Articles

ON THE EXISTENCE OF LOCAL CLASSICAL SOLUTION FOR A CLASS OF ONE-DIMENSIONAL COMPRESSIBLE NON-NEWTONIAN FLUIDS

Expand
  • CNS and Department of Mathematics, Northwest University, Xi’an 710127, China

Received date: 2013-12-10

  Revised date: 2014-04-21

  Online published: 2015-01-20

Supported by

Supported by NSFC (11201371, 1331005), and Natural Science Foundation of Shaanxi Province (2012JQ020).

Abstract

In this paper, the aim is to establish the local existence of classical solutions for a class of compressible non-Newtonian fluids with vacuum in one-dimensional bounded intervals, under the assumption that the data satisfies a natural compatibility condition. For the results, the initial density does not need to be bounded below away from zero.

Cite this article

FANG Li,LI ZiLai . ON THE EXISTENCE OF LOCAL CLASSICAL SOLUTION FOR A CLASS OF ONE-DIMENSIONAL COMPRESSIBLE NON-NEWTONIAN FLUIDS[J]. Acta mathematica scientia, Series B, 2015 , 35(1) : 157 -181 . DOI: 10.1016/S0252-9602(14)60148-X

References

[1] Bae H O. Existence, regularity and decay rate of solutions of non-Newtonian flow. J Math Anal Appl, 1999, 231: 467–491
[2] Beir˜ao da Veiga H. Long time behavior for one-dimensional motion of a general barotropic viscous fluid. Arch Ration Mech Anal, 1989, 108: 141–160
[3] Bohme G. Non-Newtonian Fluid Mechanics. New York: North-Holland, 1987
[4] Cho Y, Kim H. On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. Manuscripta Math, 2006, 120: 92–129
[5] Choe H J, Kim H. Global existence of the radially symmetric solutions of the radially symmetric solutions of the Navier-Stokes equation for the isentropic compressible fluids. Math Models Methods Appl Sci, 2005, 28: 1–28
[6] Ding S J, Wen H Y, Zhu C J. Global classical large solutions to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum. J Differ Equ, 2011, 251: 1696–1725
[7] Fang L, Guo Z H. A blow-up criterion for a class of non-Newtonian fluids with singularity and vacuum. Acta Math Appl Sin, 2013, (3): 502–515
[8] Fang L, Guo Z H. Analytical solutions to a class of non-Newtonian fluids with free boundaries. J Math Phys, 2012, 53: 103701
[9] Feireisl E, Novotn´y A, Petzeltov´a H. On the existence of globally defined weak solution to the Navier-Stokes equations. J Math Fluid Mech, 2001, (3): 358–392
[10] Galdi G P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol I. New York: Springer-Verlag, 1994
[11] Guo B L, Zhu P C. Algebraic L2-decay for the solution to a class system of non-Newtonian fluid in RN. J Math Phys, 2000, 41: 349–356
[12] Huilgol R R. Continuum Mechanics of Viscoelastic Liquids. Delhi: Hindustan Publishing Corporation, 1975
[13] Jiang S, Zhang P. On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations. Comm Math Phys, 2001, 215: 559–581
[14] Kazhikhov A V. Stabilization of solutions of an initial-boundary-value problem for the equations of motion of a barotropic viscous fluid. Differ Equ, 1979, 15: 463–467
[15] Ladyzhenskaya O A. On nonlinear problems of continuum mechanics//Proc Internat Congr Math (Moscow 1966). Moscow: Nauka, 1968: 560–573; English translation in: Amer Math Soc Translation, 1968, 70(2)
[16] Ladyzhenskaya O A. New equations for description of motion of viscous incompressible fluids and global solvability of boundary value problems for their boundary value problems. Proc Steklov Inst Math, 1967,
102: 85–104
[17] Ladyzhenskaya O A. On some modifications of the Navier-Stokes equations for large gradient of velocity. Zap Nauchn Sem Leningrad Odtel. Mat Inst Steklov (LOMI), 1968, 7: 126–154; English translation in: Sem Math V A Steklov Math Inst Leningrad, 1968, (7)
