March  2022, 42(3): 1185-1200. doi: 10.3934/dcds.2021151

Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains

1. 

Dipartimento di Scienze Matematiche, Politecnico di Torino, Italy

2. 

Dipartimento di Matematica, Politecnico di Milano, Italy

Received  April 2021 Revised  August 2021 Published  March 2022 Early access  November 2021

For the evolution Navier-Stokes equations in bounded 3D domains, it is well-known that the uniqueness of a solution is related to the existence of a regular solution. They may be obtained under suitable assumptions on the data and smoothness assumptions on the domain (at least $ C^{2,1} $). With a symmetrization technique, we prove these results in the case of Navier boundary conditions in a wide class of merely Lipschitz domains of physical interest, that we call sectors.

Citation: Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1185-1200. doi: 10.3934/dcds.2021151
References:
[1]

P. AcevedoC. AmroucheC. Conca and A. Ghosh, Stokes and Navier-Stokes equations with Navier boundary condition, C.R. Math. Acad. Sci. Paris, 357 (2019), 115-119.  doi: 10.1016/j.crma.2018.12.002.

[2]

C. Amrouche and A. Rejaiba, $L^p$-theory for Stokes and Navier-Stokes equations with Navier boundary condition, J. Differential Equations, 256 (2014), 1515-1547.  doi: 10.1016/j.jde.2013.11.005.

[3]

G. Arioli, F. Gazzola and H. Koch, Uniqueness and bifurcation branches for planar steady Navier-Stokes equations under Navier boundary conditions, J. Math. Fluid Mech., 23 (2001), 20pp. doi: 10.1007/s00021-021-00572-4.

[4]

G. S. Beavers and D. D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197-207. 

[5]

H. Beirão da Veiga, Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions, Adv. Differential Equations, 9 (2004), 1079-1114. 

[6]

H. Beirão da Veiga and L. C. Berselli, Navier-Stokes equations: Green's matrices, vorticity direction, and regularity up to the boundary, J. Differential Equations, 246 (2009), 597-628.  doi: 10.1016/j.jde.2008.02.043.

[7]

L. C. Berselli, Some results on the Navier-Stokes equations with Navier boundary conditions, Riv. Math. Univ. Parma (N.S.), 1 (2010), 1-75. 

[8]

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.  doi: 10.3934/dcdss.2010.3.199.

[9]

G. Butler and T. Rogers, A generalization of a lemma of Bihari and applications to pointwise estimates for integral equations, J. Math. Anal. Appl., 33 (1971), 77-81.  doi: 10.1016/0022-247X(71)90183-1.

[10]

A. Falocchi and F. Gazzola, Remarks on the 3D Stokes eigenvalue problem under Navier boundary conditions, Annali di Matematica, (2021). https://doi.org/10.1007/s10231-021-01165-8.

[11]

A. Friedman, Partial Differential Equations, olt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969.

[12]

G. P. Galdi and W. J. Layton, Approximation of the larger eddies in fluid motions. II. A model for space-filtered flow, Math. Models Methods Appl. Sci., 10 (2000), 343-350.  doi: 10.1142/S0218202500000203.

[13]

G. P. Galdi, An introduction to the Navier-Stokes initial-boundary value problem, Fundamental Directions in Mathematical Fluid Mechanics, (2000), 1–70.

[14]

F. Gazzola and P. Secchi, Inflow-outflow problems for Euler equations in a rectangular cylinder, NoDEA Nonlinear Differential Equations Appl., 8 (2001), 195-217.  doi: 10.1007/PL00001445.

[15]

F. Gazzola and G. Sperone, Steady Navier-Stokes equations in planar domains with obstacle and explicit bounds for unique solvability, Arch. Ration. Mech. Anal., 238 (2020), 1283-1347.  doi: 10.1007/s00205-020-01565-9.

[16]

J. G. Heywood, The Navier-Stokes equations: On the existence, regularity and decay of solutions, Indiana Univ. Math. J., 29 (1980), 639-681.  doi: 10.1512/iumj.1980.29.29048.

[17]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.

[18]

J. Leray, Essai sur les mouvements plans d'un fluide visqueux que limitent des parois, J. Math. Pures Appl., 13 (1934), 331-419. 

[19]

P.-L. LionsF. Pacella and M. Tricarico, Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions, Indiana Univ. Math., 37 (1988), 301-324.  doi: 10.1512/iumj.1988.37.37015.

[20]

C. L. M. H. Navier, Mémoire sur les lois du mouvement des fluides, Mem. Acad. Sci. Inst. Fr., 2 (1823), 389-440. 

[21]

J. Pedlosky, Geophysical Fluid Dynamics, Springer, 1979.

[22]

V. A. Solonnikov and V. E. Scadilov, A certain boundary value problem for the stationary system of Navier-Stokes equations, Indiana Univ. Math., 125 (1973), 196-210. 

[23]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and Its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[24]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2$^nd$ edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050.

