Tồn tại của các nghiệm yếu cho dòng chảy phi trạng thái của chất lỏng với độ nhớt phụ thuộc vào khả năng cắt dưới điều kiện biên trượt trong nửa không gian

Science China Mathematics - Tập 61 - Trang 727-744 - 2017
Aibin Zang1
1School of Mathematics and Computer Science, Yichun University, Yichun, China

Tóm tắt

Bài báo này đề cập đến hệ thống chuyển động của các chất lỏng không nén được và không phải Newton với tensor ứng suất được mô tả bởi hàm khả năng p dưới các điều kiện biên trượt trong ℝ+ 3. Bằng cách sử dụng phương pháp xấp xỉ Oseen cho mô hình này và phương pháp cắt L, có thể thiết lập định lý tồn tại cho các nghiệm yếu cho dòng chảy p-khả năng với p ∈ (\frac{8}{5}, 2] với điều kiện rằng các điều kiện ban đầu lớn đủ đều.

Từ khóa

#dòng chảy không nén #chất lỏng không Newton #độ nhớt phụ thuộc #điều kiện biên trượt #nghiệm yếu

Tài liệu tham khảo

Acerbi E, Mingione G, Seregin G A. Regularity results for parabolic systems related to a class of non-Newtonian fluids. Ann Inst H Poincaré Anal Non Linéaire, 2004, 21: 25–60 Adams R, Fournier J. Sobolev Spaces, 2nd ed. Singapore: Elsevier, 2009 Amann H. Stability of the rest state of a viscous incompressible fluid. Arch Ration Mech Anal, 1994, 126: 231–242 Bae H, Choe H J. Existence and regularity of solutions of non-Newtonian flow. Quart Appl Math, 2000, 58: 379–400 Beirão da Veiga H. On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions. Comm Pure Appl Math, 2005, 58: 552–577 Beirão da Veiga H. Navier-Stokes equations with shear thickening viscosity: Regularity up to boundary. J Math Fluid Mech, 2009, 11: 233–257 Beirão da Veiga H. Navier-Stokes equations with shear thinning viscosity: Regularity up to boundary. J Math Fluid Mech, 2009, 11: 258–273 Beirão da Veiga H. On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem. J Eur Math Soc (JEMS), 2009, 11: 127–167 Beirão da Veiga H. Turbulence models, p-fluid flows, and W2;l-regularity of solutions. Comm Pure Appl Anal, 2009, 8: 769–783 Beirão da Veiga H. On the global regularity of shear thinning flows in smooth domains. J Math Anal Appl, 2009, 349: 335–360 Beirão da Veiga H, Crispo F. Concerning the Wk;p-inviscid limit for 3D flows under a slip boundary condition. J Math Fluid Mech, 2011, 13: 117–135 Beirão da Veiga H, Crispo F, Grisant C R. Reducing slip boundary value problems from the half to the whole space: Applications to inviscid limits and to non-Newtonian fluids. J Math Anal Appl, 2011, 377: 216–227 Beirão da Veiga H, Kaplický P, Růžička M. Regularity theorems, up to the boundary, for shear thickening flows. C R Math Acad Sci Paris, 2010, 348: 541–544 Berselli L C, Diening L, Růžička M. Existence of strong solutions for incompressible fluids with shear dependent viscosities. J Math Fluid Mech, 2005, 12: 101–132 Bird R B, Armstrong R C, Hassager O. Dynamic of Polymer Liquids, 2nd ed. New York: John Wiley, 1987 Bothe D, Prüss J. Lp-theory for a class of non-Newtonian fluids. SIAM J Math Anal, 2007, 39: 379–421 Bulíček M, Málek J, Rajagopal K R. Navier’s slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity. Indiana Univ Math J, 2007, 56: 51–85 Bulíček M, Majdoub M, Málek J. Unsteady flows of fluids with pressure dependent viscosity in unbounded domains. Nonlinear Anal Real World Appl, 2010, 11: 3968–3983 Bulíček M, Málek J, Rajagopal K R. Mathematical analysis of unsteady flows of fluids with pressure, shear-rate, and temperature dependent material moduli that slip at solid boundaries. SIAM J Math Anal, 2009, 41: 