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Steady flows of an Oldroyd fluid with threshold slip

This work was supported by the Russian Foundation for Basic Research, project no. 16-31- 00182 mol_a

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  • We consider a mathematical model that describes 3D steady flows of an incompressible viscoelastic fluid of Oldroyd type in a bounded domain under mixed boundary conditions, including a threshold-slip boundary condition. Using the concept of weak solutions, we reduce the original slip problem to a coupled system of variational inequalities and equations for the velocity field and stresses. For arbitrary large data (forcing and boundary data) and suitable material constants, we prove the existence of weak solutions and establish some of their properties.

    Mathematics Subject Classification: Primary: 35Q35, 76A05; Secondary: 35A01.

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  •   R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2$ ^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2003.
      Yu. Ya. Agranovich  and  P. E. Sobolevskii , Motion of nonlinear visco-elastic fluid, Nonlinear Anal., 32 (1998) , 755-760.  doi: 10.1016/S0362-546X(97)00519-1.
      M. A. Artemov and E. S. Baranovskii, Mixed boundary-value problems for motion equations of a viscoelastic medium, Electron. J. Differ. Equ., 2015 (2015), Article No. 252.
      E. S. Baranovskii , On steady motion of viscoelastic fluid of Oldroyd type, Sb. Math., 205 (2014) , 763-776.  doi: 10.1070/SM2014v205n06ABEH004397.
      E. S. Baranovskii , Optimal control for steady flows of the Jeffreys fluids with slip boundary condition, J. Appl. Ind. Math., 8 (2014) , 168-176.  doi: 10.1134/S1990478914020033.
      E. S. Baranovskii  and  M. A. Artemov , Global existence results for Oldroyd fluids with wall slip, Acta Appl. Math., 147 (2017) , 197-210.  doi: 10.1007/s10440-016-0076-z.
      E. S. Baranovskii , On weak solutions to evolution equations of viscoelastic fluid flows, J. Appl. Ind. Math., 11 (2017) , 174-184.  doi: 10.1134/S199047891702003X.
      E. S. Baranovskii, On flows of viscoelastic fluids under threshold-slip boundary conditions J. Phys.: Conf. Ser., 973 (2018), Article ID 012051. doi: 10.1088/1742-6596/973/1/012051.
      O. Bejaoui  and  M. Majdoub , Global weak solutions for some Oldroyd models, J. Differ. Equ., 254 (2013) , 660-685.  doi: 10.1016/j.jde.2012.09.010.
      S. D. Besbes  and  C. Guillopé , Non-isothermal flows of viscoelastic incompressible fluids, Nonlinear Anal., 44 (2001) , 919-942.  doi: 10.1016/S0362-546X(99)00315-6.
      H. Brezis , Equations et inéquations non linéaires dans les espaces en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968) , 115-175.  doi: 10.5802/aif.280.
      J.-Y. Chemin  and  N. Masmoudi , About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001) , 84-112.  doi: 10.1137/S0036141099359317.
      Q. Chen  and  C. Miao , Global well-posedness of viscoelastic fluids of Oldroyd type in Besov spaces, Nonlinear Anal., 68 (2008) , 1928-1939.  doi: 10.1016/j.na.2007.01.042.
      L. Chupin , Some theoretical results concerning diphasic viscoelastic flows of the Oldroyd kind, Adv. Differ. Equ., 9 (2004) , 1039-1078. 
      T. M. Elgindi  and  J. Liu , Global wellposedness to the generalized Oldroyd type models in $ R^3 $, J. Differ. Equ., 259 (2015) , 1958-1966.  doi: 10.1016/j.jde.2015.03.026.
      D. Fang , M. Hieber  and  R. Zi , Global existence results for Oldroyd-B fluids in exterior domains: the case of non-small coupling parameters, Math. Ann., 357 (2013) , 687-709.  doi: 10.1007/s00208-013-0914-5.
      D. Fang  and  R. Zi , Global solutions to the Oldroyd-B model with a class of large initial data, SIAM J. Math. Anal., 48 (2016) , 1054-1084.  doi: 10.1137/15M1037020.
      E. Fernández-Cara , F. Guillén  and  R. Ortega , Some theoretical results concerning non Newtonian fluids of the Oldroyd kind, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 26 (1998) , 1-29. 
