zbMATH — the first resource for mathematics

Viscosity splitting method for three dimensional Navier-Stokes equations. (English) Zbl 0673.35085
Summary: Three dimensional initial boundary value problem of the Navier-Stokes equation is considered. The equation is split in an Euler equation and a non-stationary Stokes equation within each time step. Unlike the conventional approach, we apply a non-homogeneous Stokes equation instead of homogeneous one. Under the hypothesis that the original problem possesses a smooth solution, the estimate of the \(H^{s+1}\) norm, \(0\leq s<3/2\), of the approximate solution and the order of the \(L^ 2\) norm of the errors is obtained.

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35B35 Stability in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
Full Text: DOI
[1] Ying, L.-a., The viscosity splitting method in bounded domains, Scientia Sinica (to appear). · Zbl 0681.76034
[2] Ying, L.-a., The viscosity splitting method for the Navier-Stokes equations in bounded domains (to appear).
[3] Ying, L.-a., On the viscosity splitting method for initial boundary value problems of the Navier-Stokes equations, Chinese Annals of Math. (to appear). · Zbl 0681.76033
[4] Chorin, A.J., Numerical study of slightly viscous flow,J. Fluid Mech.,57(1973), 785–796. · doi:10.1017/S0022112073002016
[5] Chorin, A.J., Hughes, T.J.R., McCracken, M.F., Marsden, J.E., Product formulas and numerical algorithms,Comm. Pure Appl. Math.,31(1978), 205–256. · Zbl 0358.65082 · doi:10.1002/cpa.3160310205
[6] Beale, J.T., Majda, A., Rates of convergence for viscous splitting of the Navier-Stokes equations,Math. Comp. 37(1981), 243–259. · Zbl 0518.76027 · doi:10.1090/S0025-5718-1981-0628693-0
[7] Alessandrini, G., Douglis, A., Fabes, E., An approximate layering method for the Navier-Stokes equations in bounded cylinders,Annali di Matematica,135(1983), 329–347. · Zbl 0552.76030 · doi:10.1007/BF01781075
[8] Ladyzhenskaya, O.A., The Mathematical Theory of Viscous Incompressible Flow, New York, Gordon and Breach, 1969. · Zbl 0184.52603
[9] Temam, R., Navier-Stokes Equations, Theory and Numerical Analysis, 3rd ed., North Holland, 1984. · Zbl 0568.35002
[10] Lions, J.L., Magenes, E., Non-Homogeneous Boundary Value Problems and Applications, Vol.1, Springer-Verlag, 1972. · Zbl 0223.35039
[11] Girault, V., Raviart, P.A., Finite Element Approximation of Navier-Stokes Equations, Lecture Notes in Mathematics, 749, Springer-Verlag, 1979. · Zbl 0413.65081
[12] Adames, R.A., Sobolev Spaces, New York, Academic Press, 1975.
[13] Fujita, H., Morimoto, H., On fractional powers of the Stokes operator,Proc. Japan Acad.,46 (1970), 1141–1143. · Zbl 0235.35067 · doi:10.3792/pja/1195526510
[14] Temam, R., On the Euler equations of incompressible perfect fluids,J. Functional Analysis,20(1975), 32–43. · Zbl 0309.35061 · doi:10.1016/0022-1236(75)90052-X
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.