Some Remarks on the Boundary Conditions in the Theory of Navier-Stokes Equations

Amrouche, Chérif; Penel, Patrick; Seloula, Nour

Annales mathématiques Blaise Pascal, Tome 20 (2013) no. 1, pp. 37-73.

Cet article traite de quelques questions théoriques relatives au choix des conditions aux limites, essentielles pour la modélisation et la simulation numérique en mécanique des fluides mathématique. Nous marquons la différence avec le choix standard de conditions de non glissement en soulignant trois ensembles de conditions autorisant glissement, et en insistant particulièrement sur l’interaction entre cadre fonctionnel approprié et statut de ces conditions.

This article addresses some theoretical questions related to the choice of boundary conditions, which are essential for modelling and numerical computing in mathematical fluids mechanics. Unlike the standard choice of the well known non slip boundary conditions, we emphasize three selected sets of slip conditions, and particularly stress on the interaction between the appropriate functional setting and the status of these conditions.

Classification : 35Q35, 35A01, 76D05
Mots clés : Navier-Stokes, Boundary conditions, Weak solutions

[1] Abels, H. Nonstationary Stokes system with variable viscosity in bounded and unbounded domains, Discrete Contin. Dyn. Syst. Ser. S., Volume 3 (2010), pp. 141-157 | DOI | MR | Zbl

[2] Amirat, Y.; Bresch, D.; Lemoine, J.; Simon, J. Effect of rugosity on a flow governed by stationary Navier-Stokes equations, Quart. Appl. Math., Volume 59 (2001), pp. 769-785 | MR | Zbl

[3] Amrouche, C.; Bernardi, C.; Dauge, M.; Girault, V. Vector potentials in three-dimensional nonsmooth domains, Math. Meth. Applied Sc., Volume 21 (1998), pp. 823-864 | DOI | MR | Zbl

[4] Amrouche, C.; Seloula, N. On the Stokes Equations with the Navier-Type boundary conditions, Diff. Eq. Appl., Volume 3-4 (2011), pp. 581-607 | MR

[5] Amrouche, C.; Seloula, N. $L^{\text{p}}$-Theory for Vector Potentials and Sobolev’s Inequalities for Vector Fields. Application to the Stokes Equations with Pressure Boundary Condition, To appear in Math. Mod. and Meth. in App, Volume 23 (2013), pp. 37-92 | DOI | MR

[6] Bardos, C.; Golse, F.; Paillard, L. The incompressible Euler limit of the Boltzmann equation with accomodation boundary conditions, Comm. Math. Sci., Volume 10 (2012) no. 1, pp. 159-190 | DOI | MR

[7] Beavers, G.S.; Joseph, D.D. Boundary conditions at a naturally permeable wall, J. Fluid Mech., Volume 30 (1967) no. 1, pp. 197-207 | DOI

[8] Bègue, C.; Conca, C.; Murat, F.; Pironneau, O. Les équations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression, Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Volume IX (1989), pp. 179-264 | MR | Zbl

[9] Beirao da Veiga, H. On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions, Comm. Pure Appl. Math., Volume 58 (2005) no. 4, pp. 552-577 | DOI | MR | Zbl

[10] Bellout, H.; Neustupa, J.; Penel, P. On the Navier-Stokes equation with boundary conditions based on vorticity, Math. Nachr., Volume 269-270 (2004), pp. 59-72 | DOI | MR | Zbl

[11] Bellout, H.; Neustupa, J.; Penel, P. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary conditions, Discrete and Continuous Dynamical Systems, Volume 27 (2010) no. 4, pp. 1353-1373 | DOI | MR | Zbl

[12] Bernard, J. M. Non-standard Stokes and Navier-Stokes problem: existence and regularity in stationary case, Math. Meth. Appl. Sci., Volume 25 (2002), pp. 627-661 | DOI | MR | Zbl

[13] Bernard, J. M. Time-dependent Stokes and Navier-Stokes problems with boundary conditions involving pressure, existence and regularity, Nonlinear Anal. Real World Appl., Volume 4 (2003) no. 5, pp. 805-839 | DOI | MR | Zbl

[14] Berselli, L. C. Some criteria concerning the vorticity and the problem of global regularity for the 3D Navier-Stokes equations, Ann. Univ. Ferrara Sez. VII Sci. Mat., Volume 55 (2009) no. 2, pp. 209-224 | DOI | MR | Zbl

[15] Berselli., L. C. An elementary approach to the 3D Navier-Stokes equations with Navier boundary conditions: existence and uniqueness of various classes of solutions in the flat boundary case, Discrete Contin. Dyn. Syst. Ser. S, Volume 3 (2010) no. 2, pp. 199-219 | DOI | MR | Zbl

[16] Bothe, D.; Pruss, J. $L^p$-theory for a class of non-Newtonian fluids, SIAM J. Math. Anal., Volume 39 (2007) no. 2, pp. 379-421 | DOI | MR | Zbl

[17] Bucur, D.; Feireisl, E.; Necasova, S. Boundary behavior of viscous fluids: influence of wall roughness and friction-driven boundary conditions, Arch. Ration. Mech. Anal., Volume 197 (2010) no. 1, pp. 117-138 | DOI | MR

[18] Bucur, D.; Feireisl, E.; Necasova, S.; Wolf, J. On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries, J. Differential Equations, Volume 244 (2008) no. 11, pp. 2890-2908 | DOI | MR | Zbl

