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

Chérif Amrouche[1]; Patrick Penel[2]; Nour Seloula[3]

  • [1] Université de Pau et des Pays de l’Adour LMA, Avenue de l’Université 64013 Pau cedex, France
  • [2] Université du Sud, Toulon - Var 83957 La Garde cedex , France
  • [3] Laboratoire de Mathématiques Nicolas Oresme UMR 6139 CNRS BP 5186 Université de Caen Basse Normandie

Annales mathématiques Blaise Pascal (2013)

  • Volume: 20, Issue: 1, page 37-73
  • ISSN: 1259-1734

Abstract

top
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.

How to cite

top

Amrouche, Chérif, Penel, Patrick, and Seloula, Nour. "Some Remarks on the Boundary Conditions in the Theory of Navier-Stokes Equations." Annales mathématiques Blaise Pascal 20.1 (2013): 37-73. <http://eudml.org/doc/275597>.

@article{Amrouche2013,
abstract = {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.},
affiliation = {Université de Pau et des Pays de l’Adour LMA, Avenue de l’Université 64013 Pau cedex, France; Université du Sud, Toulon - Var 83957 La Garde cedex , France; Laboratoire de Mathématiques Nicolas Oresme UMR 6139 CNRS BP 5186 Université de Caen Basse Normandie},
author = {Amrouche, Chérif, Penel, Patrick, Seloula, Nour},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Navier-Stokes; Boundary conditions; Weak solutions; Navier-Stokes equations; boundary conditions; weak solutions},
language = {eng},
month = {1},
number = {1},
pages = {37-73},
publisher = {Annales mathématiques Blaise Pascal},
title = {Some Remarks on the Boundary Conditions in the Theory of Navier-Stokes Equations},
url = {http://eudml.org/doc/275597},
volume = {20},
year = {2013},
}

TY - JOUR
AU - Amrouche, Chérif
AU - Penel, Patrick
AU - Seloula, Nour
TI - Some Remarks on the Boundary Conditions in the Theory of Navier-Stokes Equations
JO - Annales mathématiques Blaise Pascal
DA - 2013/1//
PB - Annales mathématiques Blaise Pascal
VL - 20
IS - 1
SP - 37
EP - 73
AB - 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.
LA - eng
KW - Navier-Stokes; Boundary conditions; Weak solutions; Navier-Stokes equations; boundary conditions; weak solutions
UR - http://eudml.org/doc/275597
ER -

