On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane. (English) Zbl 1301.35092
Author’s abstract: We consider the Navier-Stokes equations for viscous incompressible flows in the half-plane under the no-slip boundary condition. By using the vorticity formulation we prove the local-in-time convergence of the Navier-Stokes flows to the Euler flows outside a boundary layer and to the Prandtl flows in the boundary layer in the inviscid limit when the initial vorticity is located away from the boundary.
Reviewer: Pavel Burda (Praha)
MSC:
| 35Q30 | Navier-Stokes equations |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35A10 | Cauchy-Kovalevskaya theorems |
Keywords:
Navier-Stokes equations; incompressible viscous flows; vorticity formulation; Prandl flow; weighted Banach spacesReferences:
| [1] | Abidi, Optimal bounds for the inviscid limit of Navier-Stokes equations, Asymptot. Anal. 38 (1) pp 35– (2004) · Zbl 1092.35075 |
| [2] | Alexandre, Well-posedness of the Prandtl equation in Sobolev spaces (2012) |
| [3] | Anderson, Vorticity boundary conditions and boundary vorticity generation for two-dimensional viscous incompressible flows, J. Comput. Phys. 80 (1) pp 72– (1989) · Zbl 0656.76034 · doi:10.1016/0021-9991(89)90091-0 |
| [4] | Asano, Zero-viscosity limit of the incompressible Navier-Stokes equation. II. Mathematical analysis of fluid and plasma dynamics, I (Kyoto, 1986), Sūrikaisekikenkyūsho Kōkyūroku 656 pp 105– (1988) |
| [5] | Bardos, Existence et unicité de la solution de l’équation d’Euler en dimension deux, J. Math. Anal. Appl. 40 pp 769– (1972) · Zbl 0249.35070 · doi:10.1016/0022-247X(72)90019-4 |
| [6] | Beale, Rates of convergence for viscous splitting of the Navier-Stokes equations, Math. Comp. 37 (156) pp 243– (1981) · Zbl 0518.76027 · doi:10.1090/S0025-5718-1981-0628693-0 |
| [7] | Bona, The zero-viscosity limit of the 2D Navier-Stokes equations, Stud. Appl. Math. 109 (4) pp 265– (2002) · Zbl 1141.35431 · doi:10.1111/1467-9590.t01-1-00223 |
| [8] | Carlen, Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the 2-D Navier-Stokes equation, Duke Math. J. 81 (1) pp 135– (1995) · Zbl 0859.35011 · doi:10.1215/S0012-7094-95-08110-1 |
| [9] | Chemin, A remark on the inviscid limit for two-dimensional incompressible fluids, Comm. Partial Differential Equations 21 (11-12) pp 1771– (1996) |
| [10] | Constantin, Inviscid limit for vortex patches, Nonlinearity 8 (5) pp 735– (1995) · Zbl 0832.76011 · doi:10.1088/0951-7715/8/5/005 |
| [11] | Constantin, The inviscid limit for non-smooth vorticity, Indiana Univ. Math. J. 45 (1) pp 67– (1996) · Zbl 0859.76015 · doi:10.1512/iumj.1996.45.1960 |
| [12] | Danchin, Poches de tourbillon visqueuses, J. Math. Pures Appl. (9) 76 (7) pp 609– (1997) · Zbl 0903.76020 · doi:10.1016/S0021-7824(97)89964-3 |
| [13] | Ebin, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2) 92 pp 102– (1970) · Zbl 0211.57401 · doi:10.2307/1970699 |
| [14] | Gallay, Interaction of vortices in weakly viscous planar flows, Arch. Ration. Mech. Anal. 200 (2) pp 445– (2011) · Zbl 1229.35177 · doi:10.1007/s00205-010-0362-2 |
| [15] | Gérard-Varet, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc. 23 (2) pp 591– (2010) · Zbl 1197.35204 · doi:10.1090/S0894-0347-09-00652-3 |
| [16] | Gérard-Varet, Remarks on the ill-posedness of the Prandtl equation, Asymptot. Anal. 77 (1-2) pp 71– (2012) |
