TY - CHAP

T1 - Leray's problem on existence of steady-state solutions for the navier-stokes flow

AU - Korobkov, Mikhail V.

AU - Pileckas, Konstantin

AU - Russo, Remigio

PY - 2018/4/19

Y1 - 2018/4/19

N2 - 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 two- and 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.

AB - 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 two- and 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.

UR - http://www.scopus.com/inward/record.url?scp=85054394976&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-13344-7_5

DO - 10.1007/978-3-319-13344-7_5

M3 - Chapter

AN - SCOPUS:85054394976

SN - 9783319133430

SP - 249

EP - 297

BT - Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

PB - Springer International Publishing AG

ER -