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On the blow-up problem for the Euler equations and the Liouville type results in the fluid equations
Time-averages of fast oscillatory systems
1. | School of Mathematical and Statistical Sciences, Arizona State University, Wexler Hall (PSA), Tempe, Arizona, 85287-1804, United States, United States |
References:
[1] |
W. Arendt, C. Batty, M. Hieber and F. Neubrander, "Vector-Valued Laplace Transforms and Cauchy Problems," Monographs in Mathematics, 96, Birkhäuser Verlag, Basel, 2001. |
[2] |
A. Babin, A. Mahalov and B. Nicolaenko, Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids, European J. Mechanics B Fluids, 15 (1996), 291-300. |
[3] |
A. Babin, A. Mahalov and B. Nicolaenko, Global splitting and regularity of rotating shallow-water equations, European J. Mech. B Fluids, 16 (1997), 725-754. |
[4] |
H. Beirão da Veiga, On the barotropic motion of compressible perfect fluids, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8 (1981), 317-351. |
[5] |
Bin Cheng, Singular limits and convergence rates of compressible Euler and rotating shallow water equations, SIAM J. on Mathematical Analysis, 44 (2012), 1050-1076.
doi: 10.1137/11085147X. |
[6] |
Bin Cheng and Alex Mahalov, Euler equations on a fast rotating sphere-time-averages and zonal flows, European J. Mech. - B/Fluids, 37 (2013), 48-58.
doi: 10.1016/j.euromechflu.2012.06.001. |
[7] |
D. G. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force, Ann. of Math. (2), 105 (1977), 141-200. |
[8] |
David G. Ebin, Motion of slightly compressible fluids in a bounded domain. I. Comm. Pure Appl. Math., 35 (1982), 451-485.
doi: 10.1002/cpa.3160350402. |
[9] |
B. Galperin, H. Nakano, H. Huang and S. Sukoriansky, The ubiquitous zonal jets in the atmospheres of giant planets and Earth oceans, Geophys. Res. Lett., 31 (2004), L13303. |
[10] |
B. Galperin, S. Sukoriansky, N. Dikovskaya, P. L. Read, Y. H. Yamazaki and R. Wordsworth, Anisotropic turbulence and zonal jets in rotating flows with a $\beta$-effect, Nonlinear Processes in Geophysics, 13 (2006), 83-98. |
[11] |
E. Garcýa-Melendo and A. Sánchez-Lavega, A study of the stability of Jovian zonal winds from HST images: 1995-2000, Icarus, 152 (2001), 316-330. |
[12] |
H.-P. Huang, B. Galperin and S. Sukoriansky, Anisotropic spectra in two-dimensional turbulence on the surface of a rotating sphere, Phys. Fluids, 13 (2001), 225-240. |
[13] |
N. A. Maximenko, B. Bang and H. Sasaki, Observational evidence of alternating jets in the world ocean, Geophys. Res. Lett., 32 (2005), L12607. |
[14] |
, NASA/JPL/University of Arizona, http://photojournal.jpl.nasa.gov/catalog/PIA02873. |
[15] |
Sergiu Klainerman and Andrew Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.
doi: 10.1002/cpa.3160340405. |
[16] |
H.-O. Kreiss, Problems with different time scales for partial differential equations, Comm. Pure Appl. Math., 33 (1980), 399-439.
doi: 10.1002/cpa.3160330310. |
[17] |
T. Nozawa and S. Yoden, Formation of zonal band structure in forced two-dimensional turbulence on a rotating sphere, Phys. Fluids, 9 (1997), 2081-2093.
