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September  2019, 39(9): 4945-4953. doi: 10.3934/dcds.2019202

Kato's type theorems for the convergence of Euler-Voigt equations to Euler equations with Drichlet boundary conditions

The Center of Applied Mathematics, Yichun University, Yichun, Jiangxi 336000, China

Received  March 2018 Revised  November 2018 Published  May 2019

After investigating existence and uniqueness of the global strong solutions for Euler-Voigt equations under Dirichlet conditions, we obtain the Kato's type theorems for the convergence of the Euler-Voigt equations to Euler equations. More precisely, the necessary and sufficient conditions that the solution of Euler-Voigt equation converges to the one of Euler equations, as $ \alpha\to 0 $, can be obtained.

Citation: Aibin Zang. Kato's type theorems for the convergence of Euler-Voigt equations to Euler equations with Drichlet boundary conditions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 4945-4953. doi: 10.3934/dcds.2019202
References:
[1]

C. Bardos and E. S. Titi, Mathematics and turbulence: where do we stand?, Journal of Turbulence, 14 (2013), 42-76.  doi: 10.1080/14685248.2013.771838.  Google Scholar

[2]

Y. CaoE. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.  Google Scholar

[3]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341.  doi: 10.1103/PhysRevLett.81.5338.  Google Scholar

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S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Physica D, 133 (1999), 49-65.  doi: 10.1016/S0167-2789(99)00098-6.  Google Scholar

[5]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes. The International Conference on Turbulence(Los Alamos, NM, 1998), Phys. Fluids, 11 (1999), 2343-2353.  doi: 10.1063/1.870096.  Google Scholar

[6]

A. CheskidovD. D. HolmE. Olson and E. S. Titi, On a Leray-a model of turbulence, Proc. R. Soc. Lond. Ser. A Math.Phys. Eng. Sci., 461 (2005), 629-649.  doi: 10.1098/rspa.2004.1373.  Google Scholar

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T. ClopeauA. Mikelić and R. Robert, On the vanishing viscosity limit for the $2D$ incompressible Navier-Stokes equations with the friction type boundary condition, Nonlinearity, 11 (1998), 1625-1636.  doi: 10.1088/0951-7715/11/6/011.  Google Scholar

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P. Constantin, Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations, Commun. Math. Phys., 104 (1986), 311-326.  doi: 10.1007/BF01211598.  Google Scholar

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P. Constantin, Euler equations, Navier–Stokes equations and turbulence, in Mathematical Foundation of Turbulent Viscous Flows (eds. M. Cannone and T. Miyakawa), Springer Lecture Notes in Mathematics, 1871 (2005), 1–43. doi: 10.1007/11545989_1.  Google Scholar

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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.  Google Scholar

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J. E. Dunn and R. L. Fosdick, Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Rational Mech. Anal., 56 (1974), 191-252.  doi: 10.1007/BF00280970.  Google Scholar

[12]

C. FoiasD. D. Holm and E. S. Titi, The three-dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. Differ. Equ., 14 (2002), 1-35.  doi: 10.1023/A:1012984210582.  Google Scholar

[13]

D. D. Holm and E. S. Titi, Computational models of turbulence: The LANS-a model and the role of global analysis, SIAM News, 38 (2005), 1-5.   Google Scholar

[14]

D. D. HolmJ. E. Marsden and T. S. Ratiu, The Euler-Pointcare equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721.  Google Scholar

[15]

D. D. HolmJ. E. Marsden and T. S. Ratiu, Euler-Pointcare models of ideal fluids with nonlinear dispersion., Phys. Rev. Lett., 88 (1998), 4173-4176.   Google Scholar

[16]

A. A. IlyinE. M. Lunasin and E. S. Titi, A modified-Leray-a subgrid scale model of turbulence, Nonlinearity, 19 (2006), 879-897.  doi: 10.1088/0951-7715/19/4/006.  Google Scholar

[17]

T. Kato, Remarks on zero viscosity limit for Nonstationary Navier-Stokes flows with boundary, in Seminar on Nonlinear Partial Differential Differential Equations, (Editor S.S. Chern) Mathematical Sciences Research institute Publications, New York, (1984), 85–98. doi: 10.1007/978-1-4612-1110-5_6.  Google Scholar

[18]

A. Larios and E. S. Titi, On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models, Disc.Cont. Dyn. System Ser. B, 14 (2010), 603-627.  doi: 10.3934/dcdsb.2010.14.603.  Google Scholar

[19]

A. Larios and E. S. Titi, Higher-Order Global Regularity of an Inviscid Voigt-Regularization of the Three-Dimensional Inviscid Resistive Magnetohydrodynamic Equations, J. Math. Fluid Mech., 16 (2014), 59-76.  doi: 10.1007/s00021-013-0136-3.  Google Scholar

