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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

 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.

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