Vanishing viscosity limits for axisymmetric flows with boundary

Abe, 2019, The Navier-Stokes equations with the Neumann boundary condition in an infinite cylinder, Manuscr. Math., 160, 359, 10.1007/s00229-018-01102-9 Abe, 2019, Axisymmetric flows in the exterior of a cylinder, Proc. R. Soc. Edinb. A, 10.1017/prm.2018.121 Abidi, 2008, Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes, Bull. Sci. Math., 132, 592, 10.1016/j.bulsci.2007.10.001 Adams, 2003, Sobolev Spaces, vol. 140 Agmon, 1959, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Commun. Pure Appl. Math., 12, 623, 10.1002/cpa.3160120405 Akiyama, 2004, On a resolvent estimate of a system of Laplace operators with perfect wall condition, Funkc. Ekvacioj, 47, 361, 10.1619/fesi.47.361 Bardos, 1972, Existence et unicité de la solution de l'équation d'Euler en dimension deux, J. Math. Anal. Appl., 40, 769, 10.1016/0022-247X(72)90019-4 Bardos, 2018, Onsager's conjecture for the incompressible Euler equations in bounded domains, Arch. Ration. Mech. Anal., 228, 197, 10.1007/s00205-017-1189-x Berselli, 2012, On the vanishing viscosity limit of 3D Navier-Stokes equations under slip boundary conditions in general domains, Commun. Math. Phys., 316, 171, 10.1007/s00220-012-1581-1 Bourguignon, 1974, Remarks on the Euler equation, J. Funct. Anal., 15, 341, 10.1016/0022-1236(74)90027-5 Chae, 1998, Existence of axisymmetric weak solutions of the 3-D Euler equations for near-vortex-sheet initial data, Electron. J. Differ. Equ., 26, 17 Chae, 1997, Axisymmetric weak solutions of the 3-D Euler equations for incompressible fluid flows, Nonlinear Anal., 29, 1393, 10.1016/S0362-546X(96)00186-1 Chemin, 1998, Perfect Incompressible Fluids, vol. 14 Cheskidov, 2008, Energy conservation and Onsager's conjecture for the Euler equations, Nonlinearity, 21, 1233, 10.1088/0951-7715/21/6/005 Clopeau, 1998, On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions, Nonlinearity, 11, 1625, 10.1088/0951-7715/11/6/011 Constantin, 1986, Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations, Commun. Math. Phys., 104, 311, 10.1007/BF01211598 Constantin, 2007, On the Euler equations of incompressible fluids, Bull. Am. Math. Soc. (N.S.), 44, 603, 10.1090/S0273-0979-07-01184-6 Constantin, 1994, Onsager's conjecture on the energy conservation for solutions of Euler's equation, Commun. Math. Phys., 165, 207, 10.1007/BF02099744 Danchin, 2007, Axisymmetric incompressible flows with bounded vorticity, Usp. Mat. Nauk, 62, 73 Davies, 1989, Heat Kernels and Spectral Theory, vol. 92 Delort, 1991, Existence de nappes de tourbillon en dimension deux, J. Am. Math. Soc., 4, 553, 10.1090/S0894-0347-1991-1102579-6 Delort, 1992, Une remarque sur le problème des nappes de tourbillon axisymétriques sur R3, J. Funct. Anal., 108, 274, 10.1016/0022-1236(92)90026-F DiPerna, 1987, Concentrations in regularizations for 2-D incompressible flow, Commun. Pure Appl. Math., 40, 301, 10.1002/cpa.3160400304 Duchon, 2000, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity, 13, 249, 10.1088/0951-7715/13/1/312 Ebin, 1970, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math., 92, 102, 10.2307/1970699 Evans, 2010, Partial Differential Equations, vol. 19 Evans, 1994, Hardy spaces and the two-dimensional Euler equations with nonnegative vorticity, J. Am. Math. Soc., 7, 199, 10.1090/S0894-0347-1994-1220787-3 Eyink, 1994, Energy dissipation without viscosity in ideal hydrodynamics, I. Fourier analysis and local