      H. Fujita , A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions, RIMS Kokyuroku, 888 (1994) , 199-216. 
      C. Guillopé  and  J. C. Saut , Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal., 15 (1990) , 849-869.  doi: 10.1016/0362-546X(90)90097-Z.
      L. He  and  L. Xi , Global well-posedness for viscoelastic fluid system in bounded domains, SIAM J. Math. Anal., 42 (2010) , 2610-2625.  doi: 10.1137/10078503X.
      M. Hieber , Y. Naito  and  Y. Shibata , Global existence results for Oldroyd-B fluids in exterior domains, J. Differ. Equ., 252 (2012) , 2617-2629.  doi: 10.1016/j.jde.2011.09.001.
      T. Kashiwabara , On a strong solution of the non-stationary Navier-Stokes equations under slip or leak boundary conditions of friction type, J. Differ. Equ., 254 (2013) , 756-778.  doi: 10.1016/j.jde.2012.09.015.
      C. Le Roux  and  A. Tani , Steady solutions of the Navier-Stokes equations with threshold slip boundary conditions, Math. Methods Appl. Sci., 30 (2007) , 595-624.  doi: 10.1002/mma.802.
      C. Le Roux , On flows of viscoelastic fluids of Oldroyd type with wall slip, J. Math. Fluid Mech., 16 (2014) , 335-350.  doi: 10.1007/s00021-013-0159-9.
      Z. Lei , C. Liu  and  Y. Zhou , Global solutions for incompressible viscoelastic fluids, Arch. Ration. Mech. Anal., 188 (2008) , 371-398.  doi: 10.1007/s00205-007-0089-x.
      J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969.
      P. L. Lions  and  N. Masmoudi , Global solutions for some Oldroyd models of non-Newtonian flows, Chin. Ann. Math. Ser. B, 21 (2000) , 131-146.  doi: 10.1142/S0252959900000170.
      V. G. Litvinov, Motion of A Nonlinearly Viscous Fluid, Nauka, Moscow, 1982.
      S. Maryani , Global well-posedness for free boundary problem of the Oldroyd-B model fluid flow, Math. Methods Appl. Sci., 39 (2016) , 2202-2219.  doi: 10.1002/mma.3634.
      J. T. Oden and N. Kikuchi, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1988. doi: 10.1137/1.9781611970845.
      J. G. Oldroyd , On the formulation of rheological equations of state, Proc. R. Soc. Lond. Ser. A, 200 (1950) , 523-541.  doi: 10.1098/rspa.1950.0035.
      J. G. Oldroyd , Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids, Proc. R. Soc. Lond. Ser. A, 245 (1958) , 278-297.  doi: 10.1098/rspa.1958.0083.
      K. R. Rajagopal , On some unresolved issues in non-linear fluid dynamics, Russian Math. Surveys, 58 (2003) , 319-330.  doi: 10.1070/RM2003v058n02ABEH000612.
      M. Renardy , Existence of slow steady flows of viscoelastic fluids with differential constitutive equations, Z. Angew. Math. Mech., 65 (1985) , 449-451.  doi: 10.1002/zamm.19850650919.
      M. Renardy and R. Rogers, An Introduction to Partial Differential Equations, 2$ ^{nd} $ edition, Springer-Verlag, New York, 2004.
      J.-C. Saut, Lectures on the mathematical theory of viscoelastic fluids, in Lectures on the analysis of nonlinear partial differential equations. Part 3 (eds. F. Lin and P. Zhang), Int. Press, Somerville, MA, (2013), 325-393.
      C. Truesdell, A First Course in Rational Continuum Mechanics, Academic Press, New York, 1977.
      E. M. Turganbaev , Initial-boundary value problems for the equations of a viscoelastic fluid of Oldroyd type, Sib. Math. J., 36 (1995) , 389-403.  doi: 10.1007/BF02110162.
      D. A. Vorotnikov , On the existence of weak stationary solutions of a boundary value problem in the Jeffreys model of the motion of a viscoelastic medium, Russian Math. (Iz. VUZ), 48 (2004) , 10-14. 
      Z. Ye  and  X. Xu , Global regularity for the 2D Oldroyd-B model in the corotational case, Math. Methods Appl. Sci., 39 (2016) , 3866-3879.  doi: 10.1002/mma.3834.
      V. G. Zvyagin  and  D. A. Vorotnikov , Approximating-topological methods in some problems of hydrodynamics, J. Fixed Point Theory Appl., 3 (2008) , 23-49.  doi: 10.1007/s11784-008-0056-7.
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