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Local Weak Solution of the Isentropic Compressible Navier–Stokes Equations with Variable Viscosity

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Abstract

In this paper, we consider the 3-D compressible isentropic Navier–Stokes equations with constant shear viscosity \(\mu \) and the bulk one \(\lambda =b\rho ^\beta \), here b is a positive constant, \(\beta \ge 0\). This model was first introduced and well studied by Vaigant and Kazhikhov (Sib Math J 36(6):1283–1316, 1995) in 2D domain. In this paper, under the assumption that \(\gamma >1\), the local existence of weak solutions with higher regularity for the 3D periodic domain is established in the presence of vacuum without any smallness on the initial data. This generalize the previous paper (Desjardins in Commun Partial Differ Equ 22(5):977–1008, 1997; Huang and Yan in J Math Phys 62(11):111504, 2021) to variable viscosity coefficients. Also this is the first result concerning the local weak solution with high regularity for the Kazhikhov model in 3D case.

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References

  1. Abbatiello, Anna, Feireisl, Eduard, Novotny, Antoni: Generalized solutions to models of compressible viscous fluids. Discret. Contin. Dyn. Syst. 41, 1–28 (2021)

    Article  MathSciNet  Google Scholar 

  2. Beira da Veiga, H.: Long time behavior for one-dimensional motion of a general barotropic viscous fluid. Arch. Ration. Mech. Anal. 108, 141–160 (1989)

    Article  MathSciNet  Google Scholar 

  3. Cho, Y., Choe, H.J., Kim, H.: Unique solvability of the initial boundary value problems for compressible viscous fluids. J. Math. Pures Appl. 83(2), 243–275 (2004)

    Article  MathSciNet  Google Scholar 

  4. Choe, H.J., Kim, H.: Strong solutions of the Navier–Stokes equations for nonhomogeneous incompressible fluids. Commun. Partial Differ. Equ. 28(5–6), 1183–1201 (2003)

    Article  MathSciNet  Google Scholar 

  5. Desjardins, B.: Regularity of weak solutions of the compressible isentropic Navier–Stokes equations. Commun. Partial Differ. Equ. 22(5), 977–1008 (1997)

    Article  MathSciNet  Google Scholar 

  6. Ducomet, Bernard, Necasovv, Sarka: On the 2D compressible Navier–Stokes system with density-dependent viscosities. Nonlinearity 26(6), 1783–1797 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  7. Feireisl, E., Novotny, A., Petzeltova, H.: On the existence of globally defined weak solutions to the Navier–Stokes equations. J. Math. Fluid Mech. 3, 358–392 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  8. Hoff, D.: Global solutions of the Navier–Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differ. Equ. 120(1), 215–254 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  9. Hoff, D.: Dynamics of singularity surfaces or compressible, viscous flows in two space dimensions. Commun. Pure. Appl Math. LV, 1365–1407 (2002)

  10. Duan, Q., Huang, X.: Local weak solution of the isentropic compressible Navier–Stokes equations in a half-space. Sci. China Math. 65(5), 993–1002 (2022)

    Article  MathSciNet  Google Scholar 

  11. Huang, X.: Existence and uniqueness of weak solutions of the compressible spherically symmetric Navier-Stokes equations. J. Differ. Equ. 262, 1341–1358 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  12. Huang, X., Li, J.: Existence and blowup behavior of global strong solutions to the two-dimensional barotropic compressible Navier–Stokes system with vacuum and large initial data. J. Math. Pures Appl. (9) 106(1), 123–154 (2016)

    Article  MathSciNet  Google Scholar 

  13. Huang, X., Li, J.: Global well-posedness of classical solutions to the Cauchy problem of two-dimensional barotropic compressible Navier-Stokes system with vacuum and large initial data. SIAM J. Math. Anal. 54(3), 3192–3214 (2022)

    Article  MathSciNet  Google Scholar 

  14. Huang, X., Yan, W.: Local weak solution of the isentropic compressible Navier–Stokes equations. J. Math. Phys. 62(11), 111504 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  15. Itaya, N.: On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscous fluids. Kodai Math. Sem. Rep. 23, 60–120 (1971)

