Abstract
The large-time asymptotic behavior of classical solutions to the density-dependent incompressible Navier–Stokes equations driven by an external force on bounded domains in 2-D is studied. It is shown that the velocity field and its first-order derivatives converge to zero as time goes to infinity for large initial data and external forces.
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Adams R.A.: Sobolev Spaces. Academic, New York (1975)
Antontsev S.N., Kazhikhov A.V., Monakhov V.N.: Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. North-Holland, Amsterdam (1990)
Danchin R.: Density-dependent incompressible fluids in critical spaces. Proc. Roy. Soc. Edinburgh 133, 1311–1334 (2003)
Danchin, R.: Navier–Stokes equations with variable density. Hyperbolic Problems and Related Topics, International Press, Graduate Series in Analysis. pp. 121–135 (2003)
Danchin, R.: Fluides incompressibles densit variable. Sminaire quations aux Drives Partielles, 2002–2003, Exp. No. XI, Palaiseau. (2002–2003)
Danchin R.: The inviscid limit for density-dependent incompressible fluids. Ann. Fac. Sci. Toulouse Math. 15, 637–688 (2006)
Danchin R.: Local and global well-posedness results for flows of inhomogeneous viscous fluids. Adv. Differ. Equ. 9, 353–386 (2004)
Danchin R.: Density dependent incompressible fluids in bounded domains. J. Math. Fluid Mech. 8, 333–381 (2006)
Desjardins B.: Global existence results for the incompressible density-dependent Navier–Stokes equations in the whole space. Differ. Integr. Equ. 10, 587–598 (1997)
Evans L.C.: Partial Differential Equations. AMS, Rhode Island (1998)
Itoh S., Tani A.: Solvability of nonstationary problems for nonhomogeneous incompressible fluids and the convergence with vanishing viscosity. Tokyo J. Math. 22, 17–42 (1999)
Ladyzhenskaya O.A., Solonnikov V.A.: The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids. J. Soviet Math. 9, 697–749 (1978)
Leray J.: Sur le mouvement dun liquide visqueux remplissant lespace. Acta Mathematica 63, 193–248 (1934)
Lions, P.L.: Mathematical Topics in Fluid Mechanics, vol. I–II. Oxford University Press, New York (1996–1998)
Ladyzhenskaya O.A., Solonnikov V.A., Uraltseva N.N.: Linear and Quasi-linear Equations of Parabolic Type. AMS, Rhode Island (1968)
Temam R.: Navier–Stokes Equations. North Holland, Amsterdam (1977)
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Communicated by S. Friedlander
The research of K. Zhao was partially supported by NSF through grant DMS 0807406 and under agreement no. 0635561.
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Zhao, K. Large Time Behavior of Density-Dependent Incompressible Navier–Stokes Equations on Bounded Domains. J. Math. Fluid Mech. 14, 471–483 (2012). https://doi.org/10.1007/s00021-011-0076-8
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DOI: https://doi.org/10.1007/s00021-011-0076-8