[20]

J. S. Linshitz and E. S. Titi, On the convergence rate of the Euler-$\alpha$, an inviscid second-grade complex fluid, model to Euler equations, J. Stat. Phys., 138 (2010), 305-332.  doi: 10.1007/s10955-009-9916-9.  Google Scholar

[21]

M. C. Lopes FilhoH. J. Nussenzveig Lopes and G. Planas, On the inviscid limit for 2D incompressible flow with Navier friction condition, SIAM J. Math. Anal., 36 (2005), 1130-1141.  doi: 10.1137/S0036141003432341.  Google Scholar

[22]

M. C. Lopes FilhoH. J. Nussenzveig LopesE. S. Titi and A. Zang, Convergence of the 2D Euler-$\alpha$ to Euler equations in the Dirichlet case: Indifference to boundary layers, Physica D., 292/293 (2015), 51-61.  doi: 10.1016/j.physd.2014.11.001.  Google Scholar

[23]

M. C. Lopes FilhoH. J. Nussenzveig LopesE. S. Titi and A. Zang, On the Approximation of 2D Euler Equations by Second-Grade Fluid with Dirichlet Boundary Conditions, J. Math. Fluid Mech., 17 (2015), 327-340.  doi: 10.1007/s00021-015-0207-8.  Google Scholar

[24]

N. Masmoudi, Remarks about the inviscid limit of the Navier-Stokes system, Comm. Math. Phys., 270 (2007), 777-788.  doi: 10.1007/s00220-006-0171-5.  Google Scholar

[25]

N. Masmoudi and F. Rousset, Uniform regularity for the Navier-Stokes equation with Navier boundary condition, Arch. Rational Mech. Anal., 203 (2012), 529-575.  doi: 10.1007/s00205-011-0456-5.  Google Scholar

[26]

X. Wang, A Kato type theorem on zero viscosity limit of Navier-Stokes flows, Indiana Univ. Math. J., 50 (2001), 223-241.  doi: 10.1512/iumj.2001.50.2098.  Google Scholar

[27]

L. WangZ. Xin and A. Zang, Vanishing viscous limits for 3D Navier-Stokes equations with a Navier-slip boundary condition, J. Math. Fluid Mech., 14 (2012), 791-825.  doi: 10.1007/s00021-012-0103-4.  Google Scholar

[28]

Y. Xiao and Z. Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math., 60 (2007), 1027-1055.  doi: 10.1002/cpa.20187.  Google Scholar

[29]

A. Zang, Global well-posedness for Euler-Voigt Equations (In Chinese), Pure and Applied Mathematics, 34 (2018), 1-6.   Google Scholar

show all references

References:
[1]

C. Bardos and E. S. Titi, Mathematics and turbulence: where do we stand?, Journal of Turbulence, 14 (2013), 42-76.  doi: 10.1080/14685248.2013.771838.  Google Scholar

[2]

Y. CaoE. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.  Google Scholar

[3]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341.  doi: 10.1103/PhysRevLett.81.5338.  Google Scholar

[4]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Physica D, 133 (1999), 49-65.  doi: 10.1016/S0167-2789(99)00098-6.  Google Scholar

[5]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes. The International Conference on Turbulence(Los Alamos, NM, 1998), Phys. Fluids, 11 (1999), 2343-2353.  doi: 10.1063/1.870096.  Google Scholar

[6]

A. CheskidovD. D. HolmE. Olson and E. S. Titi, On a Leray-a model of turbulence, Proc. R. Soc. Lond. Ser. A Math.Phys. Eng. Sci., 461 (2005), 629-649.  doi: 10.1098/rspa.2004.1373.  Google Scholar

[7]

T. ClopeauA. Mikelić and R. Robert, On the vanishing viscosity limit for the $2D$ incompressible Navier-Stokes equations with the friction type boundary condition, Nonlinearity, 11 (1998), 1625-1636.  doi: 10.1088/0951-7715/11/6/011.  Google Scholar

[8]

P. Constantin, Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations, Commun. Math. Phys., 104 (1986), 311-326.  doi: 10.1007/BF01211598.  Google Scholar

[9]

P. Constantin, Euler equations, Navier–Stokes equations and turbulence, in Mathematical Foundation of Turbulent Viscous Flows (eds. M. Cannone and T. Miyakawa), Springer Lecture Notes in Mathematics, 1871 (2005), 1–43. doi: 10.1007/11545989_1.  Google Scholar

[10]

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.  Google Scholar

[11]