energy transfer, Physica D, 78, 222, 10.1016/0167-2789(94)90117-1 Eyink, 2006, Onsager and the theory of hydrodynamic turbulence, Rev. Mod. Phys., 78, 87, 10.1103/RevModPhys.78.87 Feng, 2015, On the Cauchy problem for axi-symmetric vortex rings, Arch. Ration. Mech. Anal., 215, 89, 10.1007/s00205-014-0775-4 Frisch, 1995 Gallay, 2017, Infinite energy solutions of the two-dimensional Navier-Stokes equations, Ann. Fac. Sci. Toulouse Math., 26, 979, 10.5802/afst.1558 Gallay, 2014, Energy bounds for the two-dimensional Navier-Stokes equations in an infinite cylinder, Commun. Partial Differ. Equ., 39, 1741, 10.1080/03605302.2013.870575 Gallay, 2015, Uniform boundedness and long-time asymptotics for the two-dimensional Navier-Stokes equations in an infinite cylinder, J. Math. Fluid Mech., 17, 23, 10.1007/s00021-014-0188-z Gallay Gallay, 2015, Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations, Confluentes Math., 7, 67, 10.5802/cml.25 Geissert, 2013, H∞-calculus for a system of Laplace operators with mixed order boundary conditions, Discrete Contin. Dyn. Syst., Ser. S, 6, 1259 Giga, 1986, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, J. Differ. Equ., 62, 186, 10.1016/0022-0396(86)90096-3 Giga, 2001, Global existence of smooth solutions for two dimensional Navier-Stokes equations with nondecaying initial velocity, J. Math. Fluid Mech., 3, 302, 10.1007/PL00000973 Giga, 1989, Navier-Stokes flow in R3 with measures as initial vorticity and Morrey spaces, Commun. Partial Differ. Equ., 14, 577, 10.1080/03605308908820621 Giga, 1988, Two-dimensional Navier-Stokes flow with measures as initial vorticity, Arch. Ration. Mech. Anal., 104, 223, 10.1007/BF00281355 Hopf, 1951, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4, 213, 10.1002/mana.3210040121 Iftimie, 2006, Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions, Nonlinearity, 19, 899, 10.1088/0951-7715/19/4/007 Iftimie, 2011, Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions, Arch. Ration. Mech. Anal., 199, 145, 10.1007/s00205-010-0320-z Jia, 2015, Are the incompressible 3D Navier-Stokes equations locally ill-posed in the natural energy space?, J. Funct. Anal., 268, 3734, 10.1016/j.jfa.2015.04.006 Jiu, 2015, Viscous approximation and weak solutions of the 3D axisymmetric Euler equations, Math. Methods Appl. Sci., 38, 548, 10.1002/mma.3087 Jiu, 2006, On strong convergence to 3-D axisymmetric vortex sheets, J. Differ. Equ., 223, 33, 10.1016/j.jde.2005.04.001 Kato, 1972, Nonstationary flows of viscous and ideal fluids in R3, J. Funct. Anal., 9, 296, 10.1016/0022-1236(72)90003-1 Kato, 1975, Quasi-linear equations of evolution, with applications to partial differential equations, vol. 448, 25 Kato, 1984, Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary, vol. 2, 85 Kato, 1984, Strong Lp-solutions of the Navier-Stokes equation in Rm, with applications to weak solutions, Math. Z., 187, 471, 10.1007/BF01174182 Kato, 1984, Nonlinear evolution equations and the Euler flow, J. Funct. Anal., 56, 15, 10.1016/0022-1236(84)90024-7 Kelliher, 2006, Navier-Stokes equations with Navier boundary conditions for a bounded domain in the plane, SIAM J. Math. Anal., 38, 210, 10.1137/040612336 Koch, 2009, Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203, 83, 10.1007/s11511-009-0039-6 Koch, 2001, Well-posedness for the Navier-Stokes equations, Adv. Math., 157, 22, 10.1006/aima.2000.1937 Ladyženskaya, 1968, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Naučn. Semin. