In this article, we investigate an initial and boundary value problem for a class of compressible non-Newtonian fluids, provided the initial energy is small and the initial density containing the vacuum state is allowed. For , we obtain the existence and uniqueness of the global strong solution for this problem in a one-dimensional bounded interval.
Accepted:
Published online:
Jianjun Xu 1; Hongjun Yuan 1

@article{CRMECA_2021__349_1_29_0, author = {Jianjun Xu and Hongjun Yuan}, title = {Existence and uniqueness of global strong solutions for a class of {non-Newtonian} fluids with small initial energy and vacuum}, journal = {Comptes Rendus. M\'ecanique}, pages = {29--41}, publisher = {Acad\'emie des sciences, Paris}, volume = {349}, number = {1}, year = {2021}, doi = {10.5802/crmeca.68}, language = {en}, }
TY - JOUR AU - Jianjun Xu AU - Hongjun Yuan TI - Existence and uniqueness of global strong solutions for a class of non-Newtonian fluids with small initial energy and vacuum JO - Comptes Rendus. Mécanique PY - 2021 SP - 29 EP - 41 VL - 349 IS - 1 PB - Académie des sciences, Paris DO - 10.5802/crmeca.68 LA - en ID - CRMECA_2021__349_1_29_0 ER -
%0 Journal Article %A Jianjun Xu %A Hongjun Yuan %T Existence and uniqueness of global strong solutions for a class of non-Newtonian fluids with small initial energy and vacuum %J Comptes Rendus. Mécanique %D 2021 %P 29-41 %V 349 %N 1 %I Académie des sciences, Paris %R 10.5802/crmeca.68 %G en %F CRMECA_2021__349_1_29_0
Jianjun Xu; Hongjun Yuan. Existence and uniqueness of global strong solutions for a class of non-Newtonian fluids with small initial energy and vacuum. Comptes Rendus. Mécanique, Volume 349 (2021) no. 1, pp. 29-41. doi : 10.5802/crmeca.68. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.68/
[1] Non-Newtonian Fluid Mechanics, Applied Mathematics and Mechanics, North-Holland, Amsterdam, 1987 | Zbl
[2] New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems, Boundary Value Problems of Mathematical Physics V, American Mathematical Society, Providence, RI, 1970, pp. 95-118
[3] Global regularity estimates for multidimensional equations of compressible non-Newtonian fluids, Math. Notes, Volume 68 (2000), pp. 312-325 | DOI | MR | Zbl
[4] Strong solutions of the Navier–Stokes equations for isentropic compressible fluids, J. Differ. Equ., Volume 190 (2003), pp. 504-523 | DOI | MR | Zbl
[5] Existence and uniqueness of solutions for a class of non-Newtonian fluids with singularity and vacuum, J. Differ. Equ., Volume 245 (2008), pp. 2871-2916 | DOI | MR | Zbl
[6] On the existence of local classical solution for a class of one-dimensional compressible non-Newtonian fluids, Acta Math. Sci., Volume 35 (2015) no. 1, pp. 157-181 | DOI | MR
[7] Local existence of strong solution for a class of compressible non-Newtonian fluids with non-Newtonian potential, Comput. Math. Appl., Volume 65 (2013) no. 4, pp. 563-575 | DOI | MR | Zbl
[8] Local strong solutions for the compressible non-Newtonian models with density-dependent viscosity and vacuum, Chin. Ann. Math., Volume 41 (2020), pp. 371-382 | DOI | MR | Zbl
[9] Unique solvability for a class of full non-Newtonian fluids of one dimension with vacuum, Z. Angew. Math. Phys., Volume 60 (2009), pp. 868-898 | DOI | MR | Zbl
[10] Global classical solution to a one-dimensional compressible non-Newtonian fluid with large initial data and vacuum, Nonlinear Anal., Volume 174 (2018), pp. 189-208 | DOI | MR | Zbl
[11] Global Strong solutions of a class of non-Newtonian fluids with small initial energy, J. Math. Anal. Appl., Volume 474 (2019), pp. 72-93 | DOI | MR | Zbl
[12] Global classical solutions to 1D full compressible Navier–Stokes equations with the Robin boundary condition on temperature, Nonlinear Anal. Real World Appl., Volume 47 (2019), pp. 306-323 | DOI | MR | Zbl
[13] The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., Volume 20 (1980), pp. 67-104 | DOI | MR | Zbl
[14] Global solutions of the Navier–Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differ. Equ., Volume 120 (1995), pp. 215-254 | DOI | MR | Zbl
[15] Mathematical Topics in Fluid Dynamics, Compressible Models, Volume 2, Oxford Science Publication, Oxford, 1998
[16] On the existence of globally defined weak solutions to the Navier–Stokes equations of compressible isentropic fluids, J. Math. Fluid Mech., Volume 3 (2001) no. 4, pp. 358-392 | DOI
[17] Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier–Stokes equations, Commun. Pure Appl. Math., Volume 65 (2012), pp. 549-585 | DOI | MR | Zbl
[18] Global classical and weak solutions to the three-dimensional full compressible Navier–Stokes system with vacuum and large oscillations, Arch. Rational Mech. Anal., Volume 227 (2018), pp. 995-1059 | DOI | MR | Zbl
[19] Global weak solutions to a class of non-Newtonian compressible fluids, Math. Methods Appl. Sci., Volume 38 (2015) no. 16, pp. 3482-3494 | DOI | MR | Zbl
[20] A Class of Compressible non-Newtonian Fluids with Vacuum, Jilin University, Changchun, 2005
[21] Uniform estimates and stabilization of symmetric solutions of a system of quasi-linear equations, Differ. Uravn., Volume 36 (2000) no. 5, pp. 634-646 (in Russian) (Engl. transl. J. Differ. Equ. 36 (2000), no. 5, p. 701-716) | Zbl
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