Local Strong Solutions for the Compressible Non-Newtonian Models with Density-Dependent Viscosity and Vacuum

Lining Tong , Yanyan Sun

Chinese Annals of Mathematics, Series B ›› 2020, Vol. 41 ›› Issue (3) : 371 -382.

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Chinese Annals of Mathematics, Series B ›› 2020, Vol. 41 ›› Issue (3) : 371 -382. DOI: 10.1007/s11401-020-0204-y
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Local Strong Solutions for the Compressible Non-Newtonian Models with Density-Dependent Viscosity and Vacuum

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Abstract

The one-dimensional compressible non-Newtonian models are considered in this paper. The extra-stress tensor in our models satisfies a kind of power law structure which was proposed by O. A. Ladyzhenskaya in 1970s. In particular, the viscosity coefficient in our models depends on the density. By using energy-estimate, the authors obtain the existence and uniqueness of local strong solutions for which the density is non-negative.

Keywords

Compressible non-Newtonian fluid / Density dependent / Vacuum

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Lining Tong, Yanyan Sun. Local Strong Solutions for the Compressible Non-Newtonian Models with Density-Dependent Viscosity and Vacuum. Chinese Annals of Mathematics, Series B, 2020, 41(3): 371-382 DOI:10.1007/s11401-020-0204-y

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References

[1]

Ladyzhenskaya O. New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them. Trudy Mat. Inst. Steklov., 1967, 102: 85-104

[2]

Málek J, Nečas J, Rokyta M, Růžička M. Weak and Measure-Valued Solutions to Evolutionary PDEs, 1996

[3]

Lions J. Quelques Methodes de Resolution des Problemes Aux Limites Non-Line Aires, 1969

[4]

Bellout H, Bloom F, Nečas J. Young measure-valued solutions for non-non Newtonian incompressible fluids. Comm. Part. Diff. Equ., 1994, 19: 1763-1803

[5]

Málek J, Nečas J, Ružička M. On weak solutions of non-Newtonian incompressible fluids in bounded three-dimensional domains. Advan. Diff. Eqs., 2001, 6: 257-302

[6]

Málek J N J, Ružička M. On the non-Newtonian incompressible fluids. Math. Models and Methods in Appl. Sci., 1993, 3(1): 35-63

[7]

Nečas J. Theory of multipolar viscous fluids. The Mathematics of Finite Elements and Applications., 1990, 7: 233-244

[8]

Matušü-Nečasová Novotny A. Measure-valued solution for non-Newtonian compressible isothermal monopolar fluid. Acta. Appl. Math., 1994, 37: 109-128

[9]

Matušü-Nečasová Medvidova-Lukacova M. Bipolar barotropic non-Newtonian fluid. Comm. Math. Uni. Carolinae., 1994, 35(3): 467-483

[10]

Feireisl E, Liao X, Málek J. Global weak solutions to a class of non-Newtonian compressible fluids. Math. Method. App. Sci., 2015, 38(16): 3482-3494

[11]

Mamontov A. Global solvability of multidimensional Navier-Stokes equations of compressible nonlin-early viscous fluid. I. Siberian. Math., 1999, 40(2): 351-362

[12]

Mamontov A. Existence of global solutions of multidimensional equations for a compressible Bingham fluid. Matematiceskie Zametki, 2007, 82(4): 560-577

[13]

Yuan H, Xu X. Existence and uniqueness of solutions for a class of non-Newtonian fluids with singularity and vacuum. J. Diff. Eqs., 2008, 245: 2871-2916

[14]

Xu X. A class of compressible non-Newtonian fluids with vacuum, 2005

[15]

Fang L, Li Z. On the existence of local classcal solution for a class of one-dimensional compressible non-Newtonian fluids. Acta. Math. Sci., 2015, 35(1): 157-181

[16]

Yang D, Tong L. Existence and uniqueness of compressible non-Newtonian Ellis fluids for one-dimension with vacuum. Commu. Appl. Math. Comput., 2017, 31(1): 128-142

[17]

Fang L, Guo Z. Analytical solutions to a class of non-Newtonian fluids with free boundaries. J. Math. Phys., 2012, 53(10): 467-491

[18]

Wang C, Yuan H. Global strong solutions for a class of heat-conducting non-Newtonian fluids with vacuum. Nonlinear Anal. Real World Appl., 2010, 11: 3680-3703

[19]

Yuan H, Li H. Existence and uniqueness of solution of the initial boundary value problem for a class of non-Newtianian fluids with vacuum. Z. Angew. Math. Phys., 2008, 59: 457-474

[20]

Yin L, Xu X, Yuan H. Global existence and uniqueness of solution of the non-Newtonian with vacuum and danping. J. Math. Anal. Appl., 2012, 391: 223-239

[21]

Fang L, Guo Z, Wang Y. Local strong solutions to a compressible non-Newtonian fluid with density-dependent viscosity. Math. Meth. Appl. Sci, 2016, 39: 2583-2601

[22]

Chen M, Xu X. Unique solvability for the density-dependent non-Newtonian compressible fluids with vacuum. Math. Nachr., 2016, 289(4): 452-470

[23]

Melikyan A. Generalized Characteristics of First Order PDEs, 1998

[24]

Ladyzhenskaya O, Solonnikov V, Ural'ceva N. Linear and Quasilinear Equations of Parabolic Type. 1998 Amer. Math. Soc., 1968

[25]

Zhang R, Fang L. Existence and uniqueness of local strong solutions for non-Newtonian fluids with density-dependent viscosity, singularity and vacuum. Pure. Appl. Math., 2015, 31(1): 97-110

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