Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-23T09:03:50.708Z Has data issue: false hasContentIssue false
  • English
  • Français

Zero dissipation limit and stability of boundary layers for the heat conductive Boussinesq equations in a bounded domain

Published online by Cambridge University Press:  03 June 2015

Jing Wang  and
Feng Xie
Jing Wang
Affiliation:
Department of Mathematics, Shanghai Normal University, Shanghai 200234, People’s Republic of China, ([email protected])
Feng Xie
Affiliation:
Department of Mathematics and Ministry of Education–Lab of Scientific Computation, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China, ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study the zero dissipation limit of the initial boundary-value problem of the multi-dimensional Boussinesq equations with viscosity and heat conductivity. Such equations are used as models for the motion of multi-dimensional incompressible fluids in atmospheric and oceanographic turbulence. In particular, they describe the thermal convection of an incompressible flow, and constitute the relations between the velocity field, the pressure and the local temperature. Under the Navier slip boundary condition in the velocity field and the thermal isolation boundary condition for the temperature, we prove the existence of weak amplitude characteristic boundary layers. Then, by a standard energy method, we prove the L2 convergence of the solutions when both the viscosity and the heat conductivity coefficients tend to 0.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 3 , June 2015 , pp. 611 - 637
Copyright
Copyright © Royal Society of Edinburgh 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1 Berselli, L. C.. Some results on the Navier–Stokes equations with Navier boundary conditions. Riv. Mat. Univ. Parma 1 (2010), 1–75.Google Scholar
2 Cannon, J. R. and DiBenedetto, E.. The initial value problem for the Boussinesq equations with data in Lp . In Approximation Methods for Navier–Stokes Problems: Proc. Symp. Int. Union of Theoretical and Applied Mechanics, University of Paderborn, Paderborn, Germany, September 9–15, 1979. Lecture Notes in Mathematics, vol. 771, pp. 129–144 (Springer, 1980).Google Scholar
3 Chae, D.. Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv. Math. 203 (2006), 497–513.Google Scholar
4 Chae, D. and Imanuvilov, O. Y.. Generic solvability of the axisymmetric 3D Euler equations and the 2D Boussinesq equations. J. Diff. Eqns 156 (1999), 1–17.Google Scholar
5 Evans, L. C.. Partial differential equations. Graduate Studies in Mathematics, vol. 19 (Providence, RI: American Mathematical Society, 1998).Google Scholar
6 Fan, J. S., Nakamura, G. and Wang, H. B.. Blow-up criteria of smooth solutions to the 3D Boussinesq system with zero viscosity in a bounded domain. Nonlin. Analysis 75 (2012), 3436–3442.Google Scholar
7 Holmes, M. H.. Introduction to perturbation methods. Texts in Applied Mathematics, vol. 20 (Springer, 1995).Google Scholar
8 Iftimie, D. and Sueur, F.. Viscous boundary layers for the Navier–Stokes equations with the Navier slip conditions. Arch. Ration. Mech. Analysis 199 (2011), 145–175.Google Scholar
9 Ishimura, N. and Morimoto, H.. Remarks on the blow-up criterion for the 3-D Boussinesq equations. Math. Models Meth. Appl. Sci. 9 (1999), 1323–1332.Google Scholar
10 Jiang, S., Zhang, J. W. and Zhao, J. N.. Boundary-layer effects for the 2-D Boussinesq equations with vanishing diffusivity limit in the half plane. J. Diff. Eqns 250 (2011), 3907–3936.Google Scholar
11 Li, T. T. and Yu, W. C.. Boundary value problems for quasilinear hyperbolic systems. Duke University Mathematics Series, vol. V (Durham, NC: Duke University Mathematics Department, 1985).Google Scholar
12 Lorca, S. A. and Boldrini, J. L.. The initial value problem for a generalized Boussinesq model. Nonlin. Analysis 36 (1999), 457–480.Google Scholar
13 Majda, A.. Introduction to PDEs and waves for the atmosphere and ocean. Courant Lecture Notes in Mathematics, vol. 9 (New York: American Mathematical Society/Courant Institute of Mathematical Sciences, 2003).Google Scholar
14 Moffatt, H. K.. Some remarks on topological fluid mechanics. In An introduction to the geometry and topology of fluid flows (ed. Ricca, R. L.), pp. 3–10 (Dordrecht: Kluwer Academic, 2001).Google Scholar
15 Oleinik, O. A. and Samokhin, V. N.. Mathematical models in boundary layer theory. Applied Mathematics and Mathematical Computation, vol. 15 (Boca Raton, FL: Chapman and Hall/CRC, 1999).Google Scholar
16 Pedlosky, J.. Geophysical fluid dynamics (Springer, 1987).Google Scholar
17 Qin, Y. M., Yang, X. G., Wang, Y. Z. and Liu, X.. Blow-up criteria of smooth solutions to the 3D Boussinesq equations. Math. Meth. Appl. Sci. 35 (2012), 278–285.Google Scholar
18 Rousset, F.. Stability of small amplitude boundary layers for mixed hyperbolic–parabolic systems. Trans. Am. Math. Soc. 355 (2003), 2991–3008.Google Scholar
19 Rousset, F.. Characteristic boundary layers in real vanishing viscosity limits. J. Diff. Eqns 210 (2005), 25–64.Google Scholar
20 Sammartino, M. and Caflisch, R. E.. Zero viscosity limit for analytic solutions of the Navier–Stokes equation on a half-space. I. Existence for Euler and Prandtl equations. Commun. Math. Phys. 192 (1998), 433–461.Google Scholar
21 Sammartino, M. and Caflisch, R. E.. Zero viscosity limit for analytic solutions of the Navier–Stokes equation on a half-space. II. Construction of the Navier–Stokes solution. Commun. Math. Phys. 192 (1998), 463–491.Google Scholar
22 Schlichting, H.. Boundary layer theory (McGraw-Hill, 1979).Google Scholar
23 Serre, D. and Zumbrun, K.. Boundary layer stability in real vanishing viscosity limit. Commun. Math. Phys. 221 (2001), 267–292.Google Scholar
24 Temam, R. and Wang, X. M.. Asymptotic analysis of the linearized Navier–Stokes equations in a general 2D domain. Asymp. Analysis 14 (1997), 293–321.Google Scholar
25 Wang, J.. Boundary layers for compressible Navier–Stokes equations with outflow boundary condition. J. Diff. Eqns 248 (2010), 1143–1174.Google Scholar
26 Wang, J.. Zero dissipation limit of the 1D linearized Navier–Stokes equations for a compressible fluid. J. Math. Analysis Applic. 374 (2011), 693–721.Google Scholar
27 Wang, J. and Tong, L.. Stability of boundary layers for the inflow compressible Navier–Stokes equations. Discrete Contin. Dynam. Syst. B17 (2012), 2595–2613.Google Scholar
28 Wang, J. and Xie, F.. Characteristic boundary layers for parabolic perturbations of quasi-linear hyperbolic problems. Nonlin. Analysis 73 (2010), 2504–2523.Google Scholar
29 Xin, Z. P. and Yanagisawa, T.. Zero-viscosity limit of the linearized Navier–Stokes equations for a compressible viscous fluid in the half-plane. Commun. Pure Appl. Math. 52 (1999), 479–541.Google Scholar
30 Zhao, K.. 2D inviscid heat conductive Boussinesq equations on a bounded domain. Michigan Math. J. 59 (2010), 329–352.Google Scholar