Operator splitting for two-dimensional incompressible fluid equations

Holden, Helge; Karlsen, Kenneth; Karper, Trygve

doi:10.1090/S0025-5718-2012-02626-3

Operator splitting for two-dimensional incompressible fluid equations
HTML articles powered by AMS MathViewer

by Helge Holden, Kenneth H. Karlsen and Trygve Karper;

Math. Comp. 82 (2013), 719-748

DOI: https://doi.org/10.1090/S0025-5718-2012-02626-3

Published electronically: June 27, 2012

PDF | Request permission

Abstract:

We analyze splitting algorithms for a class of two-dimensional fluid equations, which includes the incompressible Navier–Stokes equations and the surface quasi-geostrophic equation. Our main result is that the Godunov and Strang splitting methods converge with the expected rates provided the initial data are sufficiently regular.

References

Luis A. Caffarelli and Alexis Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2) 171 (2010), no. 3, 1903–1930. MR 2680400, DOI 10.4007/annals.2010.171.1903
Dongho Chae, The quasi-geostrophic equation in the Triebel-Lizorkin spaces, Nonlinearity 16 (2003), no. 2, 479–495. MR 1958612, DOI 10.1088/0951-7715/16/2/307
Peter Constantin, Euler equations, Navier-Stokes equations and turbulence, Mathematical foundation of turbulent viscous flows, Lecture Notes in Math., vol. 1871, Springer, Berlin, 2006, pp. 1–43. MR 2196360, DOI 10.1007/11545989_{1}
Peter Constantin, Gautam Iyer, and Jiahong Wu, Global regularity for a modified critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J. 57 (2008), no. 6, 2681–2692. MR 2482996, DOI 10.1512/iumj.2008.57.3629
Peter Constantin, Andrew J. Majda, and Esteban Tabak, Formation of strong fronts in the $2$-D quasigeostrophic thermal active scalar, Nonlinearity 7 (1994), no. 6, 1495–1533. MR 1304437
Peter Constantin and Jiahong Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal. 30 (1999), no. 5, 937–948. MR 1709781, DOI 10.1137/S0036141098337333
Antonio Córdoba and Diego Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys. 249 (2004), no. 3, 511–528. MR 2084005, DOI 10.1007/s00220-004-1055-1
Hongjie Dong and Dapeng Du, Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space, Discrete Contin. Dyn. Syst. 21 (2008), no. 4, 1095–1101. MR 2399451, DOI 10.3934/dcds.2008.21.1095
Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie, and Nils Henrik Risebro, Splitting methods for partial differential equations with rough solutions, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2010. Analysis and MATLAB programs. MR 2662342, DOI 10.4171/078
Helge Holden, Kenneth H. Karlsen, Nils Henrik Risebro, and Terence Tao, Operator splitting for the KdV equation, Math. Comp. 80 (2011), no. 274, 821–846. MR 2772097, DOI 10.1090/S0025-5718-2010-02402-0
H. Holden, C. Lubich, N. H. Risebro. Operator splitting for partial differential equations with Burgers nonlinearity. Math. Comp., to appear.
A. Kiselev, Regularity and blow up for active scalars, Math. Model. Nat. Phenom. 5 (2010), no. 4, 225–255. MR 2662457, DOI 10.1051/mmnp/20105410
Alexander Kiselev, Nonlocal maximum principles for active scalars, Adv. Math. 227 (2011), no. 5, 1806–1826. MR 2803787, DOI 10.1016/j.aim.2011.03.019
A. Kiselev, F. Nazarov, and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 (2007), no. 3, 445–453. MR 2276260, DOI 10.1007/s00222-006-0020-3
O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Mathematics and its Applications, Vol. 2, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Second English edition, revised and enlarged; Translated from the Russian by Richard A. Silverman and John Chu. MR 254401
N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 350027
Andrew J. Majda and Andrea L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882
C. Miao and L. Xue. Global wellposedness for a modified critical dissipative quasi-geostrophic equation. J. Diff. Eq. 252:792–818, 2012.
J. Pedlosky. Geophysical Fluid Dynamics. Springer, 1987, (2nd edition).
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. MR 290095
Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925, DOI 10.1090/cbms/106