665–707 Crispo F. Shear thinning viscous fluids in cylindrical domain. Regularity up to the boundary. J Math Fluid Mech, 2008, 10: 311–325 Crispo F. Global regularity of a class of p-fluid flow in cylinders. J Math Anal Appl, 2008, 341: 559–574 Diening L, Málek J, Steinhauer M. On the Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. ESAIM Control Optim Calc Var, 2008, 14: 211–232 Diening L, Růžička M. Strong solutions for generalized Newtonian fluids. J Math Fluid Mech, 2005, 7: 413–450 Diening L, Růžička M, Wolf J. Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Ann Sc Norm Super Pisa Cl Sci (5), 2010, 9: 1–46 Ebmeyer C. Regularity in Sobolev spaces of steady flows of fluids with shear-dependent viscosity. Math Methods Appl Sci, 2006, 29: 1687–1707 Frehse J, Málek J, Steinhauer M. An existence result for fluids with shear dependent viscosity-steady flows. Nonlinear Anal, 1997, 30: 3041–3049 Frehse J, Málek J, Steinhauer M. On the analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method. SIAM J Math Anal, 2003, 34: 1064–1083 Galdi P, Grisanti R. Existence and regularity of steady flows for shear-thinning liquids in exterior two-dimensional domains. Arch Ration Mech Anal, 2011, 200: 533–559 Kato T, Lai C Y. Nonlinear evolution equations and the Euler flow. J Funct Anal, 1984, 56: 15–28 Ladyžhenskaya O A. New equation for description of motion of viscous incompressible fluids and solvability in the large boundary value problems for them. Proc Steklov Inst Math, 1967, 102: 95–118 Ladyžhenskaya O A. On some modification of the Navier-Stokes equations for large gradients of velocity. Zap Nauchn Sem S-Peterburg Otdel Mat Inst Steklov (POMI), 1968, 7: 126–154 Ladyžhenskaya O A. The Mathematical Theory of Incompressible Flow, 2nd ed. New York: Gordon and Breach, 1969 Lions J L. Quelques methodes de résolution des problèmes aus limites nonlinéaires. Paris: Dunod, 1969 Málek J, Nečas J, Rokyta M, et al. Weak and Measure-Valued Solutions to Evolutionary PDEs. London: Chapman & Hall, 1996 Málek J, Nečas J, Růžička M. On the weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: The case p ≤ 2. Adv Difference Equ, 2001, 6: 257–302 Málek J, Rajagopal K R. Mathematical issues concerning the Navier-Stokes equations and some of its generalizations. In: Evolutionary Equations. Handbook of Differential Equations, vol. 2. Amsterdam: Elsevier/North-Holland, 2005, 371–459 Maremonti R. Some theorems of existence for solutions of the Navier-Stokes equations with slip boundary conditions in half-space. Ric Mat, 1991, 40: 81–135 Naumann J, Wolf J. Interior differentiability of weak solutions to the equations of stationary motion of a class of non-Newtonian fluids. J Math Fluid Mech, 2005, 7: 298–313 Navier C L M. Mémoire sur les lois du mouvement des fluides. Mémoires de l‘Académie Royale des Sciences de Institutde France, vol. 1. Mem Acad Sci Inst France, http://cdarve.web.cern.ch/cdarve/publications cd/navier darve.pdf, 1822 Pokorný M. Cauchy problem for the non-Newtonian viscous incompressible fluid. Appl Math, 1996, 41: 169–201 Shinlkin T N. Regularity up to boundary of solutions to boundary-value problems of the theory of generalized Newtonian liquids. J Math Sci (NY), 1998, 92: 4386–4403 Wolf J. Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity. J Math Fluid Mech, 2007, 9: 104–138 Xiao Y L, Xin Z P. On the vanishing viscosity limit for the Navier-Stokes equations with a slip boundary condition. Comm Pure Appl Math, 2007, 60: 1027–1055