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On the Global Existence for a Class of Compressible Non-Newtonian Fluids with Inhomogeneous Boundary Data

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Abstract

This paper is concerned to the study of global existence of weak solutions to a class of compressible non-Newtonian fluids in three-dimensional bounded domain. More precisely, we consider an isentropic compressible non-Newtonian fluid with adiabatic constant \(\gamma>\frac{3}{2}\). We study the global existence of an initial boundary value problem with nonhomogeneous Dirichlet boundary conditions by constructing an approximation scheme, energy estimates, and a weak convergence method.

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References

  1. S. A. Antontsev and A. A. Kazhikov, Mathematical Study of Flows of Nonhomogeneous Fluids, Lecture Notes, 1973.

    Google Scholar 

  2. C. Baranger, L. Boudin, P. E. Jabin, and M. Simona, “A Modeling of Biospray for the Upper Airways”, In ESAIM Proc. EDP Sci., 14 (2005), 41–47.

    Article  MathSciNet  Google Scholar 

  3. M. E. Bogovskii, “Solution of Some Vector Analysis Problems Connected with Operators Div and Grad”, Tr. Sem. S.L. Sobolev, 80 (1980), 5–40.

    MathSciNet  Google Scholar 

  4. L. Boudin, L. Desvilletter, and M. Motte, A Modeling of Compressible Droplets in a Fluid, 2003.

    Google Scholar 

  5. J. Choe and H. Kim, Strong Solutions of the Navier–Stokes Equations for Nonhomogeneous Incompressible Fluids, 2003.

    Google Scholar 

  6. R. J. DiPerna and P. L. Lions, “Ordinary Differential Equations, Transport Theory and Sobolev Spaces”, Invent. Math., 98:3 (1989), 511–547.

    Article  ADS  MathSciNet  Google Scholar 

  7. L. Fang and Z. Guo, “Global Weak Solutions to a Three-Dimensional Compressible Non-Newtonian Fluid”, Comm. Math. Sci., 20:6 (2022), 17–33.

    Article  MathSciNet  Google Scholar 

  8. L. Fang, X. J. Kong, and J. J. Liu, “Weak Solution to a One-Dimensional Full Compressible Non-Newtonian Fluids”, Math. Models Methods Appl. Sci., 41:9 (2018), 3441–3462.

    Article  ADS  MathSciNet  Google Scholar 

  9. E. Feireisl, A. Novotný, and H. Petzeltová, “On the Existence of Globally Defined Weak Solutions to the Navier–Stokes Equations”, J. Math. Fluid Mech., 3:4 (2001), 358–392.

    Article  ADS  MathSciNet  Google Scholar 

  10. E. Feireisl, Dynamics of Viscous Compressible Fluids, vol. 26, Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, 2004.

    Google Scholar 

  11. E. Feireisl, X. Liao, and J. Málek, “Global Weak Solutions to a Class of Non-Newtonian Compressible Fluids”, Math. Methods Appl. Sci., 38:16 (2015), 3482–3494.

    Article  ADS  MathSciNet  Google Scholar 

  12. V. Girinon, “Navier–Stokes Equations with Nonhomogeneous Boundary Conditions in a Convex Bi-Dimensional Domain”, Ann. Inst. H. Poincaré, 1:26 (5) (2009), 2025–53.

    Article  MathSciNet  Google Scholar 

  13. V. Girinon, “Navier–Stokes Equations with Nonhomogeneous Boundary Conditions in a Bounded Three-Dimensional Domain”, J. Math. Fluid Mech., 13:3 (2011), 309–39.

    Article  ADS  MathSciNet  Google Scholar 

  14. S. Itoh and A. Tani, “Solvability of Nonstationary Problems for Nonhomogeneous Incompressible Fluids and the Convergence with Vanishing Viscosity”, Tokyo J. Math., 22 (1999), 17–42.

    Article  MathSciNet  Google Scholar 

  15. S. Jiang and P. Zhang, “Axisymmetric Solutions of the 3D Navier–Stokes Equations for Compressible Isentropic Fluids”, J. Math. Pures Appl., 82:8 (2003), 949–973.

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  17. O. A. Ladyzhenskaya, “New Equations for the Description of the Motions of Viscous Incompressible Fluids, and Global Solvability for Their Boundary Value Problems”, Tr. Math. Inst. Steklov, 102 (1967), 85–104.

