On the blow-up problem for the Euler equations and the Liouville type results in the fluid equations

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October 2013, 6(5): 1139-1150. doi: 10.3934/dcdss.2013.6.1139

On the blow-up problem for the Euler equations and the Liouville type results in the fluid equations

Dongho Chae ^1,

1.	Department of Mathematics, Chung-Ang University, 84 Heukseok-ro, Dongjak-gu, Seoul, South Korea

Received December 2011 Revised November 2012 Published March 2013

Abstract
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In this paper we briefly review recent results mostly by the author related to the blow-up problem of the 3D Euler equations and the Liouville type results in the various equations of the fluids.

Keywords: Euler equations, Liouville type theorem., blow-up problem, Navier-Stokes equations.

Mathematics Subject Classification: Primary: 76B03, 76D03; Secondary: 35Q3.

Citation: Dongho Chae. On the blow-up problem for the Euler equations and the Liouville type results in the fluid equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1139-1150. doi: 10.3934/dcdss.2013.6.1139

References:

[1]	C. Bardos and É. S. Titi, Euler equations of incompressible ideal fluids, Russian Math. Surveys, 62 (2007), 409-451. doi: 10.1070/RM2007v062n03ABEH004410.
[2]	J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 94 (1984), 61-66.
[3]	D. Chae, Remarks on the Liouville type results for the compressible Navier-Stokes equations in $\mathbb{R}^N2$, Nonlinearity, 25 (2012), 1345-1349.
[4]	D. Chae, The Liouville type theorems for the steady Navier-Stokes equations and the self-similar Euler equations on $\mathbb{R}^3$, preprint, arXiv:1105.3639v2.
[5]	D. Chae, Conditions on the pressure for vanishing velocity in the incompressible fluid flows, Comm. PDE, 37 (2012), 1445-1455.
[6]	D. Chae, Liouville type of theorems for the Euler and the Navier-Stokes equations, Advances in Math., 228 (2011), 2855-2868. doi: 10.1016/j.aim.2011.07.020.
[7]	D. Chae, On the self-similar solutions of the 3D Euler and the related equations, Comm. Math. Phys., 305 (2011), 333-349. doi: 10.1007/s00220-011-1266-1.
[8]	D. Chae, On the Lagrangian dynamics of the axisymmetric 3D Euler equations, J. Diff. Eqns., 249 (2010), 571-577. doi: 10.1016/j.jde.2010.03.012.
[9]	D. Chae, On the nonexistence of global weak solutions to the Navier-Stokes-Poisson equations in $\mathbb{R}^N2$, Comm. PDE, 35 (2010), 535-557. doi: 10.1080/03605300903473418.
[10]	D. Chae, On the generalized self-similar singularities for the Euler and the Navier-Stokes equations, J. Funct. Anal., 258 (2010), 2865-2883. doi: 10.1016/j.jfa.2010.02.006.
[11]	D. Chae, On the blow-up problem for the axisymmetric 3D Euler equations, Nonlinearity, 21 (2008), 2053-2060. doi: 10.1088/0951-7715/21/9/007.
[12]	D. Chae, Nonexistence of self-similar singularities for the 3D incompressible Euler equations, Comm. Math. Phys., 273 (2007), 203-215. doi: 10.1007/s00220-007-0249-8.
[13]	D. Chae, Nonexistence of asymptotically self-similar singularities in the Euler and the Navier-Stokes equations, Math. Ann., 338 (2007), 435-449. doi: 10.1007/s00208-007-0082-6.
[14]	D. Chae, On the continuation principles for the Euler equations and the quasi-geostrophic equation, J. Diff. Eqns., 227 (2006), 640-651. doi: 10.1016/j.jde.2005.12.013.
