## Incompressible Euler as a limit of complex fluid models with Navier boundary conditions.(English)Zbl 1232.35122

The authors study an incompressible second-grade fluid equation of two or three space dimensions. This a model for visco-elastic fluid, where the stress tensor is described by two parameters: the elastic response $$\alpha$$ and the viscosity $$\nu$$. When $$\nu= 0$$, this equation is called $$\alpha$$-Euler equation, and when $$\alpha =\nu =0$$, this is the Euler equation. They study the limiting process as $$\alpha ,\nu \to 0$$, from a second-grade fluid equation to the Euler equation. First they discuss the boundary conditions and show that perfect-slip boundary condition of Navier is an admissible condition for an incompressible second-grade fluid equation. In this way they formulate an initial-boundary problem, and give a definition of weak solutions for this problem. They prove that if the initial data is smooth, their weak solutions strongly converge in $$L^2$$ to a solution of incompressible Euler equation as $$\alpha ,\nu \to 0$$. They also prove an existence theorem of global solutions for the axisymmetric flow equations without swirl.

### MSC:

 35Q31 Euler equations 76A10 Viscoelastic fluids 35D30 Weak solutions to PDEs 35B07 Axially symmetric solutions to PDEs
Full Text:

### References:

 [1] Abboud, H.; Sayah, T., Upwind discretization of a time-dependent two-dimensional grade-two fluid model, Comput. math. appl., 57, 1249-1264, (2009) · Zbl 1186.76614 [2] Ahmad, I.; Sajid, M.; Hayat, T., Heat transfer in unsteady axisymmetric second grade fluid, Appl. math. comput., 215, 1685-1695, (2009) · Zbl 1179.80015 [3] Bardos, C.; Linshiz, J.; Titi, E., Global regularity for a Birkhoff-rott-α approximation of the dynamics of vortex sheets of the 2D Euler equations, Phys. D, 237, 1905-1911, (2008) · Zbl 1143.76384 [4] Clopeau, T.; Mikelic, A.; Robert, R., On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions, Nonlinearity, 11, 1625-1636, (1998) · Zbl 0911.76014 [5] Busuioc, A.V.; Iftimie, D., A non-Newtonian fluid with Navier boundary conditions, J. dynam. differential equations, 18, 2, 357-379, (2006) · Zbl 1127.76005 [6] Busuioc, A.V.; Ratiu, T., The second grade fluid and averaged Euler equations with Navier-slip boundary conditions, Nonlinearity, 16, 1119-1149, (2003) · Zbl 1026.76004 [7] Busuioc, A.V.; Ratiu, T., Some remarks on a certain class of axisymmetric fluids of differential type, Phys. D, 191, 106-120, (2004) · Zbl 1054.76006 [8] Dunn, J.E.; Fosdick, R.L., Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade, Arch. ration. mech. anal., 56, 191-252, (1974) · Zbl 0324.76001 [9] Dunn, J.E.; Rajagopal, K.R., Fluids of differential type - critical review and thermodynamic analysis, Internat. J. engrg. sci., 33, 689-729, (1995) · Zbl 0899.76062 [10] Fan, Jishan; Ozawa, Tohro, On the regularity criteria for the generalized Navier-Stokes equations and Lagrangian averaged Euler equations, Differential integral equations, 21, 443-457, (2008) · Zbl 1224.35338 [11] Holm, D.; Marsden, J.; Ratiu, T., Euler-Poincaré models of ideal fluids with nonlinear dispersion, Phys. rev. lett., 80, 4173-4176, (1998) [12] Hou, Thomas; Congming, Li, On global well-posedness of the Lagrangian averaged Euler equations, SIAM J. math. anal., 38, 782-794, (2006) · Zbl 1111.76006 [13] Iftimie, D., Remarques sur la limite $$\alpha \rightarrow 0$$ pour LES fluides de grade 2, () · Zbl 1014.35081 [14] Iftimie, D.; Planas, G., Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions, Nonlinearity, 19, 899-918, (2006) · Zbl 1169.35365 [15] Khan, M.; Hyder Ali, S.; Qi, H., Exact solutions for some oscillating flows of a second grade fluid with a fractional derivative model, Math. comput. modelling, 49, 1519-1530, (2009) · Zbl 1165.76310 [16] le Roux, C., Existence and uniqueness of the flow of second-grade fluids with slip boundary conditions, Arch. ration. mech. anal., 148, 309-356, (1999) · Zbl 0934.76005 [17] Linshiz, J.; Titi, E., On the convergence rate of the Euler-α inviscid second-grade complex fluid, model to the Euler equations, J. stat. phys., 138, 305-332, (2010) · Zbl 1375.35348 [18] Liu, Xiaofeng; Jia, Houyu, Local existence and blowup criterion of the Lagrangian averaged Euler equations in Besov spaces, Commun. pure appl. anal., 7, 845-852, (2008) · Zbl 1141.76011 [19] Liu, Xiaofeng; Wang, Meng; Zhang, Zhifei, A note on the blowup criterion of the Lagrangian averaged Euler equations, Nonlinear anal., 67, 2447-2451, (2007) · Zbl 1123.35039 [20] Lopes Filho, M.C.; Nussenzveig Lopes, H.J.; Planas, G., On the inviscid limit for 2D incompressible flow with Navier friction condition, SIAM J. math. anal., 36, 1130-1141, (2005) · Zbl 1084.35060 [21] Massoudi, Mehrdad; Tran, Phuoc X.; Wulandana, R., Convection-radiation heat transfer in a nonlinear fluid with temperature-dependent viscosity, Math. probl. eng., (2009), article ID 232670 · Zbl 1179.80026 [22] Navier, C.L.M.H., Mémoire sur LES lois du mouvement des fluides, Mém. acad. royale sci. inst. fr., VI, 389-440, (1823) [23] Xiao, Y.; Xin, Z., On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition, Comm. pure appl. math., 60, 1027-1055, (2007) · Zbl 1117.35063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.