Geodesic
Steady-state solutions to the equations of motion of second-grade fluids with general Navier-type slip boundary conditions in Hölder spaces
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 34, Tome 306 (2003), pp. 210-228 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We consider a boundary-value problem for the stationary flow of an incompressible second-grade fluid in a bounded domain. The boundary condition allows for no-slip, Navier-type slip and free slip on different parts of the boundary. We first establish the well-posedness of a linear auxiliary problem by means of a fixed-point argument in which it is decomposed into a Stokes-type problem and two transport equations. Then we use the method of successive approximations to prove the unique solvability in Hölder spaces of the nonlinear problem with a sufficiently small body force. 
@article{ZNSL_2003_306_a10,
     author = {A. Tani and C. Le Roux},
     title = {Steady-state solutions to the equations of motion of second-grade fluids with general {Navier-type} slip boundary conditions in {H\"older} spaces},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {210--228},
     year = {2003},
     volume = {306},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_306_a10/}
}
TY  - JOUR
AU  - A. Tani
AU  - C. Le Roux
TI  - Steady-state solutions to the equations of motion of second-grade fluids with general Navier-type slip boundary conditions in Hölder spaces
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2003
SP  - 210
EP  - 228
VL  - 306
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2003_306_a10/
LA  - en
ID  - ZNSL_2003_306_a10
ER  - 
%0 Journal Article
%A A. Tani
%A C. Le Roux
%T Steady-state solutions to the equations of motion of second-grade fluids with general Navier-type slip boundary conditions in Hölder spaces
%J Zapiski Nauchnykh Seminarov POMI
%D 2003
%P 210-228
%V 306
%U http://geodesic.mathdoc.fr/item/ZNSL_2003_306_a10/
%G en
%F ZNSL_2003_306_a10
A. Tani; C. Le Roux. Steady-state solutions to the equations of motion of second-grade fluids with general Navier-type slip boundary conditions in Hölder spaces. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 34, Tome 306 (2003), pp. 210-228. http://geodesic.mathdoc.fr/item/ZNSL_2003_306_a10/

[1] H. Beirão da Veiga, “On a stationary transport equation”, Ann. Univ. Ferra, Sc. Mat., 32 (1986), 79–91 | MR | Zbl

[2] D. Bresch, J. Lemoine, “Stationary solutions for second grade fluids equations”, Math. Models Methods Appl. Sci., 8 (1998), 737–748 | DOI | MR | Zbl

[3] V. Coscia, G. P. Galdi, “Existence, uniqueness and stability of regular steady motions of a second-grade fluid”, Int. J. Non-Linear Mech., 29 (1994), 493–506 | DOI | MR | Zbl

[4] M. M. Denn, “Extrusion instabilities and wall slip”, Annual review of fluid mechanics, Annu. Rev. Fluid Mech., 33, Annual Reviews, Palo Alto, CA, 2001, 265–287 | Zbl

[5] J. E. Dunn, R. L. Fosdick, “Thermodynamics, stability, and boundedness of fluids of complexity $2$ and fluids of second grade”, Arch. Ration. Mech. Anal., 56 (1974), 191–252 | DOI | MR | Zbl

[6] J. E. Dunn, K. R. Rajagopal, “Fluids of differential type: critical review and thermodynamic analysis”, Int. J. Eng. Sci., 33 (1995), 689–729 | DOI | MR | Zbl

[7] G. P. Galdi, “Mathematical theory of second-grade fluids”, Stability and Wave Propogation in Fluids and Solids, ed. G. P. Galdi, Springer-Verlag, Berlin, 1995, 67–104 | MR | Zbl

[8] S. Itoh, A. Tani, “The initial value problem for the nonhomogeneous Navier–Stokes equations with general slip boundary condition”, Proc. R. Soc. Edinb., Sect. A, Math., 130 (2000), 827–835 | DOI | MR | Zbl

[9] S. Itoh, N. Tanaka, A. Tani, “The initial value problem for the Navier–Stokes equations with general slip boundary condition in Hölder spaces”, J. Math. Fluid Mech., 5 (2003), 275–301 | MR | Zbl

[10] S. Itoh, N. Tanaka, A. Tani, Steady solution and its stability to Navier–Stokes equations with general Navier's slip boundary condition, Preprint

[11] C. le Roux, “Existence and uniqueness of flows of second-grade fluids with slip boundary conditions”, ARMA, 148 (1999), 309–356 | DOI | MR | Zbl

[12] I. Sh. Mogilevskiĭ, V. A. Solonnikov, “Problem of the stationary flow of a second-grade fluid in Hölder classes of functions”, ZNSP, 243 (1997), 154–168

[13] A. Novotny, “About steady transport equation. II: Schauder estimates in domains with smooth boundaries”, Port. Math., 54 (1997), 317–333 | MR | Zbl

[14] V. A. Solonnikov, “On general boundary value problem for the elliptic system in the sense of Douglis–Nirenberg, I”, Izv. Akad. Nauka USSR, Ser. Mat., 28:3 (1964), 665–706 ; “On general boundary value problem for the elliptic system in the sense of Douglis–Nirenberg, II”, Trudy Mat. Inst. Steklov, 92, 1966, 233–297 | MR | Zbl | MR | Zbl

[15] A. Tani, “The initial value problem for the equations of motion of general fluid with general slip boundary condition”, KRIMSKU, 734 (1990), 123–142 | MR

[16] A. Tani, Initial value problem for the equations of motion of compressible viscous fluid with general Navier's slip boundary condition, Preprint

[17] A. Tani, S. Itoh, N. Tanaka, “The initial value problem for the Navier–Stokes equations with general slip boundary condition”, Adv. Math. Sci. Appl., 4 (1994), 51–69 | MR | Zbl