[15]	D. Chae, Incompressible Euler Equations: The blow-up problem and related results, in "Handbook of Differential Equations: Evolutionary Equations," Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 1-55. doi: 10.1016/S1874-5717(08)00001-7.
[16]	D. Chae, On the well-posedness of the Euler equations in the Besov and Triebel-Lizorkin spaces, "Tosio Kato's Method and Principle for Evolution Equations in Mathematical Physics," (Sapporo, 2001), Sürikaisekikenkyüsho Kökyüroku, No. 1234 (2001), 42-57.
[17]	D. Chae, K. Kang and J. Lee, Notes on the asymptotically self-similar singularities in the Euler and the Navier-Stokes equations, DCDS-A, 25 (2009), 1181-1193. doi: 10.3934/dcds.2009.25.1181.
[18]	P. Constantin, An Eulerian-Lagrangian approach for incompressible fluids: Local theory, J. Amer. Math. Soc., 14 (2001), 263-278. doi: 10.1090/S0894-0347-00-00364-7.
[19]	P. Constantin, On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc., 44 (2007), 603-621. doi: 10.1090/S0273-0979-07-01184-6.
[20]	P. Constantin, C. Fefferman and A. Majda, Geometric constraints on potential singularity formulation in the 3-D Euler equations, Comm. P.D.E., 21 (1996), 559-571. doi: 10.1080/03605309608821197.
[21]	P. Constantin, P. Lax and A. Majda, A simple one-dimensional model for the three dimensional vorticity equation, Comm. Pure Appl. Math., 38 (1985), 715-724. doi: 10.1002/cpa.3160380605.
[22]	L. Euler, Principes généraux du mouvement des fluides, Mémoires de l'Académie des Sciences de Berlin, 11 (1755), 274-315.
[23]	U. Frisch, T. Matsumoto and J. Bec, Singularities of Euler Flow? Not Out of the Blue!, J. Stat. Phys., 113 (2003), 761-781. doi: 10.1023/A:1027308602344.
[24]	G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems," Springer Tracts in Natural Philosophy, 39, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.
[25]	Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319. doi: 10.1002/cpa.3160380304.
[26]	T. Kato, Nonstationary flows of viscous and ideal fluids in $\mathbb{R}^3$, J. Funct. Anal., 9 (1972), 296-305.
[27]	R. M. Kerr, Vortex collapse and turbulence, Fluid Dynamics Research, 36 (2005), 249-260. doi: 10.1016/j.fluiddyn.2004.09.003.
[28]	G. Koch, N. Nadirashvili, G. Seregin and V. Šverák, Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203 (2009), 83-105. doi: 10.1007/s11511-009-0039-6.
[29]	H. Kozono and Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with applications to the Euler equations, Comm. Math. Phys., 214 (2000), 191-200. doi: 10.1007/s002200000267.
[30]	H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations, Math Z., 242 (2002), 251-278. doi: 10.1007/s002090100332.
[31]	J. Leray, Sur le mouvement d'un fluide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.
[32]	A. Majda and A. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, 27, Cambridge Univ. Press., Cambridge, 2002.
[33]	J. Nečas, M. Ružička and V. Šverák, On Leray's self-similar solutions of the Navier-Stokes equations, Acta Math., 176 (1996), 283-294. doi: 10.1007/BF02551584.
[34]	T.-P. Tsai, On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch. Rat. Mech. Anal., 143 (1998), 29-51. doi: 10.1007/s002050050099.