[25]

J. Watanabe, On incompressible viscous fluid flows with slip boundary conditions, J. Comput. Appl. Math., 159 (2003), 161-172.  doi: 10.1016/S0377-0427(03)00568-5.

[26]

J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9781139171755.

show all references

References:
[1]

P. AcevedoC. AmroucheC. Conca and A. Ghosh, Stokes and Navier-Stokes equations with Navier boundary condition, C.R. Math. Acad. Sci. Paris, 357 (2019), 115-119.  doi: 10.1016/j.crma.2018.12.002.

[2]

C. Amrouche and A. Rejaiba, $L^p$-theory for Stokes and Navier-Stokes equations with Navier boundary condition, J. Differential Equations, 256 (2014), 1515-1547.  doi: 10.1016/j.jde.2013.11.005.

[3]

G. Arioli, F. Gazzola and H. Koch, Uniqueness and bifurcation branches for planar steady Navier-Stokes equations under Navier boundary conditions, J. Math. Fluid Mech., 23 (2001), 20pp. doi: 10.1007/s00021-021-00572-4.

[4]

G. S. Beavers and D. D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197-207. 

[5]

H. Beirão da Veiga, Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions, Adv. Differential Equations, 9 (2004), 1079-1114. 

[6]

H. Beirão da Veiga and L. C. Berselli, Navier-Stokes equations: Green's matrices, vorticity direction, and regularity up to the boundary, J. Differential Equations, 246 (2009), 597-628.  doi: 10.1016/j.jde.2008.02.043.

[7]

L. C. Berselli, Some results on the Navier-Stokes equations with Navier boundary conditions, Riv. Math. Univ. Parma (N.S.), 1 (2010), 1-75. 

[8]

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.  doi: 10.3934/dcdss.2010.3.199.

[9]

G. Butler and T. Rogers, A generalization of a lemma of Bihari and applications to pointwise estimates for integral equations, J. Math. Anal. Appl., 33 (1971), 77-81.  doi: 10.1016/0022-247X(71)90183-1.

[10]

A. Falocchi and F. Gazzola, Remarks on the 3D Stokes eigenvalue problem under Navier boundary conditions, Annali di Matematica, (2021). https://doi.org/10.1007/s10231-021-01165-8.

[11]

A. Friedman, Partial Differential Equations, olt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969.

[12]

G. P. Galdi and W. J. Layton, Approximation of the larger eddies in fluid motions. II. A model for space-filtered flow, Math. Models Methods Appl. Sci., 10 (2000), 343-350.  doi: 10.1142/S0218202500000203.

[13]

G. P. Galdi, An introduction to the Navier-Stokes initial-boundary value problem, Fundamental Directions in Mathematical Fluid Mechanics, (2000), 1–70.

[14]

F. Gazzola and P. Secchi, Inflow-outflow problems for Euler equations in a rectangular cylinder, NoDEA Nonlinear Differential Equations Appl., 8 (2001), 195-217.  doi: 10.1007/PL00001445.

[15]

F. Gazzola and G. Sperone, Steady Navier-Stokes equations in planar domains with obstacle and explicit bounds for unique solvability, Arch. Ration. Mech. Anal., 238 (2020), 1283-1347.  doi: 10.1007/s00205-020-01565-9.

[16]

J. G. Heywood, The Navier-Stokes equations: On the existence, regularity and decay of solutions, Indiana Univ. Math. J., 29 (1980), 639-681.  doi: 10.1512/iumj.1980.29.29048.

[17]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.

[18]

J. Leray, Essai sur les mouvements plans d'un fluide visqueux que limitent des parois, J. Math. Pures Appl., 13 (1934), 331-419. 

[19]

P.-L. LionsF. Pacella and M. Tricarico, Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions, Indiana Univ. Math., 37 (1988), 301-324.  doi: 10.1512/iumj.1988.37.37015.

[20]

C. L. M. H. Navier, Mémoire sur les lois du mouvement des fluides, Mem. Acad. Sci. Inst. Fr., 2 (1823), 389-440. 

[21]

J. Pedlosky, Geophysical Fluid Dynamics, Springer, 1979.

[22]

V. A. Solonnikov and V. E. Scadilov, A certain boundary value problem for the stationary system of Navier-Stokes equations, Indiana Univ. Math., 125 (1973), 196-210. 

[23]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and Its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[24]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2$^nd$ edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050.

[25]

J. Watanabe, On incompressible viscous fluid flows with slip boundary conditions, J. Comput. Appl. Math., 159 (2003), 161-172.  doi: 10.1016/S0377-0427(03)00568-5.

[26]

J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9781139171755.

Figure 1.  From left to right: a pipe bifurcation, a joint, a vein, a drop, a tunnel, a bottle
Figure 2.  Some sectors obtained as subdomains of a sphere
Figure 3.  Some Lipschitz domains that are not sectors, according to Definition 3
Figure 4.  Sectors of type $ (B) $
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