### AIMS Mathematics

2019, Issue 1: 1-11. doi: 10.3934/Math.2019.1.1
Research article Topical Sections

# A note on the Euler–Voigt system in a 3D bounded domain: Propagation of singularities and absence of the boundary layer

• Received: 24 September 2018 Accepted: 09 December 2018 Published: 18 December 2018
• MSC : 35Q35(76F02, 76B99, 76D05)

• We consider the Euler–Voigt equations in a smooth bounded domain as an approximation for the 3D Euler equations. We show that the solutions of the Voigt equations are global, do not smooth out the data, and converge to the solutions of the Euler equations. For these reasons they represent a good model, also for computations of turbulent flows.

Citation: Luigi C. Berselli, Davide Catania. A note on the Euler–Voigt system in a 3D bounded domain: Propagation of singularities and absence of the boundary layer[J]. AIMS Mathematics, 2019, 4(1): 1-11. doi: 10.3934/Math.2019.1.1

### Related Papers:

• We consider the Euler–Voigt equations in a smooth bounded domain as an approximation for the 3D Euler equations. We show that the solutions of the Voigt equations are global, do not smooth out the data, and converge to the solutions of the Euler equations. For these reasons they represent a good model, also for computations of turbulent flows.

 [1] C.Amrouche and A.Rejaiba, Lp-theory for Stokes and Navier-Stokes equations with Navier boundary condition, J. Differ. Equations, 256 (2014), 1515-1547. [2] H.Beirão da Veiga, Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions, Advances in Differential Equations, 9 (2004), 1079-1114. [3] H.Beirão da Veiga, On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions, Comm. Pure Appl. Math., 58 (2005), 552-577. [4] H.Beirão da Veiga, Regularity of solutions to a non-homogeneous boundary value problem for general Stokes systems in $R_+^n$, Math. Ann., 331 (2005), 203-217. [5] H.Beirão da Veiga and F.Crispo, Concerning the Wk, p-inviscid limit for 3-D flows under a slip boundary condition, J. Math. Fluid Mech., 13 (2011), 117-135. [6] H.Beirão da Veiga and F.Crispo, A missed persistence property for the Euler equations and its effect on inviscid limits, Nonlinearity, 25 (2012), 1661-1669. [7] L.C. Berselli, Some results on the Navier-Stokes equations with Navier boundary conditions, Riv. Math. Univ. Parma (N.S.), 1 (2010), 1-75. [8] L.C. Berselli and L.Bisconti, On the structural stability of the Euler-Voigt and Navier-Stokes-Voigt models, Nonlinear Anal-Theor, 75 (2012), 117-130. [9] L.C. Berselli, T.Iliescu and W.J. Layton, Mathematics of Large Eddy Simulation of turbulent flows, Scientific Computation, Springer-Verlag, Berlin, 2006. [10] L.C. Berselli and M.Romito, On the existence and uniqueness of weak solutions for a vorticity seeding model, SIAM J. Math. Anal., 37 (2006), 1780-1799. [11] J.P. Bourguignon and H.Brezis, Remarks on the Euler equation, J. Funct. Anal., 15 (1974), 341-363. [12] A.V. Busuioc, D.Iftimie, M.C. LopesFilho, et al. Incompressible Euler as a limit of complex fluid models with Navier boundary conditions, J. Differ. Equations, 252 (2012), 624-640. [13] Y.Cao, E.M. 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. [14] R.W. Carroll and R.E. Showalter, Singular and degenerate Cauchy problems, Vol. 127, Academic Press, 1977. [15] T.ChacónRebollo and R.Lewandowski, Mathematical and numerical foundations of turbulence models and applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, New York, 2014. [16] P.Constantin and C.Foias, Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. [17] G.P. Galdi and W.J. Layton, Approximation of the larger eddies in fluid motions. Ⅱ. A model for space-filtered flow, Math. Mod. Meth. Appl. S., 10 (2000), 343-350. [18] D.Iftimie and G.Planas, Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions, Nonlinearity, 19 (2006), 899-918. [19] A.Larios, The inviscid voigt-regularization for hydrodynamic models: Global regularity, boundary conditions, and blow-up phenomena, Ph.D. thesis, Univ. California, Irvine, 2011. [20] A.Larios and E. S. Titi, On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 603-627. [21] 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. [22] A.Larios, B.Wingate, M.Petersen, et al. The equations and a computational investigation of the finite-time blow-up of solutions to the 3D Euler Equations, Theor. Comp. Fluid Dyn., 3 (2018), 23-34. [23] W.J. Layton, Advanced models for large eddy simulation, Computational Fluid Dynamics-Multiscale Methods (H.Deconinck, ed.), Von Karman Institute for Fluid Dynamics, Rhode-Saint-Genèse, Belgium, 2002. [24] W.J. Layton and R.Lewandowski, On a well-posed turbulence model, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 111-128. [25] A.P. Oskolkov, On the theory of Voight fluids, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 96 (1980), 233-236. [26] C.Parés, Existence, uniqueness and regularity of solution of the equations of a turbulence model for incompressible fluids, Appl. Anal., 43 (1992), 245-296. [27] V.A. Solonnikov and V.E. Ščadilov, A certain boundary value problem for the stationary system of Navier-Stokes equations, Trudy Mat. Inst. Steklov., 125 (1973), 196-210. [28] R.Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal., 20 (1975), 32-43. [29] R.Temam, Navier-Stokes equations. Theory and numerical analysis, Vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. [30] 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. [31] Y.Xiao and Z.Xin, Remarks on vanishing viscosity limits for the 3D Navier-Stokes equations with a slip boundary condition, Chinese Ann. Math. B, 32 (2011), 321-332.
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