J. E. Dunn and R. L. Fosdick, Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Rational Mech. Anal., 56 (1974), 191-252.  doi: 10.1007/BF00280970.  Google Scholar

[12]

C. FoiasD. D. Holm and E. S. Titi, The three-dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. Differ. Equ., 14 (2002), 1-35.  doi: 10.1023/A:1012984210582.  Google Scholar

[13]

D. D. Holm and E. S. Titi, Computational models of turbulence: The LANS-a model and the role of global analysis, SIAM News, 38 (2005), 1-5.   Google Scholar

[14]

D. D. HolmJ. E. Marsden and T. S. Ratiu, The Euler-Pointcare equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721.  Google Scholar

[15]

D. D. HolmJ. E. Marsden and T. S. Ratiu, Euler-Pointcare models of ideal fluids with nonlinear dispersion., Phys. Rev. Lett., 88 (1998), 4173-4176.   Google Scholar

[16]

A. A. IlyinE. M. Lunasin and E. S. Titi, A modified-Leray-a subgrid scale model of turbulence, Nonlinearity, 19 (2006), 879-897.  doi: 10.1088/0951-7715/19/4/006.  Google Scholar

[17]

T. Kato, Remarks on zero viscosity limit for Nonstationary Navier-Stokes flows with boundary, in Seminar on Nonlinear Partial Differential Differential Equations, (Editor S.S. Chern) Mathematical Sciences Research institute Publications, New York, (1984), 85–98. doi: 10.1007/978-1-4612-1110-5_6.  Google Scholar

[18]

A. Larios and E. S. Titi, On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models, Disc.Cont. Dyn. System Ser. B, 14 (2010), 603-627.  doi: 10.3934/dcdsb.2010.14.603.  Google Scholar

[19]

A. Larios and E. S. Titi, Higher-Order Global Regularity of an Inviscid Voigt-Regularization of the Three-Dimensional Inviscid Resistive Magnetohydrodynamic Equations, J. Math. Fluid Mech., 16 (2014), 59-76.  doi: 10.1007/s00021-013-0136-3.  Google Scholar

[20]

J. S. Linshitz and E. S. Titi, On the convergence rate of the Euler-$\alpha$, an inviscid second-grade complex fluid, model to Euler equations, J. Stat. Phys., 138 (2010), 305-332.  doi: 10.1007/s10955-009-9916-9.  Google Scholar

[21]

M. C. Lopes FilhoH. J. Nussenzveig Lopes and G. Planas, On the inviscid limit for 2D incompressible flow with Navier friction condition, SIAM J. Math. Anal., 36 (2005), 1130-1141.  doi: 10.1137/S0036141003432341.  Google Scholar

[22]

M. C. Lopes FilhoH. J. Nussenzveig LopesE. S. Titi and A. Zang, Convergence of the 2D Euler-$\alpha$ to Euler equations in the Dirichlet case: Indifference to boundary layers, Physica D., 292/293 (2015), 51-61.  doi: 10.1016/j.physd.2014.11.001.  Google Scholar

[23]

M. C. Lopes FilhoH. J. Nussenzveig LopesE. S. Titi and A. Zang, On the Approximation of 2D Euler Equations by Second-Grade Fluid with Dirichlet Boundary Conditions, J. Math. Fluid Mech., 17 (2015), 327-340.  doi: 10.1007/s00021-015-0207-8.  Google Scholar

[24]

N. Masmoudi, Remarks about the inviscid limit of the Navier-Stokes system, Comm. Math. Phys., 270 (2007), 777-788.  doi: 10.1007/s00220-006-0171-5.  Google Scholar

[25]

N. Masmoudi and F. Rousset, Uniform regularity for the Navier-Stokes equation with Navier boundary condition, Arch. Rational Mech. Anal., 203 (2012), 529-575.  doi: 10.1007/s00205-011-0456-5.  Google Scholar

[26]

X. Wang, A Kato type theorem on zero viscosity limit of Navier-Stokes flows, Indiana Univ. Math. J., 50 (2001), 223-241.  doi: 10.1512/iumj.2001.50.2098.  Google Scholar

[27]

L. WangZ. Xin and A. Zang, Vanishing viscous limits for 3D Navier-Stokes equations with a Navier-slip boundary condition, J. Math. Fluid Mech., 14 (2012), 791-825.  doi: 10.1007/s00021-012-0103-4.  Google Scholar

[28]

Y. Xiao and Z. Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math., 60 (2007), 1027-1055.  doi: 10.1002/cpa.20187.  Google Scholar

[29]

A. Zang, Global well-posedness for Euler-Voigt Equations (In Chinese), Pure and Applied Mathematics, 34 (2018), 1-6.   Google Scholar

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