Convergence of the 2D Euler-α to Euler equations in the Dirichlet case: Indifference to boundary layers

Abstract In this article we consider the Euler- α system as a regularization of the incompressible Euler equations in a smooth, two-dimensional, bounded domain. For the limiting Euler system we consider the usual non-penetration boundary condition, while, for the Euler- α regularization, we use velocity vanishing at the boundary. We also assume that the initial velocities for the Euler- α system approximate, in a suitable sense, as the regularization parameter α → 0 , the initial velocity for the limiting Euler system. For small values of α , this situation leads to a boundary layer, which is the main concern of this work. Our main result is that, under appropriate regularity assumptions, and despite the presence of this boundary layer, the solutions of the Euler- α system converge, as α → 0 , to the corresponding solution of the Euler equations, in L 2 in space, uniformly in time. We also present an example involving parallel flows, in order to illustrate the indifference to the boundary layer of the α → 0 limit, which underlies our work.

[1]  Giovanni P. Galdi,et al.  Further existence results for classical solutions of the equations of a second-grade fluid , 1994 .

[2]  E Weinan,et al.  Boundary Layer Theory and the Zero-Viscosity Limit of the Navier-Stokes Equation , 2000 .

[3]  Valentina Busuioc,et al.  On second grade fluids with vanishing viscosity , 1999 .

[4]  Gerd Baumann,et al.  Navier–Stokes Equations on R3 × [0, T] , 2016 .

[5]  A. Ilyin.  ATTRACTORS FOR NAVIER-STOKES EQUATIONS IN DOMAINS WITH FINITE MEASURE , 1996 .

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

[7]  Dragoş Iftimie.  Remarques sur la limite a=0 pour les fluides de grade 2 , 2002 .

[8]  Yasunori Maekawa,et al.  On the Inviscid Limit Problem of the Vorticity Equations for Viscous Incompressible Flows in the Half‐Plane , 2012 .

[9]  R. Rivlin,et al.  Stress-Deformation Relations for Isotropic Materials , 1955 .

[10]  M. L. Filho.  Boundary layers and the vanishing viscosity limit for incompressible 2D flow , 2007, 0712.0875.

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

[12]  R. Temam,et al.  The convergence of the solutions of the Navier-Stokes equations to that of the Euler equations , 1997 .

[13]  J. E. Marsden,et al.  The geometry and analysis of the averaged Euler equations and a new diffeomorphism group , 1999 .

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

[15]  R. Temam.  On the Euler equations of incompressible perfect fluids , 1975 .

[16]  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 .

[17]  Jerrold E. Marsden,et al.  EULER-POINCARE MODELS OF IDEAL FLUIDS WITH NONLINEAR DISPERSION , 1998 .

[18]  Darryl D. Holm,et al.  Camassa-Holm Equations as a Closure Model for Turbulent Channel and Pipe Flow , 1998, chao-dyn/9804026.

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

[20]  S. Shkoller.  Analysis on Groups of Diffeomorphisms of Manifolds with Boundary and the Averaged Motion of a Fluid , 2000 .

[21]  Anna L. Mazzucato,et al.  Vanishing viscosity plane parallel channel flow and related singular perturbation problems , 2008 .

[22]  Darryl D. Holm,et al.  On a Leray–α model of turbulence , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[23]  P. Lions,et al.  Ordinary differential equations, transport theory and Sobolev spaces , 1989 .

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

[25]  The Lagrangian Averaged Euler (LAE-α) Equations with Free-Slip or Mixed Boundary Conditions , 2002 .

[26]  E. Titi,et al.  On the Convergence Rate of the Euler-α, an Inviscid Second-Grade Complex Fluid, Model to the Euler Equations , 2009, 0911.1846.

[27]  Darryl D. Holm,et al.  The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories , 1998, chao-dyn/9801015.

[28]  Igor Kukavica,et al.  On the inviscid limit of the Navier-Stokes equations , 2014, 1403.5748.

[29]  G. Galdi.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations : Volume I: Linearised Steady Problems , 1994 .

[30]  Darryl D. Holm,et al.  The Three Dimensional Viscous Camassa–Holm Equations, and Their Relation to the Navier–Stokes Equations and Turbulence Theory , 2001, nlin/0103039.

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

[32]  E. Titi,et al.  Global regularity and convergence of a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations , 2009, 0902.3356.

[33]  Darryl D. Holm,et al.  The Camassa-Holm equations and turbulence , 1999 .

[34]  Tudor S. Ratiu,et al.  The second grade fluid and averaged Euler equations with Navier-slip boundary conditions , 2003 .

[35]  S. Shkoller,et al.  THE VORTEX BLOB METHOD AS A SECOND-GRADE NON-NEWTONIAN FLUID , 1999, math/9910088.

[36]  A. Mazzucato,et al.  Vanishing Viscosity Limits for a Class of Circular Pipe Flows , 2010 .

[37]  Darryl D. Holm,et al.  The Navier–Stokes-alpha model of fluid turbulence , 2001, nlin/0103037.

[38]  J. E. Dunn,et al.  Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade , 1974 .

[39]  P. Hartman.  Ordinary Differential Equations , 1965 .

[40]  E. Titi,et al.  Global regularity for a Birkhoff–Rott-α approximation of the dynamics of vortex sheets of the 2D Euler equations , 2007, 0709.4626.

[41]  Anna L. Mazzucato,et al.  Vanishing viscosity limit for incompressible flow inside a rotating circle , 2008 .

[42]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[43]  Darryl D. Holm,et al.  A connection between the Camassa–Holm equations and turbulent flows in channels and pipes , 1999, chao-dyn/9903033.

[44]  R. Schwarzenberger.  ORDINARY DIFFERENTIAL EQUATIONS , 1982 .

[45]  F. Toschi,et al.  Acceleration and vortex filaments in turbulence , 2005, nlin/0501041.

[46]  Marcel Oliver,et al.  Global well-posedness for the averaged Euler equations in two dimensions , 2000 .

[47]  Tosio Kato,et al.  Nonlinear evolution equations and the Euler flow , 1984 .

[48]  Edriss S. Titi,et al.  Mathematics and turbulence: where do we stand? , 2013, 1301.0273.

[49]  Doina Cioranescu,et al.  Weak and classical solutions of a family of second grade fluids , 1997 .

[50]  Dragoş Iftimie.  Remarques sur la limite pour les fluides de grade , 2002 .

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

[52]  Darryl D. Holm,et al.  On the Clark–α model of turbulence: global regularity and long-time dynamics , 2004, nlin/0412007.

[53]  E. Titi,et al.  On the Rate of Convergence of the Two-Dimensional α-Models of Turbulence to the Navier–Stokes Equations , 2009, 0902.4247.

[54]  E. Titi,et al.  Global regularity and convergence of a Birkhoff‐Rott‐α approximation of the dynamics of vortex sheets of the two‐dimensional Euler equations , 2010 .

[55]  E. Titi,et al.  Approximation of 2D Euler Equations by the Second-Grade Fluid Equations with Dirichlet Boundary Conditions , 2014, 1412.6587.

[56]  D. Acheson.  Elementary Fluid Dynamics , 1990 .

[57]  D. Whiffen.  Thermodynamics , 1973, Nature.

[58]  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 .

[59]  Giovanni P. Galdi,et al.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems , 2011 .

[60]  A. V. Busuioc,et al.  Incompressible Euler as a limit of complex fluid models with Navier boundary conditions , 2012 .