The goal of this note is to present a simple proof of existence and uniqueness of the solution of the Leray problem for high viscosities and homogeneous or non homogeneous boundary conditions. Furthermore we address the issue of uniqueness of the Poiseuille flow in a pipe.
C.J.Amick, Existence of solutions to the nonhomogeneous steady Navier–Stokes equations, Indiana Univ. Math. J.33 (1984), 817–830. doi:10.1512/iumj.1984.33.33043.
M.Chipot, Asymptotic Issues for Some Partial Differential Equations, Imperial College Press, 2016.
4.
M.Chipot and S.Mardare, Asymptotic behaviour of the Stokes problem in cylinders becoming unbounded in one direction, J. Math. Pures Appl.90(2) (2008), 133–159. doi:10.1016/j.matpur.2008.04.002.
5.
R.Dautray and J.-L.Lions, in: Analyse Mathématique et Calcul Numérique Pour les Sciences et les Techniques. Tome 1, Masson, Paris, 1984.
6.
L.C.Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, 1998.
7.
H.Fujita, On stationary solutions to Navier–Stokes equation in symmetric plane domain under general outflow condition, in: Proceedings of International Conference on Navier–Stokes Equations. Theory and Numerical Methods, Pitman Research Notes in Mathematics, Vol. 388, Varenna, Italy, 1997, pp. 16–30.
8.
H.Fujita and H.Morimoto, A remark on the existence of the Navier–Stokes flow with non-vanishing outflow condition, GAKUTO Internat. Ser. Math. Sci. Appl.10 (1997), 53–61.
9.
G.P.Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Steady-State Problems, 2nd edn, Springer, 2011.
10.
D.Gilbarg and N.S.Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, Berlin, Heidelberg, New-York, 2001, Reprint of the 1998 edition.
11.
D.Kinderlehrer and G.Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classic Appl. Math., Vol. 31, SIAM, Philadelphia, 2000.
12.
M.V.Korobkov, K.Pileckas and R.Russo, On the flux problem in the theory of steady Navier–Stokes equations with nonhomogeneous boundary conditions, Arch. Ration. Mech. Anal.207(1) (2013), 185–213. doi:10.1007/s00205-012-0563-y.
13.
M.V.Korobkov, K.Pileckas and R.Russo, Solution of Leray’s problem for stationary Navier–Stokes equations in plane and axially symmetric spatial domains, Ann. of Math. (2)181(2) (2015), 769–807. doi:10.4007/annals.2015.181.2.7.
14.
H.Kozono and T.Yanagisawa, Leray’s problem on the stationary Navier–Stokes equations with inhomogeneous boundary data, Math. Z.262(1) (2009), 27–39. doi:10.1007/s00209-008-0361-2.
15.
O.A.Ladyzhenskaya, New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them, Proc. Steklov Inst. Math.102 (1967), 95–118.
16.
O.A.Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, 1969.
17.
J.Leray, Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique, J. Math. Pures Appl.12 (1933), 1–82.
18.
J.L.Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.
19.
H.Morimoto, Stationary Navier–Stokes flow in 2-D channels involving the general outflow condition, in: Handbook of Differential Equations: Stationary Partial Differential Equations 4, Ch. 5, Elsevier, 2007, pp. 299–353. doi:10.1016/S1874-5733(07)80008-8.
20.
H.Morimoto, A remark on the existence of 2-D steady Navier–Stokes flow in bounded symmetric domain under general outflow condition, J. Math. Fluid Mech.9(3) (2007), 411–418. doi:10.1007/s00021-005-0206-2.
21.
R.Temam, Navier–Stokes Equations, North Holland, Amsterdam, 1979.