References

top
  1. H. Abels, Nonstationary Stokes system with variable viscosity in bounded and unbounded domains, Discrete Contin. Dyn. Syst. Ser. S. 3 (2010), 141-157 Zbl1191.76038MR2610556
  2. Y. Amirat, D. Bresch, J. Lemoine, J. Simon, Effect of rugosity on a flow governed by stationary Navier-Stokes equations, Quart. Appl. Math. 59 (2001), 769-785 Zbl1019.76014MR1866556
  3. C. Amrouche, C. Bernardi, M. Dauge, V. Girault, Vector potentials in three-dimensional nonsmooth domains, Math. Meth. Applied Sc. 21 (1998), 823-864 Zbl0914.35094MR1626990
  4. C. Amrouche, N. Seloula, On the Stokes Equations with the Navier-Type boundary conditions, Diff. Eq. Appl. 3-4 (2011), 581-607 Zbl1259.35092MR2918930
  5. C. Amrouche, N. Seloula, L 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 23 (2013), 37-92 Zbl1260.35101MR2997467
  6. C. Bardos, F. Golse, L. Paillard, The incompressible Euler limit of the Boltzmann equation with accomodation boundary conditions, Comm. Math. Sci. 10 (2012), 159-190 Zbl1291.35169MR2901306
  7. G.S. Beavers, D.D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech. 30 (1967), 197-207 
  8. C. Bègue, C. Conca, F. Murat, O. Pironneau, 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 IX (1989), 179-264 Zbl0687.35069MR992649
  9. H. Beirao da Veiga, On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions, Comm. Pure Appl. Math. 58 (2005), 552-577 Zbl1075.35045MR2119869
  10. H. Bellout, J. Neustupa, P. Penel, On the Navier-Stokes equation with boundary conditions based on vorticity, Math. Nachr. 269-270 (2004), 59-72 Zbl1061.35073MR2074773
  11. H. Bellout, J. Neustupa, P. Penel, On a ν -continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary conditions, Discrete and Continuous Dynamical Systems 27 (2010), 1353-1373 Zbl1200.35209MR2629528
  12. J. M. Bernard, Non-standard Stokes and Navier-Stokes problem: existence and regularity in stationary case, Math. Meth. Appl. Sci. 25 (2002), 627-661 Zbl1027.35091MR1900648
  13. J. M. Bernard, Time-dependent Stokes and Navier-Stokes problems with boundary conditions involving pressure, existence and regularity, Nonlinear Anal. Real World Appl. 4 (2003), 805-839 Zbl1037.35053MR1978563
  14. L. C. Berselli, Some criteria concerning the vorticity and the problem of global regularity for the 3D Navier-Stokes equations, Ann. Univ. Ferrara Sez. VII Sci. Mat. 55 (2009), 209-224 Zbl1205.35186MR2563656
  15. L. C. Berselli., 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 3 (2010), 199-219 Zbl1193.35125MR2610559
  16. D. Bothe, J. Pruss, L p -theory for a class of non-Newtonian fluids, SIAM J. Math. Anal. 39 (2007), 379-421 Zbl1172.35052MR2338412
  17. D. Bucur, E. Feireisl, S. Necasova, Boundary behavior of viscous fluids: influence of wall roughness and friction-driven boundary conditions, Arch. Ration. Mech. Anal. 197 (2010), 117-138 Zbl1273.76073MR2646816
  18. D. Bucur, E. Feireisl, S. Necasova, J. Wolf, On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries, J. Differential Equations 244 (2008), 2890-2908 Zbl1143.35080MR2418180
  19. M. Bulicek, J. Malek, K. R. Rajagopal, Navier’s slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity, Indiana Univ. Math. J. 56 (2007), 51-85 Zbl1129.35055MR2305930
  20. G. Q. Chen, D. Osborne, Z. Qian, The Navier-Stokes equations with the kinematic and vorticity boundary conditions on non-flat boundaries, Acta Math. Sci. Ser. B Engl. Ed. 29 (2009), 919-948 Zbl1212.35346MR2509999
  21. C. Ebmeyer, J. Frehse, Steady Navier-Stokes equations with mixed boundary value conditions in three-dimensional Lipschitzian domains, Math. Ann 319 (2001), 349-381 Zbl0997.35049MR1815115
  22. L. T. Hoang, Incompressible fluids in thin domains with Navier friction boundary conditions (I), J. Math. Fluid Mech. 12 (2010), 435-472 Zbl1261.35107MR2674072
  23. L. T. Hoang, G. R. Sell, Navier-Stokes equations with Navier boundary conditions for an oceanic model, J. Dynam. Differential Equations 22 (2010), 653-616 Zbl1204.86010MR2719921
  24. D. Iftimie, G. Raugel, Some results on the Navier-Stokes equations in thin 3D domains, J. Diff. Eq. 169 (2001), 281-331 Zbl0972.35085MR1808469
  25. D. Iftimie, G. Raugel, G. Sell, Navier-Stokes equations in thin 3D domains with Navier boundary conditions, Indiana Univ. Math. J. 56 (2007), 1083-1156 Zbl1129.35056MR2333468
  26. D. Iftimie, F. Sueur, Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions, Arch. Ration. Mech. Anal. 199 (2011), 145-175 Zbl1229.35184MR2754340
  27. W. Jager, A. Mikelic, On the interface boundary condition of Beavers, Joseph, and Saffman, SIAM J. Appl. Math. 60 (2000), 1111-1127 Zbl0969.76088MR1760028
  28. W. Jager, A. Mikelic, On the roughness-induced effective boundary conditions for an incompressible viscous flow, J. Differential Equations 170 (2001), 96-122 Zbl1009.76017MR1813101
  29. T. Kato, Remarks on zero viscosity limit for non-stationary Navier-Stokes flows with boundary, (1984), Springer, Seminar on nonlinear PDE (Berkeley) Zbl0559.35067MR765230
  30. H. Kozono, T. Yanagisawa, L r -variational Inequality for Vector Fields and the Helmholtz-Weyl Decomposition in Bounded domains, Indiana Univ. Math. J. 58 (2009), 1853-1920 Zbl1179.35147MR2542982
  31. J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, (1969), Dunod, Gauthier-Villars, Paris Zbl0189.40603MR259693
  32. S. Marusic, On the Navier-Stokes system with pressure boundary condition, Ann. Univ. Ferrara Sez. VII Sci. Mat. 53 (2007), 319-331 Zbl1248.76034MR2358233
  33. M. Mitrea, S. Monniaux, The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains, Differential Integral Equations 22 (2009), 339-356 Zbl1240.35412MR2492825
  34. C.L.M.H. Navier, Sur les lois de l’équilibre et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst. 6 (1827) 
  35. J. Neustupa, P. Penel, 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 348 (2010), 1093-1097 Zbl1205.35206MR2735014
  36. J. Serrin, Mathematical principles of classical fluid mechanics, (1959), Handbuch der Physik, Springer-Verlag MR108116
  37. R. Shimada, On the L p - L q maximal regularity for Stokes equations with Robin boundary condition in a bounded domain, Math. Methods Appl. Sci. 30 (2007), 257-289 Zbl1107.76029MR2285430
  38. V. A. Solonnikov, L p -estimates for solutions to the initial-boundary-value problem for the generalized Stokes system in a bounded domain, J. Math. Sci. 105 (2001), 2448-2484 Zbl0986.35084MR1855442
  39. V. A. Solonnikov, Estimates of the solution of model evolution generalized Stokes problem in weighted Hölder spaces, J. Math. Sci. (N. Y.) 143 (2007), 2969-2986 Zbl1127.35051MR2270886
  40. V. A. Solonnikov, V. E. Scadilov, A certain boundary value problem for the stationary system of Navier-Stokes equations, Boundary value problems of mathematical physics, Trudy Mat. Inst. Steklov. 8 (1973), 196-210 Zbl0313.35063MR364910
  41. R. Temam, Theory and Numerical Analysis of the Navier-Stokes Equations, (1977), North-Holland, Amsterdam Zbl0383.35057MR769654
  42. V.I. Yudovich, A two-dimensional non-stationary problem on the flow of an ideal compressible fluid through a given region, Mat. Sb. 4 (1964), 562-588 MR177577

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.