| [17] | Giga, Nonlinear partial differential equations. Asymptotic behavior of solutions and self-similar solutions (2010) · Zbl 1215.35001 |
| [18] | Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math. 53 (9) pp 1067– (2000) · Zbl 1048.35081 · doi:10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.0.CO;2-Q |
| [19] | Guo, A note on Prandtl boundary layers, Comm. Pure Appl. Math. 64 (10) pp 1416– (2011) · Zbl 1232.35126 · doi:10.1002/cpa.20377 |
| [20] | Hmidi, Régularité höldérienne des poches de tourbillon visqueuses. (French) [Hölder regularity of viscous vortex patches], J. Math. Pures Appl. (9) 84 (11) pp 1455– (2005) · Zbl 1095.35024 · doi:10.1016/j.matpur.2005.01.004 |
| [21] | Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz. 3 pp 1032– (1963) |
| [22] | Kano, Sur les ondes de surface de l’eau avec une justification mathématique des équations des ondes en eau peu profonde, J. Math. Kyoto Univ. 19 (2) pp 335– (1979) · Zbl 0419.76013 |
| [23] | Kato, On classical solutions of the two-dimensional nonstationary Euler equation, Arch. Rational Mech. Anal. 25 pp 188– (1967) · Zbl 0166.45302 · doi:10.1007/BF00251588 |
| [24] | Kato, Nonstationary flows of viscous and ideal fluids in R3, J. Functional Analysis 9 pp 296– (1972) · Zbl 0229.76018 · doi:10.1016/0022-1236(72)90003-1 |
| [25] | Kato, Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983) pp 85– (1984) · doi:10.1007/978-1-4612-1110-5_6 |
| [26] | Kelliher, On Kato’s conditions for vanishing viscosity, Indiana Univ. Math. J. 56 (4) pp 1711– (2007) · doi:10.1512/iumj.2007.56.3080 |
| [27] | Kelliher, On the vanishing viscosity limit in a disk, Math. Ann. 343 (3) pp 701– (2009) · Zbl 1155.76020 · doi:10.1007/s00208-008-0287-3 |
| [28] | Kramer, Vorticity dynamics of a dipole colliding with a no-slip wall, Phys. Fluids 19 pp 126603– (2007) · Zbl 1182.76399 · doi:10.1063/1.2814345 |
| [29] | Kukavica, On the local existence of analytic solutions to the Prandtl boundary layer equations, Commun. Math. Sci. 11 (1) pp 269– (2013) · Zbl 1291.35224 · doi:10.4310/CMS.2013.v11.n1.a8 |
| [30] | Lombardo, Well-posedness of the boundary layer equations, SIAM J. Math. Anal. 35 (4) pp 987– (2003) · Zbl 1053.76013 · doi:10.1137/S0036141002412057 |
| [31] | Lopes Filho, Vanishing viscosity limit for incompressible flow inside a rotating circle, Phys. D 237 (10-12) pp 1324– (2008) · Zbl 1143.76416 · doi:10.1016/j.physd.2008.03.009 |
| [32] | Lopes Filho, Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows, Bull. Braz. Math. Soc. (N.S.) 39 (4) pp 471– (2008) · Zbl 1178.35288 · doi:10.1007/s00574-008-0001-9 |
| [33] | Maekawa, Solution formula for the vorticity equations in the half plane with application to high vorticity creation at zero viscosity limit, Adv. Differential Equations 18 (1-2) pp 101– (2013) · Zbl 1261.35111 |
| [34] | Majda, Cambridge Texts in Applied Mathematics 27 (2002) |
| [35] | Marchioro, On the vanishing viscosity limit for two-dimensional Navier-Stokes equations with singular initial data, Math. Methods Appl. Sci. 12 (6) pp 463– (1990) · Zbl 0703.76020 · doi:10.1002/mma.1670120602 |
| [36] | Marchioro, On the inviscid limit for a fluid with a concentrated vorticity, Comm. Math. Phys. 196 (1) pp 53– (1998) · Zbl 0911.35086 · doi:10.1007/s002200050413 |
| [37] | Marchioro, Vortices and localization in Euler flows, Comm. Math. Phys. 154 (1) pp 49– (1993) · Zbl 0774.35058 · doi:10.1007/BF02096831 |