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7, 155 Ladyženskaya, 1969, An example of nonuniqueness in Hopf's class of weak solutions of the Navier-Stokes equations, Izv. Akad. Nauk SSSR, Ser. Mat., 33, 240 Leonardi, 1999, On axially symmetric flows in R3, Z. Anal. Anwend., 18, 639, 10.4171/ZAA/903 Leray, 1934, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63, 193, 10.1007/BF02547354 Lions, 1969 Lions, 1996, Mathematical Topics in Fluid Mechanics, vol. 3 Lopes Filho, 2005, On the inviscid limit for two-dimensional incompressible flow with Navier friction condition, SIAM J. Math. Anal., 36, 1130, 10.1137/S0036141003432341 Lunardi, 1995, Analytic semigroups and optimal regularity in parabolic problems, vol. 16 Majda, 1993, Remarks on weak solutions for vortex sheets with a distinguished sign, Indiana Univ. Math. J., 42, 921, 10.1512/iumj.1993.42.42043 Majda, 2002, Vorticity and Incompressible Flow, vol. 27 Masmoudi, 2007, Remarks about the inviscid limit of the Navier-Stokes system, Commun. Math. Phys., 270, 777, 10.1007/s00220-006-0171-5 Masmoudi, 2012, Uniform regularity for the Navier-Stokes equation with Navier boundary condition, Arch. Ration. Mech. Anal., 203, 529, 10.1007/s00205-011-0456-5 Miyakawa, 1992, Planar Navier-Stokes flows in a bounded domain with measures as initial vorticities, Hiroshima Math. J., 22, 401, 10.32917/hmj/1206392908 Onsager, 1949, Statistical hydrodynamics, Nuovo Cimento, 9, 279, 10.1007/BF02780991 Sawada, 2007, A remark on L∞ solutions to the 2-D Navier-Stokes equations, J. Math. Fluid Mech., 9, 533, 10.1007/s00021-005-0212-4 Seeley, 1971, Norms and domains of the complex powers ABz, Am. J. Math., 93, 299, 10.2307/2373377 Shirota, 1994, Note on global existence for axially symmetric solutions of the Euler system, Proc. Jpn. Acad., Ser. A, Math. Sci., 70, 299, 10.3792/pjaa.70.299 Sohr, 1998, Imaginary powers of second order differential operators and Lq-Helmholtz decomposition in the infinite cylinder, Math. Ann., 311, 577, 10.1007/s002080050201 Spirito, 2014, On inviscid limits for the Navier-Stokes equations with slip boundary conditions involving the vorticity, vol. 8, 967 Stein, 1993, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 Swann, 1971, The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in R3, Transl. Am. Math. Soc., 157, 373 Temam, 1975, On the Euler equations of incompressible perfect fluids, J. Funct. Anal., 20, 32, 10.1016/0022-1236(75)90052-X Temam, 1977 Ukhovskii, 1968, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., 32, 52, 10.1016/0021-8928(68)90147-0 Beirão da Veiga, 2010, Sharp inviscid limit results under Navier type boundary conditions. An Lp theory, J. Math. Fluid Mech., 12, 397, 10.1007/s00021-009-0295-4 Beirão da Veiga, 2011, Concerning the Wk,p-inviscid limit for 3-D flows under a slip boundary condition, J. Math. Fluid Mech., 13, 117, 10.1007/s00021-009-0012-3 Vishik, 1981, Individual and statistical solutions of a two-dimensional Euler system, Dokl. Akad. Nauk SSSR, 261, 780 Xiao, 2007, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition, Commun. Pure Appl. Math., 60, 1027, 10.1002/cpa.20187 Yudovich, 1962, Some bounds for solutions of elliptic equations, Mat. Sb. (N.S.), 59, 229 Yudovich, 1963, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. Mat. Fiz., 1032 Zelik, 2013, Infinite energy solutions for damped Navier-Stokes equations in R2, J. Math. Fluid Mech., 15, 717, 10.1007/s00021-013-0144-3