[18] Ladyzhenskaya O A, Seregin G A. On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. J Math Fluid Mech, 1999, (1): 356–387
[19] Ladyzhenskaya O A. New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them//Boundary Value Problems of Mathematical Physics V. Providence RI: Amer Math Soc, 1970[20] Lions P L. Mathematical Topics in Fluid Mechanics, Vol 2: Compressible Fluids. Oxford Lecture Ser Math Appl, Vol 10. Oxford: Clarendon Press, 1998
[21] Liu T P, Xin Z P, Yang T. Vacuum states of compressible flow. Discrete Contin Dyn Syst, 1998, (4): 1–32
[22] M´alek J, Neˇcas J, Rokyta M, et al. Weak and Measure-Valued Solutions to Evolutionary PDEs. New York: Chapman and Hall, 1996
[23] Mamontov A E. Existence of global solutions of multidimensional Burger’s equations of a compressible viscous fluid. Mat Sb, 1999, 19: 61–80
[24] Mamontov A E. Global solvability of the multidiensional Navier-Stokes equations of a compressible fluid with nonlinear viscosity. I. Sib Mat Zh, 1999, 40: 408–420
[25] Mamontov A E. Global solvability of the multidiensional Navier-Stokes equations of a compressible fluid with nonlinear viscosity. II. Sib Mat Zh, 1999, 40: 635–649
[26] Matsumura A, Yanagi S. Uniform boundedness of the solutions for a one dimensional isentropic model system of compressible viscous gas. Comm Math Phys, 1996, 175: 259–274
[27] Neˇcasov´a ˇ S, Luk´aˇcov´a M. Bipolar isothermal non-Newtonian compressible fluids. J Math Anal Appl, 1998, 225: 168–192
[28] Neˇcasov´a ˇ S, Luk´aˇcov´a M. Bipolar barotropic non-Newtonian compressible fluids. Comm Math Univ Carolin, 1994, 35: 467–483
[29] Neˇcas J, Novotn´y A. Some qualitative properties of the viscous compressible heat conductive multipolar fluid. Comm Partrial Differential Equations, 1991, 16: 197–220
[30] Neˇcas J, Novotn´y A, ˇ Silhav´y M. Global solution to the compressible isothermal multipolar fluid. J Math Anal Appl, 1991, 162: 223–241
[31] Schwalter W R. Mechanics of Non-Newtonian Fluid. New York: Pergamon Press, 1978
[32] Solonnikov V A, Kazhikhov A V. Existence theorem for the equations of motion of a compressible viscous fluid. Annu Rev Fluid Mech, 1981, 13: 79–95
[33] Whitaker S. Introduction to Fluid Mechanics. Melbourne, FL: Krieger, 1981
[34] Xin Z P. Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density. Comm Pure Appl Math, 1998, 51: 229–240
[35] Yang T, Yao Z A, Zhu C J. Compressible Navier-Stokes equaitons with density-dependent viscosity and vacuum. Comm Partial Differential Equaitons, 2001, 26: 965–981
[36] Yang T, Zhu C J. Compressible Navier-Stokes equaitons with degenerate viscosity coefficient and vacuum. bComm Partial Differential Equaitons, 2002, 230: 329–363
[37] Yin L, Xu X J, Yuan H J. Global existence and uniqueness of solution of the initial boundary value problem for a class of non-Newtonian fluids with vacuum. Z Angew Math Phys, 2008, 59: 457–474
[38] Yuan H J, Xu X J. Existence and uniqueness of solutions for a class of non-Newtonian fluds with singularity and vacuum. J Differential Equations, 2008, 245: 2871–2916
[39] Yuan H J, Li H P. Existence and uniqueness of solutions for a class of non-Newtonian fluds with vacuum and damping. J Math Anal Appl, 2012, 391: 223–239
[40] Yuan H J, Li H Z, Qiao J Z, Li F P. Global existence of strong solutions of Navier-Stokes equations withnon-Newtonian potential for one-dimensional isentropic compressible fluids. Acta Math Sci, 2012, 32B:1467–1486

Outlines

/