Stationary solutions of the Navier--Stokes equations in periodic tubes

L. V. Kapitanskii

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 14, Tome 115 (1982), pp. 104-113

Résumé

Let $\Omega$ be a tubular domain in $R^n$, $n=2,3$, with a Lipschitz boundary $\partial\Omega$, invariant with respect to a translation by the vector $\vec l\in R^n$. It is proven that, for any prescribed real number $\rho_0$, there exists at least one solutin $[\vec v, \rho]$ of the nonhomologeneous boundary-value problem for a stationary Navier–Stokes system with a pereodic $\vec v$ and pressure $\rho$, having the drop $\rho_0$ over the period. (The exterior forces and the boundary values of the velocity field are assumed to be pereodic.) In addition, one proves the existence of a "critical" nonnegative number $\rho^*$, depending only on the geometry of the domain $\Omega$, the viscosity coefficient, the exterior forces and the boundary values of $\vec v$, such that for $|\rho_0|>\rho^*$ “the fluid flows along the direction of the decrease of the preassure.”