Remigio Russo

The Liouville Theorem for the Steady-State Navier–Stokes Problem for Axially Symmetric 3D Solutions in Absence of Swirl

We study the Navier–Stokes equations of steady motion of a viscous incompressible fluid in

The flux problem for the Navier-Stokes equations

{\mathbb{R}^{3}}

On the Flux Problem in the Theory of Steady Navier–Stokes Equations with Nonhomogeneous Boundary Conditions

R3. We prove that there are no nontrivial solution of these equations defined in the whole space

Steady Navier–Stokes system with nonhomogeneous boundary conditions in the axially symmetric case

{\mathbb{R}^{3}}

The existence of a solution with finite Dirichlet integral for the steady Navier–Stokes equations in a plane exterior symmetric domain

R3 for axially symmetric case with no swirl (the Liouville theorem). Also we prove the conditional Liouville type theorem for axial symmetric solutions to the Euler system.

The existence theorem for the steady Navier–Stokes problem in exterior axially symmetric 3D domains

This is a survey of results on the Leray problem (1933) for the nonhomogeneous boundary value problem for the steady Navier–Stokes equations in a bounded domain with multiple boundary components. The boundary conditions are assumed only to satisfy the necessary requirement of zero total flux. The authors have proved that the problem is solvable in arbitrary bounded planar or threedimensional axially symmetric domains. The proof uses Bernoulli’s law for weak solutions of the Euler equations and a generalization of the Morse–Sard theorem for functions in Sobolev spaces. Similar existence results (without any restrictions on fluxes) are proved for steady Navier–Stokes system in twoand three-dimensional exterior domains with multiply connected boundary under assumptions of axial symmetry. In particular, it was shown that in domains with two axes of symmetry and for symmetric boundary datum, the two-dimensional exterior problem has a symmetric solution vanishing at infinity.