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doi: 10.3934/dcds.2021151
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Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains

Alessio Falocchi 1, and  Filippo Gazzola 2,

1. 

Dipartimento di Scienze Matematiche, Politecnico di Torino, Italy
2. 

Dipartimento di Matematica, Politecnico di Milano, Italy

Received  April 2021 Revised  August 2021 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.

Keywords: Navier-Stokes equations, Navier boundary conditions, regularity and uniqueness.
Mathematics Subject Classification: 35Q30, 35K20.
Citation: Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021151
References:
[1]

P. Acevedo, C. Amrouche, C. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[19]

P.-L. Lions, F. 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.  Google Scholar

[20]

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

[21]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

show all references

References:
[1]

P. Acevedo, C. Amrouche, C. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[19]

P.-L. Lions, F. 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.  Google Scholar

[20]

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

[21]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

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