    MathSciNet  Google Scholar 

  18. O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, NY. Gordon and Breach, New York, 1969.

    Google Scholar 

  19. O. A. Ladyzhenskaya and V. A. Solonnikov, “Unique Solvability of an Initial-And Boundary-Value Problem for Viscous Incompressible Nonhomogeneous Fluids”, J. Soviet Mathematics, 9 (1978), 697–749.

    Article  Google Scholar 

  20. J. Leray, “Sur le mouvement d’un liquide visqueux emplissant l’espace”, Acta Math., 63 (1934), 193–248.

    Article  MathSciNet  Google Scholar 

  21. P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1, Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, 3, Oxford Science Publications, New York, NY. The Clarendon Press, Oxford University Press, 1996.

    Google Scholar 

  22. P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2, Compressible Models, Oxford Lecture Series in Mathematics and its Applications, 10, Oxford Science Publications, New York, NY. The Clarendon Press, Oxford University Press, 1998.

    Google Scholar 

  23. A. E. Mamontov, “Global Regularity Estimates for Multi-Dimensional Equations of Compressible Non-Newtonian Fluid”, Dinamika Sploshn. Sredy, 116 (2000), 50–54.

    Google Scholar 

  24. A. E. Mamontov, “Existence of Global Solutions to Multi-Dimensional Equations for Bingham Fluids”, Mat. Zametki, 82:4 (2007), 560–577; Math. Notes, 82:3–4 (2007), 501–517.

    Google Scholar 

  25. J. Muhammad, L. Fang, and Z. Guo, “Global Weak Solutions to a Class of Compressible Non-Newtonian Fluids with Vacuum”, Math. Methods Appl. Sci., (2020), 1–16.

    MathSciNet  Google Scholar 

  26. S. Novo, “Compressible Navier–Stokes Model with Inflow-Outflow Boundary Conditions”, J. Math. Fluid Mech., 7 (2005), 485–514.

    Article  ADS  MathSciNet  Google Scholar 

  27. A. Novotny and I. Straskraba, Introduction to the Mathematical Theory of Compressible Flow, OUP Oxford, 2004.

    Book  Google Scholar 

  28. M. Padula, “On the Existence and Uniqueness of Non-Homogeneous Motions in Exterior Domains”, Math. Z., 203 (1990), 581–604.

    Article  MathSciNet  Google Scholar 

  29. M. Paicu and P. Zhang, “Global Solutions to the 3D Incompressible Inhomogeneous Navier–Stokes System”, J. Func. Anal., 262:8 (2012), 3556–84.

    Article  Google Scholar 

  30. J. Simon, “Nonhomogeneous Viscous Incompressible Fluids: Existence of Velocity, Density, and Pressure”, SIAM J. Math. Anal., 21:5 (1990), 1093–1117.

    Article  MathSciNet  Google Scholar 

  31. A. Valli and W. M. Zajaczkowski, “Navier–Stokes Equations for Compressible Fluids: Global Existence and Qualitative Properties of the Solutions in the General Case”, Comm. Math. Phys., 103 (1986), 259–96.

    Article  ADS  MathSciNet  Google Scholar 

  32. F. A. Williams, “Spray Combustion and Atomization”, the Phys. Fluids, 1:6 (1958), 541–545.

    Article  ADS  Google Scholar 

  33. J. Zhang, “Global Well-Posedness for The Incompressible Navier–Stokes Equations with Density-Dependent Viscosity Coefficient”, J. Differ. Equations, 259:5 (2015), 1722–42.

    Article  ADS  MathSciNet  Google Scholar 

  34. V. V. Zhikov and S. E. Pastukhova, “On the Solvability of the Navier–Stokes System for a Compressible Non-Newtonian Fluid”, Dokl. Akad. Nauk, 427:3 (2009), 303–307.

    MathSciNet  Google Scholar 

  35. V. V. Zhikov, “On the Weak Convergence of Fluxes to a Flux”, Dokl. Akad. Nauk, 81 (2010), 58–62.

    MathSciNet  Google Scholar 

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  1. Department of Mathematics, Shanghai University, Shanghai, 200444, P. R. China

    J. Muhammad

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Muhammad, J. On the Global Existence for a Class of Compressible Non-Newtonian Fluids with Inhomogeneous Boundary Data. Russ. J. Math. Phys. 31, 276–298 (2024). https://doi.org/10.1134/S1061920824020109

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