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

Alessio Falocchi; Filippo Gazzola

doi:10.3934/dcds.2021151

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2022, Volume 42, Issue 3: 1185-1200. Doi: 10.3934/dcds.2021151

This issue Previous Article Topological mild mixing of all orders along polynomials Next Article Global $ C^2 $-estimates for smooth solutions to uniformly parabolic equations with Neumann boundary condition

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 1, 2021

Revised: August 1, 2021

Early access: November 1, 2021

Published online: March 1, 2022

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Abstract

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.

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Mathematics Subject Classification: 35Q30, 35K20.

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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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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.
[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. 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.
[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.