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September  2012, 11(5): 1809-1823. doi: 10.3934/cpaa.2012.11.1809

On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion

1. 

Department of Mathematical and Statistical Sciences, University of Alberta, 632 CAB, Edmonton, AB T6G 2G1, Canada

2. 

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1

Received  March 2011 Revised  August 2011 Published  March 2012

In this article we study local and global well-posedness of the Lagrangian Averaged Euler equations. We show local well-posedness in Triebel-Lizorkin spaces and further prove a Beale-Kato-Majda type necessary and sufficient condition for global existence involving the stream function. We also establish new sufficient conditions for global existence in terms of mixed Lebesgue norms of the generalized Clebsch variables.
Citation: Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809
References:
[1]

Robert A. Adams, "Sobolev Spaces,", Academic Press New York, (1975). Google Scholar

[2]

Claude Bardos, Jasmine S. Linshiz and Edriss S. Titi, Global regularity and convergence of a Birkhoff-Rott-$\alpha$ approximation of the dynamics of vortex sheets of the 2D Euler equaitons,, Comm. Pure. Appl. Math., 63 (2010), 697. doi: 10.1002/cpa.20305. Google Scholar

[3]

Dongho Chae, On the well-posedness of the Euler equations in the Besov and Triebel-Lizorkin spaces,, in: Tosio Kato's Method and Principle for Evolution Equations in Mathematical Physics, (2002), 27. Google Scholar

[4]

Dongho Chae, On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces,, Comm. Pure Appl. Math., 55 (2002), 654. doi: 10.1002/cpa.10029. Google Scholar

[5]

Shiyi Chen, Ciprian Foias, Darryl D. Holm, Eric J. Olson, Edriss S. Titi and Shannon Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flows,, {Phys. Fluids, 11 (1999), 2343. doi: 10.1063/1.870096. Google Scholar

[6]

Shiyi Chen, Darryl D. Holm, Len G. Margolin and Raoyang Zhang, Direct numerical simulations of the Navier-Stokes alpha model,, {Phys. D., 133 (1999), 66. doi: 10.1016/S0167-2789(99)00099-8. Google Scholar

[7]

Qionglei Chen, Changxing Miao and Zhifei Zhang, On the well-posedness of the ideal MHD equations in the Triebel-Lizorkin spaces,, {Arch. Rational Mech. Anal., 195 (2010), 561. doi: 10.1007/s00205-008-0213-6. Google Scholar

[8]

Alexandre J. Chorin and Jerrold E. Marsden, "A Mathemtaical Introduction to Fluid Mechanics,", {3rd ed., (1993). Google Scholar

[9]

Jian Deng, Thomas Y. Hou and Xinwei Yu, A level set formulation for the 3D incompressible Euler equations,, {Methods Appl. Anal., 12 (2005), 427. Google Scholar

[10]

Ciprian Foias, Darryl D. Holm and Edriss S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory,, {J. Dynam. Differential Equations, 14 (2002), 1. doi: 10.1023/A:1012984210582. Google Scholar

[11]

C. Robin Graham and Frank S. Henyey, Clebsch representation near points where the vorticity vanishes,, {Physics of Fluids, 12 (2000), 744. doi: 10.1063/1.870331. Google Scholar

[12]

Darryl D. Holm, Jerrold E. Marsden and Tudor S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion,, {Phys. Rev. Lett., 349 (1998), 4173. doi: 10.1103/PhysRevLett.80.4173. Google Scholar

[13]

Darryl D. Holm, Jerrold E. Marsden and Tudor S. Ratiu, Euler-Poincaré equations and semidirect products with applications to continuum theories,, {Adv. Math., 137 (1998), 1. doi: 10.1006/aima.1998.1721. Google Scholar

[14]

Darryl D. Holm, Monika Nitsche and Vakhtang Putkaradze, Euler-alpha and vortex blob regularization of vortex filament and vortex sheet motion,, {J. Fluid Mech., 555 (2006), 149. doi: 10.1017/S0022112006008846. Google Scholar

[15]

Thomas Y. Hou and Congming Li, On global well-posedness of the Lagrangian averaged Euler equations,, {SIAM J. Math. Anal.}, 38 (2006), 782. Google Scholar

[16]

Quansen Jiu, Dongjuan Niu, Edriss S. Titi and Zhouping Xin, Axisymmetric Euler-$\alpha$ equations without swirl: existence, uniqueness, and rodon measure valued solutions,, preprint, (). Google Scholar

[17]

Ram P. Kanwal, "Generalized Functons Theory and Technique,", {Academic press}, (1983). Google Scholar

