.76014 [4] Folland, G., () [5] Grenier, E.; Masmoudi, N., Ekman layers of rotating fluids, the case of well prepared initial data, Comm. partial differential equations, 22, 5-6, 953-975, (1997) · Zbl 0880.35093 [6] Henry, D., () [7] Iftimie, D.; Planas, G., Inviscid limits for the navier – stokes equations with Navier friction boundary conditions, Nonlinearity, 19, 899-918, (2006) · Zbl 1169.35365 [8] Kato, T., Remarks on zero viscosity limit for nonstationary navier – stokes flows with boundary, () · Zbl 0559.35067 [9] Kelliher, J., On kato’s conditions for vanishing viscosity, Indiana univ. math. J., 56, 4, 1711-1721, (2007) · Zbl 1125.76014 [10] J. Kelliher, On the vanishing viscosity limit in a disk, preprint, 2007 [11] Lax, P.D., Functional analysis, () · Zbl 1009.47001 [12] M. Lopes Filho, A. Mazzucato, H. Nussenzveig Lopes, M. Taylor, Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows, Bull. Braz. Math. Soc. (in press) · Zbl 1178.35288 [13] Lopes Filho, M.; Nussenzveig Lopes, H.; Planas, G., On the inviscid limit for two-dimensional incompressible flow with Navier friction condition, SIAM J. appl. math., 36, 4, 1130-1141, (2005) · Zbl 1084.35060 [14] Lopes Filho, M.; Nussenzveig Lopes, H.; Zheng, Yuxi, Convergence of the vanishing viscosity approximation for superpositions of confined eddies, Comm. math. phys., 201, 2, 291-304, (1999) · Zbl 0942.76012 [15] Masmoudi, N., The Euler limit of the navier – stokes equations and rotating fluids with boundary, Arch. ration. mech. anal., 142, 4, 375-394, (1998) · Zbl 0915.76017 [16] Matsui, S., Example of zero viscosity limit for two-dimensional nonstationary navier – stokes flows with boundary, Japan J. indust. appl. math., 11, 1, 155-170, (1994) · Zbl 0797.76011 [17] Pazy, A., () [18] Reed, M.; Simon, B., Methods of modern mathematical physics. II, () · Zbl 0517.47006 [19] Sammartino, M.; Caflisch, R., Zero viscosity limit for analytic solutions of the navier – stokes equation on a half-space I, existence for Euler and Prandtl equations, Comm. math. phys., 192, 2, 433-461, (1998) · Zbl 0913.35102 [20] Sammartino, M.; Caflisch, R., Zero viscosity limit for analytic solutions of the navier – stokes equation on a half-space II, construction of the navier – stokes solution, Comm. math. phys., 192, 2, 463-491, (1998) · Zbl 0913.35103 [21] Schlichting, H.; Gersten, K., Boundary layer theory, (2000), Springer Verlag Berlin [22] Temam, R.; Wang, Xiaoming, On the behavior of the solutions of the navier – stokes equations at vanishing viscosity, Annali Della scuola norm. sup. Pisa serie IV XXV, 807-828, (1998), (Vol. dedicated to the memory of E. De Giorgi) · Zbl 1043.35127 [23] Temam, R.; Wang, Xiaoming, Boundary layer associated with the incompressible navier – stokes equations: the non-characteristic boundary case, J. differential equations, 179, 647-686, (2002) · Zbl 0997.35042 [24] Wang, Xiaoming, A Kato type theorem on zero viscosity limit of navier – stokes flows, Indiana univ. math. J., 50, 1, 223-241, (2001) · Zbl 0991.35059
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.