Abstract
We investigate the incompressible Navier–Stokes equations with variable density. The aim is to prove existence and uniqueness results in the case of discontinuous initial density. In dimension n = 2,3, assuming only that the initial density is bounded and bounded away from zero, and that the initial velocity is smooth enough, we get the local-in-time existence of unique solutions. Uniqueness holds in any dimension and for a wider class of velocity fields. In particular, all those results are true for piecewise constant densities with arbitrarily large jumps. Global results are established in dimension two if the density is close enough to a positive constant, and in n dimensions if, in addition, the initial velocity is small. The Lagrangian formulation for describing the flow plays a key role in the analysis that is proposed in the present paper.
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Rferences
Abels H.: The initial-value problem for the Navier–Stokes equations with a free surface in L q-Sobolev spaces. Adv. Differ. Equ. 10(1), 45–64 (2005)
Ammar Khodja F., Santos M.M.: 2D density-dependent Leray problem with a discontinuous density. Methods Appl. Anal. 13(4), 321–335 (2006)
Antontsev S., Kazhikhov A., Monakhov V.: Boundary value problems in mechanics of nonhomogeneous fluids. Studies in Mathematics and its Applications Vol. 22. North-Holland, Amsterdam, 1990
Archer A.J.: Dynamical density functional theory for molecular and colloidal fluids: A microscopic approach to fluid mechanics. J. Chem. Phys. 130, 014509 (2009). doi:10.1063/1.3054633
Besov O.V. I’lin V.P., Nikolskij S.M.: Integral Function Representation and Imbedding Theorem. Nauka, Moscow, 1975
Cho Y., Kim H.: Unique solvability for the density-dependent Navier–Stokes equations. Nonlinear Anal. 59(4), 465–489 (2004)
Danchin R.: Density-dependent incompressible fluids in bounded domains. J. Math. Fluid Mech. 8, 333–381 (2006)
Danchin R.: Local and global well-posedness results for flows of inhomogeneous viscous fluids. Adv. Differ. Equ. 9(3–4), 353–386 (2004)
Danchin R.: Density-dependent incompressible viscous fluids in critical spaces. Proc. R. Soc. Edinb. Sect. A 133(6), 1311–1334 (2003)
Danchin R., Mucha P.B.: The divergence equation in rough spaces. J. Math. Anal. Appl. 386, 10–31 (2012)
Danchin R., Mucha P.B.: A Lagrangian approach for the incompressible Navier–Stokes equations with variable density. Commun. Pure Appl. Math. 65, 1458–1480 (2012)
Danchin R., Mucha P.B.: Divergence. Discrete Contin. Dyn. Syst. S (in press)
Desjardins B.: Regularity results for two-dimensional flows of multiphase viscous fluids. Arch. Ration. Mech. Anal. 137, 135–158 (1997)
Galdi G.: An introduction to the mathematical theory of the Navier–Stokes equations. Vol. I. Linearized steady problems. Springer Tracts in Natural Philosophy, Vol. 38. Springer, New York, 1994
Germain P.: Strong solutions and weak-strong uniqueness for the nonhomogeneous Navier–Stokes equation. J. Anal. Math. 105, 169–196 (2008)
Giga Y., Sohr H.: Abstract L p estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains. J. Funct. Anal. 102, 72–94 (1991)
Hoff D.: Uniqueness of weak solutions of the Navier–Stokes equations of multidimensional compressible flow. SIAM J. Math. Anal. 37(6), 1742–1760 (2006)
Huang J., Paicu M., Zhang P.: Global solutions to 2D inhomogeneous Navier–Stokes system with general velocity (preprint)
Ku D.N.: Blood flow in arteries. Annu. Rev. Fluid Mech. 29, 399–434 (1997)
Lions P.-L.: Mathematical Topics in Fluid Dynamics, Vol. 1. Incompressible Models. Oxford University Press, Oxford (1996)
Ladyzhenskaya O., Solonnikov V.: The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids. J. Sov. Math. 9, 697–749 (1978)
Liu Y., Zhang L., Wang X., Liu W.K.: Coupling of Navier–Stokes equations with protein molecular dynamics and its application to hemodynamics. Int. J. Numer. Methods Fluids 46(12), 1237–1252, (2004) (Special Issue: International Conference on Finite Element Methods in Fluids)
Maremonti P., Solonnikov V.A.: On nonstationary Stokes problem in exterior domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 24(3), 395–449 (1997)
Mucha P.B.: On an estimate for the linearized compressible Navier–Stokes equations in the L p-framework. Colloq. Math. 87(2), 159–169 (2001)
Mucha P.B.: Stability of nontrivial solutions of the Navier–Stokes system on the three dimensional torus, J. Differ. Equ. 172(2), 359–375 (2001)
Mucha P.B., Zaja̧czkowski W.M.: On local existence of solutions of free boundary problem for incompressible viscous self-gravitating fluid motion. Applicationes Mathematicae 27(3), 319–333 (2000)
Mucha P.B., Zaja̧czkowski W.M.: On a L p-estimate for the linearized compressible Navier–Stokes equations with the Dirichlet boundary conditions. J. Differ. Equ. 186(2), 377–393 (2002)
Santos M.M.: Stationary solution of the Navier–Stokes equations in a 2D bounded domain for incompressible flow with discontinuous density. Z. Angew. Math. Phys. 53(4), 661–675 (2002)
Shibata Y., Shimizu S.: On a resolvent estimate of the Stokes system in a half space arising from a free boundary problem for the Stokes equations. Math. Nachr. 282, 482–499 (2009)
Simon J.: Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J. Math. Anal. 21(5), 1093–1117 (1990)
Solonnikov V.A.: On the nonstationary motion of isolated value of viscous incompressible fluid. Izv. AN SSSR 51(5), 1065–1087 (1987)
Tice I., Wang Y.: The viscous surface-internal wave problem: nonlinear Rayleigh–Taylor instability. (2011, arXiv:1109.5657v1)
Triebel H.: Interpolation theory, function spaces, differential operators. North-Holland Mathematical Library Vol. 18. North-Holland, Amsterdam, 1978
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Danchin, R., Mucha, P.B. Incompressible Flows with Piecewise Constant Density. Arch Rational Mech Anal 207, 991–1023 (2013). https://doi.org/10.1007/s00205-012-0586-4
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DOI: https://doi.org/10.1007/s00205-012-0586-4