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On the Navier-Stokes equations with temperature-dependent transport coefficients. (English) Zbl 1133.35419
Summary: We establish long-time and large-data existence of a weak solution to the problem describing three-dimensional unsteady flows of an incompressible fluid, where the viscosity and heat-conductivity coefficients vary with the temperature. The approach reposes on considering the equation for the total energy rather than the equation for the temperature. We consider the spatially periodic problem.

35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
76M10 Finite element methods applied to problems in fluid mechanics
Full Text: DOI EuDML
[1] L. Caffarelli, R. Kohn, and L. Nirenberg, “Partial regularity of suitable weak solutions of the Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 35, no. 6, pp. 771-831, 1982. · Zbl 0509.35067 · doi:10.1002/cpa.3160350604
[2] L. Consiglieri, “Weak solutions for a class of non-Newtonian fluids with energy transfer,” Journal of Mathematical Fluid Mechanics, vol. 2, no. 3, pp. 267-293, 2000. · Zbl 0974.35090 · doi:10.1007/PL00000952
[3] L. Consiglieri, J. F. Rodrigues, and T. Shilkin, “On the Navier-Stokes equations with the energy-dependent nonlocal viscosities,” Zapiski Nauchnykh Seminarov Sankt-Peterburgskoe Otdelenie. Matematicheskiĭ Institut im. V. A. Steklova. (POMI), vol. 306, pp. 71-91, 2003. · Zbl 1148.35342
[4] J. Duchon and R. Robert, “Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations,” Nonlinearity, vol. 13, no. 1, pp. 249-255, 2000. · Zbl 1009.35062 · doi:10.1088/0951-7715/13/1/312
[5] D. G. Ebin, “Viscous fluids in a domain with frictionless boundary,” in Global Analysis-Analysis on Manifolds, H. Kurke, J. Mecke, H. Triebel, and R. Thiele, Eds., vol. 57 of Teubner-Texte zur Mathematik, pp. 93-110, Teubner, Leipzig, 1983. · Zbl 0525.58030
[6] G. L. Eyink, “Local 4/5-law and energy dissipation anomaly in turbulence,” Nonlinearity, vol. 16, no. 1, pp. 137-145, 2003. · Zbl 1138.76358 · doi:10.1088/0951-7715/16/1/309
[7] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2003.
[8] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, vol. 23 of Trans. Math. Monograph, American Mathematical Society, Rhode Island, 1968. · Zbl 0174.15403
[9] J. Leray, “Sur le mouvement d’un liquide visqueux emplissant l’espace,” Acta Mathematica, vol. 63, pp. 193-248, 1934. · JFM 60.0726.05 · doi:10.1007/BF02547354
[10] J. Málek and K. R. Rajagopal, “Mathematical issues concerning the Navier-Stokes equations and some of its generalizations,” in Handbook of Differential Equations: Evolutionary Equations. Vol. II, C. Dafermos and E. Feireisl, Eds., pp. 371-459, North-Holland, Amsterdam, 2005. · Zbl 1095.35027
[11] T. Nagasawa, “A new energy inequality and partial regularity for weak solutions of Navier-Stokes equations,” Journal of Mathematical Fluid Mechanics, vol. 3, no. 1, pp. 40-56, 2001. · Zbl 0991.35060 · doi:10.1007/PL00000963
[12] J. Ne\vcas and T. Roubí\vcek, “Buoyancy-driven viscous flow with L1-data,” Nonlinear Analysis, vol. 46, no. 5, pp. 737-755, 2001. · Zbl 1031.76004 · doi:10.1016/S0362-546X(01)00676-9
[13] T. Shilkin, “Classical solvability of the coupled system modelling a heat-convergent Poiseuille-type flow,” Journal of Mathematical Fluid Mechanics, vol. 7, no. 1, pp. 72-84, 2005. · Zbl 1065.35135 · doi:10.1007/s00021-004-0112-z
[14] J. Simon, “Compact sets in the space Lp(0,T;B),” Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 146, pp. 65-96, 1987. · Zbl 0629.46031 · doi:10.1007/BF01762360
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