| [38] | Masmoudi, Remarks about the inviscid limit of the Navier-Stokes system, Comm. Math. Phys. 270 (3) pp 777– (2007) · Zbl 1118.35030 · doi:10.1007/s00220-006-0171-5 |
| [39] | Masmoudi, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods (2012) |
| [40] | Matsui, Example of zero viscosity limit for two-dimensional nonstationary Navier-Stokes flows with boundary, Japan J. Indust. Appl. Math. 11 (1) pp 155– (1994) · Zbl 0797.76011 · doi:10.1007/BF03167219 |
| [41] | Matsui, On separation points of solutions to Prandtl boundary layer problem, Hokkaido Math. J. 13 (1) pp 92– (1984) · Zbl 0569.76048 · doi:10.14492/hokmj/1381757739 |
| [42] | Mazzucato, Vanishing viscosity limits for a class of circular pipe flows, Comm. Partial Differential Equations 36 (2) pp 328– (2011) · Zbl 1220.35121 · doi:10.1080/03605302.2010.505973 |
| [43] | Nguyen van yen, Energy dissipating structures produced by walls in two-dimensional flows at vanishing viscosity, Phys. Rev. Lett. 106 pp 184502– (2011) · doi:10.1103/PhysRevLett.106.184502 |
| [44] | Nirenberg, An abstract form of the nonlinear Cauchy-Kowalewski theorem, J. Differential Geometry 6 pp 561– (1972) · Zbl 0257.35001 |
| [45] | Nishida, A note on a theorem of Nirenberg, J. Differential Geom. 12 (4) pp 629– (1977) |
| [46] | Oleĭnik, On the mathematical theory of boundary layer for an unsteady flow of incompressible fluid, J. Appl. Math. Mech. 30 pp 951– (1966) · Zbl 0149.44804 · doi:10.1016/0021-8928(66)90001-3 |
| [47] | Orlandi, Vortex dipole rebound from a wall, Phys. Fluids A 2 pp 1429– (1990) · doi:10.1063/1.857591 |
| [48] | Safonov, The abstract Cauchy-Kovalevskaya theorem in a weighted Banach space, Comm. Pure Appl. Math. 48 (6) pp 629– (1995) · Zbl 0836.35004 · doi:10.1002/cpa.3160480604 |
| [49] | Sammartino, Zero viscosity limit for analytic solutions, of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations, Comm. Math. Phys. 192 (2) pp 433– (1998) · Zbl 0913.35102 · doi:10.1007/s002200050304 |
| [50] | Sammartino, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution, Comm. Math. Phys. 192 (2) pp 463– (1998) · Zbl 0913.35103 · doi:10.1007/s002200050305 |
| [51] | Solonnikov, Estimates of the solutions of a nonstationary linearized system of Navier-Stokes equations, Amer. Math. Soc. Transl. 75 (2) pp 1– (1968) · Zbl 0187.03402 · doi:10.1090/trans2/075/01 |
| [52] | Swann, The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in R3, Trans. Amer. Math. Soc. 157 pp 373– (1971) · Zbl 0218.76023 |
| [53] | Temam, On the behavior of the solutions of the Navier-Stokes equations at vanishing viscosity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (3-4) pp 807– (1997) · Zbl 1043.35127 |
| [54] | Ukai, A solution formula for the Stokes equation in R+n, Comm. Pure Appl. Math. 40 (5) pp 611– (1987) · Zbl 0638.76040 · doi:10.1002/cpa.3160400506 |
| [55] | Wang, A Kato type theorem on zero viscosity limit of Navier-Stokes flows, Indiana Univ. Math. J. 50 (Special Issue) pp 223– (2001) · Zbl 0991.35059 · doi:10.1512/iumj.2001.50.2098 |
| [56] | Wolibner, Un theorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long, Math. Z. 37 (1) pp 698– (1933) · JFM 59.1447.02 · doi:10.1007/BF01474610 |
| [57] | Xin, On the global existence of solutions to the Prandtl’s system, Adv. Math. 181 (1) pp 88– (2004) · Zbl 1052.35135 · doi:10.1016/S0001-8708(03)00046-X |
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