[18]

Jasmine S. Linshiz and Edriss S. Titi, On the convergence rate of the Euler-$\alpha$, an inviscid second-grade complex fluid, model to the Euler equations,, {J. Stat. Phys., 138 (2010), 305. doi: 10.1007/s10955-009-9916-9. Google Scholar

[19]

Xiaofeng Liu, Meng Wang and Zhifei Zhang, A note on the blowup criterion of the Lagrangian averaged Euler equations,, {Nonlinear Anal., 67 (2007), 2447. doi: 10.1016/j.na.2006.08.051. Google Scholar

[20]

Xiaofeng Liu and Houyu Jia, Local existence and blow-up criterion of the Lagrangian averaged Euler equations in Besov spaces,, {Comm. Pure and Appl. Anal., 7 (2008), 845. doi: 10.3934/cpaa.2008.7.845. Google Scholar

[21]

Jerrold E. Marsden and Steve Shkoller, Global well-posedness for the Lagrangian Navier-Stokes (LANS-$\alpha$) equations on bounded domains,, {R. Soc. Lond. Philos. Tran. Ser. A Math. Phys. Eng. Sci., 359 (2001), 1449. doi: 10.1098/rsta.2001.0852. Google Scholar

[22]

Jerrold E. Marseden, Tudor S. Ratiu and Steve Shkoller, The geometry and analysis of the averaged Euler equations and a new differmophism group,, {Geom. Funct. Anal.}, 10 (2000), 582. doi: 10.1007/PL00001631. Google Scholar

[23]

Nader Masmoudi, Remarks about the inviscid limit of the Navier-Stokes system,, {Comm. Math. Phys., 270 (2007), 777. doi: 10.1007/s00220-006-0171-5. Google Scholar

[24]

Marcel Oliver and Steve Shkoller, The vortex blob method as a second-grade non-Newtononian fluid,, {Comm. Partial Differential Equations, 26 (2001), 295. doi: 10.1081/PDE-100001756. Google Scholar

[25]

Jaak Peetre, "New Thoughts on Besov Spaces,", {Duke University Press}, (1976). Google Scholar

[26]

Thomas Runst and Winfried Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Operators,", {de Gruyter, (1996). Google Scholar

[27]

Elias M. Stein, "Singular Integrals and Differentiability Properties of Functions,", {Princeton University Press, (1970). Google Scholar

[28]

Hans Triebel, "Theorey of Function Spaces,", {Monograohs in Mathematics, (1983). Google Scholar

[29]

Hans Triebel, "Theorey of Function Spaces II,", {Monograohs in Mathematics, (1992). Google Scholar

show all references

References:
[1]

Robert A. Adams, "Sobolev Spaces,", Academic Press New York, (1975). Google Scholar

[2]

Claude Bardos, Jasmine S. Linshiz and Edriss S. Titi, Global regularity and convergence of a Birkhoff-Rott-$\alpha$ approximation of the dynamics of vortex sheets of the 2D Euler equaitons,, Comm. Pure. Appl. Math., 63 (2010), 697. doi: 10.1002/cpa.20305. Google Scholar

[3]

Dongho Chae, On the well-posedness of the Euler equations in the Besov and Triebel-Lizorkin spaces,, in: Tosio Kato's Method and Principle for Evolution Equations in Mathematical Physics, (2002), 27. Google Scholar

[4]

Dongho Chae, On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces,, Comm. Pure Appl. Math., 55 (2002), 654. doi: 10.1002/cpa.10029. Google Scholar

[5]

Shiyi Chen, Ciprian Foias, Darryl D. Holm, Eric J. Olson, Edriss S. Titi and Shannon Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flows,, {Phys. Fluids, 11 (1999), 2343. doi: 10.1063/1.870096. Google Scholar

[6]

Shiyi Chen, Darryl D. Holm, Len G. Margolin and Raoyang Zhang, Direct numerical simulations of the Navier-Stokes alpha model,, {Phys. D., 133 (1999), 66. doi: 10.1016/S0167-2789(99)00099-8. Google Scholar

[7]

Qionglei Chen, Changxing Miao and Zhifei Zhang, On the well-posedness of the ideal MHD equations in the Triebel-Lizorkin spaces,, {Arch. Rational Mech. Anal., 195 (2010), 561. doi: 10.1007/s00205-008-0213-6. Google Scholar

[8]

Alexandre J. Chorin and Jerrold E. Marsden, "A Mathemtaical Introduction to Fluid Mechanics,", {3rd ed., (1993). Google Scholar

[9]

