Regularity of the solutions of the steady-state Boussinesq equations with thermocapillarity effects on the surface of the liquid

✍ Scribed by Luc Paquet

• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 169 KB
• Volume: 22
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/(sici)1099-1476(199908)22:12<1043::aid-mma69>3.0.co;2-3

✦ Synopsis

In this paper we show that every variational solution of the steady-state Boussinesq equations (u, p, ) with thermocapillarity e!ect on the surface of the liquid has the following regularity: u3H( ), p3H( ), 3H( ) under appropriate hypotheses on the angles of the &2-D' container (a cross-section of the 3-D container in fact) and of the horizontal surface of the liquid with the inner surface of the container. The di$culty comes from the boundary condition on the surface of the liquid (e.g. water) which modelizes the thermocapillarity e!ect on the surface of the liquid (equation (68.10) of Levich [7]). More precisely we will show that u3P ( ) and that 3P ( ), where P ( ) denotes the usual Kondratiev space. This result will be used in a forthcoming paper to prove convergence results for "nite element methods intended to compute approximations of a non-singular solution [1] of this problem.