doi: 10.1063/1.869327. |
[18] |
C. Porco, et al., Cassini imaging of Jupiter atmosphere, satellites and rings, Science, 299 (2003), 1541-1547. |
[19] |
G. Roden, Upper ocean thermohaline, oxygen, nutrients, and flow structure near the date line in the summer of 1993, J. Geophys. Res., 103 (1998), 12919-12939. |
[20] |
G. Roden, Flow and water property structures between the Bering Sea and Fiji in the summer of 1993, J. Geophys. Res., 105 (2000), 28595-28612. |
[21] |
S. Sukoriansky, B. Galperin and N. Dikovskaya, Universal spectrum of two-dimensional turbulence on a rotating sphere and some basic features of atmospheric circulation on giant planets, Phys. Rev. Lett., 89 (2002), 124501. |
[22] |
G. Vallis and M. Maltrud, Generation of mean flows and jets on a beta plane and over topography, J. Phys. Oceanogr., 23 (1993), 1346-1362. |
show all references
References:
[1] |
W. Arendt, C. Batty, M. Hieber and F. Neubrander, "Vector-Valued Laplace Transforms and Cauchy Problems," Monographs in Mathematics, 96, Birkhäuser Verlag, Basel, 2001. |
[2] |
A. Babin, A. Mahalov and B. Nicolaenko, Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids, European J. Mechanics B Fluids, 15 (1996), 291-300. |
[3] |
A. Babin, A. Mahalov and B. Nicolaenko, Global splitting and regularity of rotating shallow-water equations, European J. Mech. B Fluids, 16 (1997), 725-754. |
[4] |
H. Beirão da Veiga, On the barotropic motion of compressible perfect fluids, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8 (1981), 317-351. |
[5] |
Bin Cheng, Singular limits and convergence rates of compressible Euler and rotating shallow water equations, SIAM J. on Mathematical Analysis, 44 (2012), 1050-1076.
doi: 10.1137/11085147X. |
[6] |
Bin Cheng and Alex Mahalov, Euler equations on a fast rotating sphere-time-averages and zonal flows, European J. Mech. - B/Fluids, 37 (2013), 48-58.
doi: 10.1016/j.euromechflu.2012.06.001. |
[7] |
D. G. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force, Ann. of Math. (2), 105 (1977), 141-200. |
[8] |
David G. Ebin, Motion of slightly compressible fluids in a bounded domain. I. Comm. Pure Appl. Math., 35 (1982), 451-485.
doi: 10.1002/cpa.3160350402. |
[9] |
B. Galperin, H. Nakano, H. Huang and S. Sukoriansky, The ubiquitous zonal jets in the atmospheres of giant planets and Earth oceans, Geophys. Res. Lett., 31 (2004), L13303. |
[10] |
B. Galperin, S. Sukoriansky, N. Dikovskaya, P. L. Read, Y. H. Yamazaki and R. Wordsworth, Anisotropic turbulence and zonal jets in rotating flows with a $\beta$-effect, Nonlinear Processes in Geophysics, 13 (2006), 83-98. |
[11] |
E. Garcýa-Melendo and A. Sánchez-Lavega, A study of the stability of Jovian zonal winds from HST images: 1995-2000, Icarus, 152 (2001), 316-330. |
[12] |
H.-P. Huang, B. Galperin and S. Sukoriansky, Anisotropic spectra in two-dimensional turbulence on the surface of a rotating sphere, Phys. Fluids, 13 (2001), 225-240. |
[13] |
N. A. Maximenko, B. Bang and H. Sasaki, Observational evidence of alternating jets in the world ocean, Geophys. Res. Lett., 32 (2005), L12607. |
[14] |
, NASA/JPL/University of Arizona, http://photojournal.jpl.nasa.gov/catalog/PIA02873. |
[15] |
Sergiu Klainerman and Andrew Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.
doi: 10.1002/cpa.3160340405. |
[16] |
H.-O. Kreiss, Problems with different time scales for partial differential equations, Comm. Pure Appl. Math., 33 (1980), 399-439.
doi: 10.1002/cpa.3160330310. |
[17] |
T. Nozawa and S. Yoden, Formation of zonal band structure in forced two-dimensional turbulence on a rotating sphere, Phys. Fluids, 9 (1997), 2081-2093.
doi: 10.1063/1.869327. |
[18] |
C. Porco, et al., Cassini imaging of Jupiter atmosphere, satellites and rings, Science, 299 (2003), 1541-1547. |
[19] |
G. Roden, Upper ocean thermohaline, oxygen, nutrients, and flow structure near the date line in the summer of 1993, J. Geophys. Res., 103 (1998), 12919-12939. |
[20] |
G. Roden, Flow and water property structures between the Bering Sea and Fiji in the summer of 1993, J. Geophys. Res., 105 (2000), 28595-28612. |
[21] |
S. Sukoriansky, B. Galperin and N. Dikovskaya, Universal spectrum of two-dimensional turbulence on a rotating sphere and some basic features of atmospheric circulation on giant planets, Phys. Rev. Lett., 89 (2002), 124501. |
[22] |
G. Vallis and M. Maltrud, Generation of mean flows and jets on a beta plane and over topography, J. Phys. Oceanogr., 23 (1993), 1346-1362. |
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