• Previous Article
    Statistical properties of one-dimensional expanding maps with singularities of low regularity
  • DCDS Home
  • This Issue
  • Next Article
    Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem
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

[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

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

[1]

Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 557-586. doi: 10.3934/cpaa.2012.11.557

[2]

Frederic Rousset. The residual boundary conditions coming from the real vanishing viscosity method. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 605-625. doi: 10.3934/dcds.2002.8.606

[3]

Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015

[4]

Thomas Strömberg. A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit. Communications on Pure & Applied Analysis, 2011, 10 (2) : 479-506. doi: 10.3934/cpaa.2011.10.479

[5]

Pierluigi Colli, Gianni Gilardi, Pavel Krejčí, Jürgen Sprekels. A vanishing diffusion limit in a nonstandard system of phase field equations. Evolution Equations & Control Theory, 2014, 3 (2) : 257-275. doi: 10.3934/eect.2014.3.257

[6]

Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503

[7]

Fabio Ramos, Edriss S. Titi. Invariant measures for the $3$D Navier-Stokes-Voigt equations and their Navier-Stokes limit. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 375-403. doi: 10.3934/dcds.2010.28.375

[8]

Ciprian G. Gal, Maurizio Grasselli. Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1581-1610. doi: 10.3934/dcdsb.2013.18.1581

[9]

Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673

[10]

Shu Wang, Chundi Liu. Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2177-2199. doi: 10.3934/cpaa.2017108

[11]

Motohiro Sobajima. On the threshold for Kato's selfadjointness problem and its $L^p$-generalization. Evolution Equations & Control Theory, 2014, 3 (4) : 699-711. doi: 10.3934/eect.2014.3.699

[12]

Jing Wang, Lining Tong. Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers. Communications on Pure & Applied Analysis, 2019, 18 (2) : 887-910. doi: 10.3934/cpaa.2019043

[13]

Huahui Li, Zhiqiang Shao. Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2373-2400. doi: 10.3934/cpaa.2016041

[14]

Luigi C. Berselli, Placido Longo. Classical solutions for the system $\bf {\text{curl}\, v = g}$, with vanishing Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 215-229. doi: 10.3934/dcdss.2019015

[15]

Jiang Xu, Ting Zhang. Zero-electron-mass limit of Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4743-4768. doi: 10.3934/dcds.2013.33.4743

[16]

Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. Dynamic boundary conditions as limit of singularly perturbed parabolic problems. Conference Publications, 2011, 2011 (Special) : 737-746. doi: 10.3934/proc.2011.2011.737

[17]

Alberto Bressan, Yilun Jiang. The vanishing viscosity limit for a system of H-J equations related to a debt management problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 793-824. doi: 10.3934/dcdss.2018050

[18]

Mapundi K. Banda, Michael Herty, Axel Klar. Coupling conditions for gas networks governed by the isothermal Euler equations. Networks & Heterogeneous Media, 2006, 1 (2) : 295-314. doi: 10.3934/nhm.2006.1.295

[19]

Sasho Popov, Jean-Marie Strelcyn. The Euler-Poisson equations: An elementary approach to integrability conditions. Journal of Geometric Mechanics, 2018, 10 (3) : 293-329. doi: 10.3934/jgm.2018011

[20]

Yanmin Niu, Xiong Li. An application of Moser's twist theorem to superlinear impulsive differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 431-445. doi: 10.3934/dcds.2019017

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (108)
  • HTML views (160)
  • Cited by (0)

Other articles
by authors

[Back to Top]