Vanishing viscosity limit for incompressible flow inside a rotating circle

Abstract In this article 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.

[1]  Russel E. Caflisch,et al.  Zero Viscosity Limit for Analytic Solutions of the Navier-Stokes Equation on a Half-Space.¶ II. Construction of the Navier-Stokes Solution , 1998 .

[2]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[3]  Xiaoming Wang,et al.  A Kato type theorem on zero viscosity limit of Navier-Stokes flows , 2001 .

[4]  Tosio Kato,et al.  Remarks on Zero Viscosity Limit for Nonstationary Navier- Stokes Flows with Boundary , 1984 .

[5]  Nader Masmoudi,et al.  The Euler Limit of the Navier‐Stokes Equations, and Rotating Fluids with Boundary , 1998 .

[6]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[7]  Kellen Petersen August.  Real Analysis , 2009 .

[8]  R. Temam.  Navier-Stokes Equations , 1977 .

[9]  H. J. Nussenzveig Lopes,et al.  Convergence of the Vanishing Viscosity Approximation for Superpositions of Confined Eddies , 1999 .

[10]  L. Hedberg,et al.  Function Spaces and Potential Theory , 1995 .

[11]  James P Kelliher,et al.  On the vanishing viscosity limit in a disk , 2006, math-ph/0612027.

[12]  James P. Kelliher,et al.  On Kato's conditions for vanishing viscosity , 2007 .

[13]  Helena J. Nussenzveig Lopes,et al.  On the Inviscid Limit for Two-Dimensional Incompressible Flow with Navier Friction Condition , 2005, SIAM J. Math. Anal..

[14]  M. Reed,et al.  Fourier Analysis, Self-Adjointness , 1975 .

[15]  Gerald B. Folland,et al.  Real Analysis: Modern Techniques and Their Applications , 1984 .

[16]  Jerry L. Bona,et al.  The Zero‐Viscosity Limit of the 2D Navier–Stokes Equations , 2002 .

[17]  Russel E. Caflisch,et al.  Zero Viscosity Limit for Analytic Solutions, of the Navier-Stokes Equation on a Half-Space.¶I. Existence for Euler and Prandtl Equations , 1998 .

[18]  H. Fernholz.  Boundary Layer Theory , 2001 .

[19]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[20]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[21]  D. Iftimie,et al.  Inviscid limits for the Navier–Stokes equations with Navier friction boundary conditions , 2006 .

[22]  E. Grenier,et al.  Ekman layers of rotating fluids, the case of well prepared initial data , 1997 .

[23]  Michael Taylor,et al.  Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows , 2007, 0709.2056.

[24]  R. Temam,et al.  Boundary Layers Associated with Incompressible Navier–Stokes Equations: The Noncharacteristic Boundary Case , 2002 .

[25]  Shin’ya Matsui,et al.  Example of zero viscosity limit for two dimensional nonstationary Navier-Stokes flows with boundary , 1991 .

[26]  Roger Temam,et al.  On the behavior of the solutions of the Navier-Stokes equations at vanishing viscosity , 1997 .

[27]  Andro Mikelić,et al.  On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions , 1998 .