    Article  MathSciNet  Google Scholar 

  16. Jiang, S., Zhang, P.: On Spherically Symmetric Solutions of the Compressible Isentropic Navier–Stokes Equations. Commun. Math. Phys. 215, 559–581 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  17. Jiu, Q., Wang, Y., Xin, Z.: Global well-posedness of 2D compressible Navier–Stokes equations with large data and vacuum. J. Math. Fluid Mech. 16(3), 483–521 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  18. Vaigant, V.A., Kazhikhov, A.V.: On existence of global solutions to the two-dimensional Navier–Stokes equations for a compressible viscous fluid. Sib. Math. J. 36(6), 1283–1316 (1995)

    Article  MathSciNet  Google Scholar 

  19. Kazhikhov, A.V.: Stabilization of solutions of an initial-boundary-value problem for the equations of motion of a barotropic viscous fluid. Differ. Equ. 15, 463–467 (1979)

    Google Scholar 

  20. Lions, P.L.: Mathematical topics in fluid mechanics, vol. 2. Compressible models. Oxford University Press, New York (1998)

  21. Li, J., Liang, Z.L.: On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier–Stokes equations with vacuum. J. Math. Pures Appl. 102(9), 640–671 (2014)

    Article  MathSciNet  Google Scholar 

  22. Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Jpn. Acad. Ser. A Math. Sci. 55(9), 337–342 (1979)

    Article  MathSciNet  Google Scholar 

  23. Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20(1), 67–104 (1980)

    MathSciNet  Google Scholar 

  24. Matsumura, A., Nishida, T.: Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys. 89(4), 445–464 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  25. Nash, J.: Le probleme de Cauchy pour les equations differentielles dn fluide general. Bull. Soc. Math. France 90, 487–497 (1962)

    Article  MathSciNet  Google Scholar 

  26. Salvi, R., Straskraba, I.: Global existence for viscous compressible fluids and their behavior as \(t\rightarrow \infty \). J. Fac. Sci. Univ. Tokyo Sect. IA Math. 40, 17–51 (1993)

    MathSciNet  Google Scholar 

  27. Serrin, J.: On the uniqueness of compressible fluid motions. Arch. Ration. Mech. Anal. 3, 271–288 (1959)

    Article  MathSciNet  Google Scholar 

  28. Solonnikov, V.A.: Solvability of the initial boundary value problem for the equation of a viscous compressible fluid. J. Sov. Math. 14, 1120–1133 (1980)

    Article  Google Scholar 

  29. Valli, A.: An existence theorem for compressible viscous fluids. Ann. Mat. Pura Appl. (IV) 130, 197–213 (1982)

    Article  MathSciNet  Google Scholar 

  30. Zhang, P.X., Zhao, J.N.: The existence of local solutions for the compressible Navier–Stokes equations with the density-dependent viscosities. Commun. Math. Sci. 12(7), 1277–1302 (2014)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Q.Duan is partially supported by the National Natural Science Foundation of China, Nos.12371235, 12271369. X.-D. Huang is partially supported by NNSFC Grant Nos. 11971464, 11688101 and CAS Project for Young Scientists in Basic ResearchGrant No.YSBR-031, National Key R &D Program of China, Grant No.2021YFA1000800.

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Authors and Affiliations

  1. School of Mathematical Sciences, Shenzhen University, Shenzhen, 518060, China

    Qin Duan

  2. Institute of Mathematics, Academy of Mathematics and Systems Sciences Chinese Academy of Sciences, Beijing, 100190, China

    Xiangdi Huang

  3. Hua Loo-Keng Center for Mathematical Sciences, Chinese Academy of Sciences, Beijing, China

    Xiangdi Huang

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  1. Qin Duan
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Correspondence to Xiangdi Huang.

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Duan, Q., Huang, X. Local Weak Solution of the Isentropic Compressible Navier–Stokes Equations with Variable Viscosity. J. Math. Fluid Mech. 26, 45 (2024). https://doi.org/10.1007/s00021-024-00871-6

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