Jian Deng, Thomas Y. Hou and Xinwei Yu, A level set formulation for the 3D incompressible Euler equations,, {Methods Appl. Anal., 12 (2005), 427. Google Scholar

[10]

Ciprian Foias, Darryl D. Holm and Edriss S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory,, {J. Dynam. Differential Equations, 14 (2002), 1. doi: 10.1023/A:1012984210582. Google Scholar

[11]

C. Robin Graham and Frank S. Henyey, Clebsch representation near points where the vorticity vanishes,, {Physics of Fluids, 12 (2000), 744. doi: 10.1063/1.870331. Google Scholar

[12]

Darryl D. Holm, Jerrold E. Marsden and Tudor S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion,, {Phys. Rev. Lett., 349 (1998), 4173. doi: 10.1103/PhysRevLett.80.4173. Google Scholar

[13]

Darryl D. Holm, Jerrold E. Marsden and Tudor S. Ratiu, Euler-Poincaré equations and semidirect products with applications to continuum theories,, {Adv. Math., 137 (1998), 1. doi: 10.1006/aima.1998.1721. Google Scholar

[14]

Darryl D. Holm, Monika Nitsche and Vakhtang Putkaradze, Euler-alpha and vortex blob regularization of vortex filament and vortex sheet motion,, {J. Fluid Mech., 555 (2006), 149. doi: 10.1017/S0022112006008846. Google Scholar

[15]

Thomas Y. Hou and Congming Li, On global well-posedness of the Lagrangian averaged Euler equations,, {SIAM J. Math. Anal.}, 38 (2006), 782. Google Scholar

[16]

Quansen Jiu, Dongjuan Niu, Edriss S. Titi and Zhouping Xin, Axisymmetric Euler-$\alpha$ equations without swirl: existence, uniqueness, and rodon measure valued solutions,, preprint, (). Google Scholar

[17]

Ram P. Kanwal, "Generalized Functons Theory and Technique,", {Academic press}, (1983). Google Scholar

[18]

Jasmine S. Linshiz and Edriss S. Titi, On the convergence rate of the Euler-$\alpha$, an inviscid second-grade complex fluid, model to the Euler equations,, {J. Stat. Phys., 138 (2010), 305. doi: 10.1007/s10955-009-9916-9. Google Scholar

[19]

Xiaofeng Liu, Meng Wang and Zhifei Zhang, A note on the blowup criterion of the Lagrangian averaged Euler equations,, {Nonlinear Anal., 67 (2007), 2447. doi: 10.1016/j.na.2006.08.051. Google Scholar

[20]

Xiaofeng Liu and Houyu Jia, Local existence and blow-up criterion of the Lagrangian averaged Euler equations in Besov spaces,, {Comm. Pure and Appl. Anal., 7 (2008), 845. doi: 10.3934/cpaa.2008.7.845. Google Scholar

[21]

Jerrold E. Marsden and Steve Shkoller, Global well-posedness for the Lagrangian Navier-Stokes (LANS-$\alpha$) equations on bounded domains,, {R. Soc. Lond. Philos. Tran. Ser. A Math. Phys. Eng. Sci., 359 (2001), 1449. doi: 10.1098/rsta.2001.0852. Google Scholar

[22]

Jerrold E. Marseden, Tudor S. Ratiu and Steve Shkoller, The geometry and analysis of the averaged Euler equations and a new differmophism group,, {Geom. Funct. Anal.}, 10 (2000), 582. doi: 10.1007/PL00001631. Google Scholar

[23]

Nader Masmoudi, Remarks about the inviscid limit of the Navier-Stokes system,, {Comm. Math. Phys., 270 (2007), 777. doi: 10.1007/s00220-006-0171-5. Google Scholar

[24]

Marcel Oliver and Steve Shkoller, The vortex blob method as a second-grade non-Newtononian fluid,, {Comm. Partial Differential Equations, 26 (2001), 295. doi: 10.1081/PDE-100001756. Google Scholar

[25]

Jaak Peetre, "New Thoughts on Besov Spaces,", {Duke University Press}, (1976). Google Scholar

[26]

Thomas Runst and Winfried Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Operators,", {de Gruyter, (1996). Google Scholar

[27]

Elias M. Stein, "Singular Integrals and Differentiability Properties of Functions,", {Princeton University Press, (1970). Google Scholar

[28]

Hans Triebel, "Theorey of Function Spaces,", {Monograohs in Mathematics, (1983). Google Scholar

[29]

Hans Triebel, "Theorey of Function Spaces II,", {Monograohs in Mathematics, (1992). Google Scholar

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