[19] Bulicek, M.; Malek, J.; Rajagopal, K. R. Navier’s slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity, Indiana Univ. Math. J., Volume 56 (2007) no. 1, pp. 51-85 | DOI | MR | Zbl

[20] Chen, G. Q.; Osborne, D.; Qian, Z. The Navier-Stokes equations with the kinematic and vorticity boundary conditions on non-flat boundaries, Acta Math. Sci. Ser. B Engl. Ed., Volume 29 (2009) no. 4, pp. 919-948 | DOI | MR | Zbl

[21] Ebmeyer, C.; Frehse, J. Steady Navier-Stokes equations with mixed boundary value conditions in three-dimensional Lipschitzian domains, Math. Ann, Volume 319 (2001) no. 2, pp. 349-381 | DOI | MR | Zbl

[22] Hoang, L. T. Incompressible fluids in thin domains with Navier friction boundary conditions (I), J. Math. Fluid Mech., Volume 12 (2010) no. 3, pp. 435-472 | DOI | MR

[23] Hoang, L. T.; Sell, G. R. Navier-Stokes equations with Navier boundary conditions for an oceanic model, J. Dynam. Differential Equations, Volume 22 (2010) no. 3, p. 653-616 | DOI | MR | Zbl

[24] Iftimie, D.; Raugel, G. Some results on the Navier-Stokes equations in thin 3D domains, J. Diff. Eq., Volume 169 (2001), pp. 281-331 | DOI | MR | Zbl

[25] Iftimie, D.; Raugel, G.; Sell, G. Navier-Stokes equations in thin 3D domains with Navier boundary conditions, Indiana Univ. Math. J., Volume 56 (2007) no. 3, pp. 1083-1156 | DOI | MR | Zbl

[26] Iftimie, D.; Sueur, F. Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions, Arch. Ration. Mech. Anal., Volume 199 (2011) no. 1, pp. 145-175 | DOI | MR | Zbl

[27] Jager, W.; Mikelic, A. On the interface boundary condition of Beavers, Joseph, and Saffman, SIAM J. Appl. Math., Volume 60 (2000) no. 4, pp. 1111-1127 | DOI | MR | Zbl

[28] Jager, W.; Mikelic, A. On the roughness-induced effective boundary conditions for an incompressible viscous flow, J. Differential Equations, Volume 170 (2001) no. 1, pp. 96-122 | DOI | MR | Zbl

[29] Kato, T. Remarks on zero viscosity limit for non-stationary Navier-Stokes flows with boundary, Springer, Seminar on nonlinear PDE (Berkeley), 1984 | MR | Zbl

[30] Kozono, H.; Yanagisawa, T. $L^r$-variational Inequality for Vector Fields and the Helmholtz-Weyl Decomposition in Bounded domains, Indiana Univ. Math. J., Volume 58 (2009) no. 4, pp. 1853-1920 | DOI | MR | Zbl

[31] Lions, J. L. Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969 | MR | Zbl

[32] Marusic, S. On the Navier-Stokes system with pressure boundary condition, Ann. Univ. Ferrara Sez. VII Sci. Mat., Volume 53 (2007) no. 2, pp. 319-331 | DOI | MR | Zbl

[33] Mitrea, M.; Monniaux, S. The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains, Differential Integral Equations, Volume 22 (2009) no. 3-4, pp. 339-356 | MR | Zbl

[34] Navier, C.L.M.H. Sur les lois de l’équilibre et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst., Volume 6 (1827) no. 369

[35] Neustupa, J.; Penel, P. Local in time strong solvability of the non-steady Navier-Stokes equations with Navier’s boundary conditions and the question of the inviscid limit, C.R.A.S. Paris, Volume 348 (2010) no. 19-20, pp. 1093-1097 | DOI | MR | Zbl

[36] Serrin, J. Mathematical principles of classical fluid mechanics, Handbuch der Physik, Springer-Verlag, 1959 | MR

[37] Shimada, R. On the $L^p-L^q$ maximal regularity for Stokes equations with Robin boundary condition in a bounded domain, Math. Methods Appl. Sci., Volume 30 (2007) no. 3, pp. 257-289 | DOI | MR | Zbl

[38] Solonnikov, V. A. $L^p$-estimates for solutions to the initial-boundary-value problem for the generalized Stokes system in a bounded domain, J. Math. Sci., Volume 105 (2001), pp. 2448-2484 | DOI | MR | Zbl

[39] Solonnikov, V. A. Estimates of the solution of model evolution generalized Stokes problem in weighted Hölder spaces, J. Math. Sci. (N. Y.), Volume 143 (2007) no. 2, pp. 2969-2986 | DOI | MR | Zbl

[40] Solonnikov, V. A.; Scadilov, V. E. A certain boundary value problem for the stationary system of Navier-Stokes equations, Boundary value problems of mathematical physics, Trudy Mat. Inst. Steklov., Volume 8 (1973), pp. 196-210 | MR | Zbl

[41] Temam, R. Theory and Numerical Analysis of the Navier-Stokes Equations, North-Holland, Amsterdam, 1977 | MR | Zbl

[42] Yudovich, V.I. A two-dimensional non-stationary problem on the flow of an ideal compressible fluid through a given region, Mat. Sb., Volume 4 (1964) no. 64, pp. 562-588 | MR