Strong Solutions of the Navier–Stokes Equations for Nonhomogeneous Incompressible Fluids

Abstract We study strong solutions of the Navier–Stokes equations for nonhomogeneous incompressible fluids in Ω ⊂ R 3. Deriving higher a priori estimates independent of the lower bounds of the density, we prove the existence and uniqueness of local strong solutions to the initial value problem (for Ω =R 3) or the initial boundary value problem (for Ω ⊂ ⊂ R 3) even though the initial density vanishes in an open subset of Ω, i.e., an initial vacuum exists. As an immediate consequence of the a priori estimates, we obtain a continuation theorem for the local strong solutions.

[1]  Jean Leray,et al.  Sur le mouvement d'un liquide visqueux emplissant l'espace , 1934 .

[2]  川口 光年,et al.  O. A. Ladyzhenskaya: The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Sci. Pub. New York-London, 1963, 184頁, 15×23cm, 3,400円. , 1964 .

[3]  Hyunseok Kim,et al.  Unique solvability for the density-dependent Navier–Stokes equations , 2004 .

[4]  P. Lions.  Mathematical topics in fluid mechanics , 1996 .

[5]  P. Lions,et al.  Ordinary differential equations, transport theory and Sobolev spaces , 1989 .

[6]  R. Temam.  Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

[7]  S. N. Antont︠s︡ev,et al.  Boundary Value Problems in Mechanics of Nonhomogeneous Fluids , 1990 .

[8]  Mariarosaria Padula,et al.  On the existence and uniqueness of non-homogeneous motions in exterior domains , 1990 .

[9]  V. A. Solonnikov,et al.  Unique solvability of an initial- and boundary-value problem for viscous incompressible nonhomogeneous fluids , 1978 .

[10]  J. U. Kim,et al.  Weak solutions of an initial boundary value problem for an incompressible viscous fluid with nonnegative density , 1987 .

[11]  Hi Jun Choe,et al.  Strong solutions of the Navier-Stokes equations for isentropic compressible fluids , 2003 .

[12]  Jacques Simon,et al.  Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure , 1990 .

[13]  G. Galdi.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations : Volume I: Linearised Steady Problems , 1994 .

[14]  Rodolfo Salvi.  The equations of viscous incompressible non-homogeneous fluids: on the existence and regularity , 1991, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.