A Fluid Problem with Navier-Slip Boundary Conditions
We study the equations governing the motion of second grade fluids in a bounded domain of ℝ d , d = 2, 3, with Navier slip boundary conditions with and without viscosity (averaged Euler equations). The main results concern the global existence and uniqueness of H 3 solutions in dimension two and, for dimension three, local existence of H 3 solutions for arbitrary initial data and global existence for small initial data and positive viscosity. The last part discusses Liapunov stability conditions for stationary solutions for the averaged Euler equations similar to the Rayleigh-Arnold stability result for the classical Euler equations. This paper presents only the main ideas of the proofs that can be found in .
KeywordsGlobal Existence Potential Vorticity Fluid Problem Small Initial Data Grade Fluid
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- V.I. Arnold. Conditions for nonlinear stability of the stationary plane curvilinear flows of an ideal fluid. Dokl. Akad. Nauk 162, n° 5, 1965, 773–777.Google Scholar
- V.I. Arnold. An a priori estimate in the theory of hydrodynamic stability. [English translation] Amer. Math. Soc. Transi. 19, 1969, 267–269.Google Scholar
- A.V. Busuioc and T.S. Ratiu The second grade fluid and averaged Euler equations with Navier-slip boundary conditions,submitted.Google Scholar
- D. Cioranescu and E. H. Ouazar. Existence and uniqueness for fluids of second grade. Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. VI (Paris, 1982/1983), pp. 178–197. Boston, MA, Pitman, 1984.Google Scholar
- C. Foias and R. Temam. Remarques sur les équations de NavierStokes stationnaires et les phénomènes successifs de bifurcation. Ann. Sc. Norm. Pisa, V, 1978, pp. 29–63.Google Scholar
- C. Foias, D.D. Holm, E.S.Titi. The three dimensional viscous Camassa-Holm Equations and their relation to the Navier-Stokes equation and turbulence theory. To appear in J. Dyn. Diff. Eq.Google Scholar
- T. Kato. On classical solutions of the two-dimensional non—stationary Euler equations. Arch. Rat. Mech. Anal. 24 n° 3, 1967, 302–324.Google Scholar
- V. A. Solonnikov. General boundary value problems for DouglisNirenberg elliptic systems. Proc. Steklov Inst. Math. 92, 1966, pp. 269–339.Google Scholar