show all references

References:

[1]	C. Bardos and É. S. Titi, Euler equations of incompressible ideal fluids, Russian Math. Surveys, 62 (2007), 409-451. doi: 10.1070/RM2007v062n03ABEH004410.
[2]	J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 94 (1984), 61-66.
[3]	D. Chae, Remarks on the Liouville type results for the compressible Navier-Stokes equations in $\mathbb{R}^N2$, Nonlinearity, 25 (2012), 1345-1349.
[4]	D. Chae, The Liouville type theorems for the steady Navier-Stokes equations and the self-similar Euler equations on $\mathbb{R}^3$, preprint, arXiv:1105.3639v2.
[5]	D. Chae, Conditions on the pressure for vanishing velocity in the incompressible fluid flows, Comm. PDE, 37 (2012), 1445-1455.
[6]	D. Chae, Liouville type of theorems for the Euler and the Navier-Stokes equations, Advances in Math., 228 (2011), 2855-2868. doi: 10.1016/j.aim.2011.07.020.
[7]	D. Chae, On the self-similar solutions of the 3D Euler and the related equations, Comm. Math. Phys., 305 (2011), 333-349. doi: 10.1007/s00220-011-1266-1.
[8]	D. Chae, On the Lagrangian dynamics of the axisymmetric 3D Euler equations, J. Diff. Eqns., 249 (2010), 571-577. doi: 10.1016/j.jde.2010.03.012.
[9]	D. Chae, On the nonexistence of global weak solutions to the Navier-Stokes-Poisson equations in $\mathbb{R}^N2$, Comm. PDE, 35 (2010), 535-557. doi: 10.1080/03605300903473418.
[10]	D. Chae, On the generalized self-similar singularities for the Euler and the Navier-Stokes equations, J. Funct. Anal., 258 (2010), 2865-2883. doi: 10.1016/j.jfa.2010.02.006.
[11]	D. Chae, On the blow-up problem for the axisymmetric 3D Euler equations, Nonlinearity, 21 (2008), 2053-2060. doi: 10.1088/0951-7715/21/9/007.
[12]	D. Chae, Nonexistence of self-similar singularities for the 3D incompressible Euler equations, Comm. Math. Phys., 273 (2007), 203-215. doi: 10.1007/s00220-007-0249-8.
[13]	D. Chae, Nonexistence of asymptotically self-similar singularities in the Euler and the Navier-Stokes equations, Math. Ann., 338 (2007), 435-449. doi: 10.1007/s00208-007-0082-6.
[14]	D. Chae, On the continuation principles for the Euler equations and the quasi-geostrophic equation, J. Diff. Eqns., 227 (2006), 640-651. doi: 10.1016/j.jde.2005.12.013.
[15]	D. Chae, Incompressible Euler Equations: The blow-up problem and related results, in "Handbook of Differential Equations: Evolutionary Equations," Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 1-55. doi: 10.1016/S1874-5717(08)00001-7.
[16]	D. Chae, On the well-posedness of the Euler equations in the Besov and Triebel-Lizorkin spaces, "Tosio Kato's Method and Principle for Evolution Equations in Mathematical Physics," (Sapporo, 2001), Sürikaisekikenkyüsho Kökyüroku, No. 1234 (2001), 42-57.
[17]	D. Chae, K. Kang and J. Lee, Notes on the asymptotically self-similar singularities in the Euler and the Navier-Stokes equations, DCDS-A, 25 (2009), 1181-1193. doi: 10.3934/dcds.2009.25.1181.
[18]	P. Constantin, An Eulerian-Lagrangian approach for incompressible fluids: Local theory, J. Amer. Math. Soc., 14 (2001), 263-278. doi: 10.1090/S0894-0347-00-00364-7.
[19]	P. Constantin, On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc., 44 (2007), 603-621. doi: 10.1090/S0273-0979-07-01184-6.
[20]	P. Constantin, C. Fefferman and A. Majda, Geometric constraints on potential singularity formulation in the 3-D Euler equations, Comm. P.D.E., 21 (1996), 559-571. doi: 10.1080/03605309608821197.
[21]	P. Constantin, P. Lax and A. Majda, A simple one-dimensional model for the three dimensional vorticity equation, Comm. Pure Appl. Math., 38 (1985), 715-724. doi: 10.1002/cpa.3160380605.
[22]	L. Euler, Principes généraux du mouvement des fluides, Mémoires de l'Académie des Sciences de Berlin, 11 (1755), 274-315.
[23]	U. Frisch, T. Matsumoto and J. Bec, Singularities of Euler Flow? Not Out of the Blue!, J. Stat. Phys., 113 (2003), 761-781. doi: 10.1023/A:1027308602344.
[24]	G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems," Springer Tracts in Natural Philosophy, 39, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.
[25]	Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319. doi: 10.1002/cpa.3160380304.
[26]	T. Kato, Nonstationary flows of viscous and ideal fluids in $\mathbb{R}^3$, J. Funct. Anal., 9 (1972), 296-305.
[27]	R. M. Kerr, Vortex collapse and turbulence, Fluid Dynamics Research, 36 (2005), 249-260. doi: 10.1016/j.fluiddyn.2004.09.003.
[28]	G. Koch, N. Nadirashvili, G. Seregin and V. Šverák, Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203 (2009), 83-105. doi: 10.1007/s11511-009-0039-6.
[29]	H. Kozono and Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with applications to the Euler equations, Comm. Math. Phys., 214 (2000), 191-200. doi: 10.1007/s002200000267.
[30]	H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations, Math Z., 242 (2002), 251-278. doi: 10.1007/s002090100332.
[31]	J. Leray, Sur le mouvement d'un fluide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.
[32]	A. Majda and A. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, 27, Cambridge Univ. Press., Cambridge, 2002.
[33]	J. Nečas, M. Ružička and V. Šverák, On Leray's self-similar solutions of the Navier-Stokes equations, Acta Math., 176 (1996), 283-294. doi: 10.1007/BF02551584.
[34]	T.-P. Tsai, On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch. Rat. Mech. Anal., 143 (1998), 29-51. doi: 10.1007/s002050050099.

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