The vanishing diffusivity limit for the 2-D Boussinesq equations with boundary effect

Abstract In this paper, we tackle the issue of the vanishing diffusivity limit of the 2-D incompressible Boussinesq equations in the half plane. Our main purpose is to study the boundary layer effect and the convergence rates as the thermal diffusion parameter ϵ goes to zero. Under the homogeneous Dirichlet boundary condition of velocity and the nonhomogeneous Dirichlet boundary condition in the x -direction for temperature, we show that the boundary layer thickness is of the order O ( ϵ β ) with ( 0 β 2 3 ) . In contrast with Jiang et al. (2011), the BL-thickness we got is thinner than that in Jiang et al. (2011). Moreover, we prove that as diffusivity tends to zero, the solutions of the fully viscous equations converge strongly to those of zero diffusion equations. We also obtain the convergence rates of the vanishing diffusivity limit.

[1]  Boundary layer for nonlinear evolution equations with damping and diffusion , 2011 .

[2]  P. Lions.  Mathematical topics in fluid mechanics , 1996 .

[3]  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 .

[4]  D. Serre,et al.  Boundary Layer Stability¶in Real Vanishing Viscosity Limit , 2001 .

[5]  Jianwen Zhang,et al.  The zero limits of angular and micro-rotational viscosities for the two-dimensional micropolar fluid equations with boundary effect , 2014 .

[6]  Dongho Chae,et al.  Global regularity for the 2D Boussinesq equations with partial viscosity terms , 2006 .

[7]  Hongyun Peng,et al.  A note on boundary layer of a nonlinear evolution system with damping and diffusions , 2015 .

[8]  O. Ladyzhenskaya,et al.  On Coercive Estimates for Solutions of Linear Systems of Hydrodynamic Type , 2004 .

[9]  Zhifei Zhang,et al.  Global regularity for the initial–boundary value problem of the 2-D Boussinesq system with variable viscosity and thermal diffusivity , 2013 .

[10]  Ting Zhang,et al.  Boundary layers for compressible Navier–Stokes equations with density-dependent viscosity and cylindrical symmetry , 2011 .

[11]  Paul C. Fife,et al.  Second-Order Equations With Nonnegative Characteristic Form , 1973 .

[12]  Hermano Frid,et al.  Boundary Layers for the Navier–Stokes Equations¶of Compressible Fluids , 1999 .

[13]  F. Motallebi,et al.  Influence of Boundary-Layer Thickness on Base Pressure and Vortex Shedding Frequency , 2001 .

[14]  Kun Zhao,et al.  2D Inviscid heat conductive boussinesq equations on a bounded domain , 2010 .

[15]  Song Jiang,et al.  On the non-resistive limit and the magnetic boundary-layer for one-dimensional compressible magnetohydrodynamics , 2015, 1505.03596.

[16]  Thomas Y. Hou,et al.  GLOBAL WELL-POSEDNESS OF THE VISCOUS BOUSSINESQ EQUATIONS , 2004 .

[17]  Song Jiang,et al.  Boundary-layer effects for the 2-D Boussinesq equations with vanishing diffusivity limit in the half plane , 2011 .

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

[19]  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 .

[20]  Zhifei Zhang,et al.  Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity , 2011 .

[21]  Zhouping Xin,et al.  On the global existence of solutions to the Prandtl's system , 2004 .

[22]  Ming-Jun Lai,et al.  Initial Boundary Value Problem for Two-Dimensional Viscous Boussinesq Equations , 2011 .

[23]  J. Pedlosky.  Geophysical Fluid Dynamics , 1979 .

[24]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[25]  José Luiz Boldrini,et al.  The initial value problem for a generalized Boussinesq model , 1999 .