We prove existence of weak solutions to the compressible Navier-Stokes equations in barotropic regime (adiabatic coefficient γ > 3/2, in three dimensions, γ > 1 in two dimensions) with large velocity prescribed at the boundary and large density prescribed at the inflow boundary of a bounded sufficiently smooth domain, without any restriction neither on the shape of the inflow/outflow boundaries nor on the shape of the domain. The result applies also to pressure laws that are non monotone on a compact portion of interval [0, ∞).

No abstract

We establish that the elliptic equation Du þ KðxÞu p þ mf ðxÞ ¼ 0 in R n has infinitely many positive entire solutions for small m50 under suitable conditions on K; p; and f : # 2002 Elsevier Science (USA)

A maximum modulus estimate for the nonstationary Stokes equations in C 2 domain is found. The singular part and regular part of Poisson kernel are analyzed. The singular part consists of the gradient of single layer potential and the gradient of composite potential defined on only normal component of the boundary data. Furthermore, the normal velocity near the boundary is bounded if the boundary data is bounded. If the normal component of the boundary data is Dinicontinuous and the tangential component of the boundary data is bounded, then the maximum modulus of velocity is bounded in whole domain.

We are concerned with the non-stationary Stokes system with non-homogeneous external force and non-zero initial data in R n + × (0, T ). We obtain new estimates of solutions including pressure in terms of mixed anisotropic Sobolev spaces. As an application, some anisotropic Sobolev estimates are presented for weak solutions of the Navier-Stokes equations in a halfspace in dimension three. 2000 Mathematics Subject Classification. 35K51, 76D07.