On the Equation div u = g and Bogovskii’s Operator in Sobolev Spaces of Negative Order

Consider the divergence problem with homogeneous Dirichlet data on a Lipschitz domain. Two approaches for its solutions in the scale of Sobolev spaces are presented. The first one is based on Calderon-Zygmund theory, whereas the second one relies on the Stokes equation with inhomogeneous data.

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

[2]  M. E. Bogovskii.  Solution of Some Vector Analysis Problems connected with Operators div and grad , 1980 .

[3]  M. E. Bogovskii.  Solution of the first boundary value problem for the equation of continuity of an incompressible medium , 1979 .

[4]  Hantaek Bae,et al.  On the Navier-Stokes equations , 2009 .

[5]  Jacques-Louis Lions,et al.  Nonlinear partial differential equations and their applications , 1998 .

[6]  Reinhard Farwig,et al.  Generalized resolvent estimates for the Stokes system in bounded and unbounded domains , 1994 .

[7]  H. Kozono,et al.  New a priori estimates for the Stokes equations in exterior domains , 1991 .

[8]  H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[9]  Timothy S. Murphy,et al.  Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , 1993 .

[10]  R. A. Silverman,et al.  The Mathematical Theory of Viscous Incompressible Flow , 1972 .

[11]  G. Galdi,et al.  Existence, uniqueness and Lq-estimates for the stokes problem in an exterior domain , 1990 .

[12]  Lamberto Cattabriga,et al.  Su un problema al contorno relativo al sistema di equazioni di Stokes , 1961 .

[13]  W. Wahl.  On necessary and sufficient conditions for the solvability of the equations rot μ=γ and div μ=ε with μ vanishing on the boundary , 1990 .

[14]  Three-dimensional solenoidal vectors , 1983 .

[15]  W. Borchers,et al.  On the equations rot v=g and div u=f with zero boundary conditions , 1990 .

[16]  A. Zygmund.  On Singular Integrals , 1956 .

[17]  P. Clément,et al.  Abstract parabolic quasilinear equations and applications to a groundwater flow problem , 1993 .

[18]  Matthias Geissert,et al.  L p -theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle , 2006 .