We formulate some conditions when non-uniqueness of approximate solutions of the stationary Navier-Stokes equations occurs. Introduction. In this paper we will be concerned with the stationary Navier-Stokes equations for incompressible liquid in a bounded domain in Rn (n ∈ {2, 3}). Since a long time, it has been known that there exists a weak solution of this problem. For example, general exist...

We consider the initial value problem for multi-dimensional bipolar compressible Navier– Stokes–Poisson equations, and show the global existence and uniqueness of the strong solution in hybrid Besov spaces with the initial data close to an equilibrium state.

In this paper we prove that any weak solution v to the incom-pressible Navier-Stokes/Euler equations in N 2 (s − 2). Similar result also holds for the viscous/inviscid MHD equations in R N with N ≥ 3.

We consider the Cauchy problem for the incompressible Navier-Stokes equations in R3, and provide a sufficient condition to ensure the smoothness of the solution. It involves only two entries of the velocity Hessian.

We characterize the solution of Navier-Stokes equation as a stochastic geodesic on the diffeomorphisms group, thus generalizing Arnold’s description of the Euler flow. Mathematics Subject Classification (2000). Primary 37L55; Secondary 35Q30, 58E30, 58J65, 60J60, 76D05.

In this paper, we consider the three-dimensional Navier-Stokes equations, and show that if the Ḃ−1 ∞,∞norm of the velocity field is sufficiently small, then the solution is in fact classical.

We consider the Navier–Stokes equations for compressible isothermal flow in the steady two dimensional case and show the existence of a weak solution in the case of periodic and of mixed boundary conditions.