×

High-temperature viscous flows. A case of pudding model. (English) Zbl 1382.35236

Summary: We investigate the Navier-Stokes-Fourier system for incompressible heat conducting inhomogeneous fluid. The main result concerns existence of global in time regular large solutions, provided the initial temperature is sufficiently large. Magnitudes of norms of initial velocity and gradients of initial temperature and density are not limited. The system may be viewed as a model of pudding, as we assume the viscosity grows with the temperature.

MSC:

35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
80A20 Heat and mass transfer, heat flow (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Amann, H., (Linear and Quasilinear Parabolic Problems. Vol. I. Linear and Quasilinear Parabolic Problems. Vol. I, Monographs in Mathematics, vol. 89 (1995), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA), Abstract linear theory · Zbl 0819.35001
[2] Bardos, C.; Lopes Filho, M. C.; Niu, D.; Nussenzveig Lopes, H. J.; Titi, E. S., Stability of two-dimensional viscous incompressible flows under three-dimensional perturbations and inviscid symmetry breaking, SIAM J. Math. Anal., 45, 3, 1871-1885 (2013) · Zbl 1291.35207
[3] Bulíček, M.; Feireisl, E.; Málek, J., A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients, Nonlinear Anal. RWA, 10, 2, 992-1015 (2009) · Zbl 1167.76316
[4] Bulíček, M.; Kaplický, P.; Málek, J., An \(L^2\)-maximal regularity result for the evolutionary Stokes-Fourier system, Appl. Anal., 90, 1, 31-45 (2011) · Zbl 1209.35102
[5] Bulíček, M.; Lewandowski, R.; Málek, J., On evolutionary Navier-Stokes-Fourier type systems in three spatial dimensions, Comment. Math. Univ. Carolin., 52, 1, 89-114 (2011) · Zbl 1240.35378
[6] Bulíček, M.; Málek, J.; Shilkin, T. N., On the regularity of two-dimensional unsteady flows of heat-conducting generalized Newtonian fluids, Nonlinear Anal. RWA, 19, 89-104 (2014) · Zbl 1367.76016
[7] Danchin, R., Global existence in critical spaces for flows of compressible viscous and heat-conductive gases, Arch. Ration. Mech. Anal., 160, 1, 1-39 (2001) · Zbl 1018.76037
[8] Danchin, R.; Mucha, P. B., A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space, J. Funct. Anal., 256, 3, 881-927 (2009) · Zbl 1160.35004
[9] Danchin, R.; Mucha, P. B., A Lagrangian approach for the incompressible Navier-Stokes equations with variable density, Comm. Pure Appl. Math., 65, 10, 1458-1480 (2012) · Zbl 1247.35088
[10] Danchin, R.; Mucha, P. B., Incompressible flows with piecewise constant density, Arch. Ration. Mech. Anal., 207, 3, 991-1023 (2013) · Zbl 1260.35107
[11] Giovangigli, V., Multicomponent flow modeling, (Modeling and Simulation in Science, Engineering and Technology (1999), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA) · Zbl 0956.76003
[12] Ladyženskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N., Linear and quasilinear equations of parabolic type, (Translations of Mathematical Monographs, Vol. 23 (1968), American Mathematical Society: American Mathematical Society Providence, R.I), Translated from the Russian by S. Smith · Zbl 0174.15403
[13] Ladyzhenskaya, O. A., The mathematical theory of viscous incompressible flow, (Mathematics and its Applications, Vol. 2 (1969), Gordon and Breach, Science Publishers: Gordon and Breach, Science Publishers New York-London-Paris), Translated from the Russian by Richard A. Silverman and John Chu. · Zbl 0184.52603
[14] Lewandowski, R., The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity, Nonlinear Anal., 28, 2, 393-417 (1997) · Zbl 0863.35077
[15] Lions, P.-L., (Mathematical Topics in Fluid Mechanics. Vol. 1. Mathematical Topics in Fluid Mechanics. Vol. 1, Oxford Lecture Series in Mathematics and its Applications, vol. 3 (1996), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York), Incompressible models, Oxford Science Publications · Zbl 0866.76002
[16] Mucha, P. B., Stability of nontrivial solutions of the Navier-Stokes system on the three dimensional torus, J. Differential Equations, 172, 2, 359-375 (2001) · Zbl 1016.35056
[17] Mucha, P. B., The Cauchy problem for the compressible Navier-Stokes equations in the \(L_p\)-framework, Nonlinear Anal., 52, 4, 1379-1392 (2003) · Zbl 1048.35065
[18] Mucha, P. B.; Zaja̧czkowski, W., On the existence for the Cauchy-Neumann problem for the Stokes system in the \(L_p\)-framework, Studia Math., 143, 1, 75-101 (2000) · Zbl 0970.35107
[19] Mucha, P. B.; Zaja̧czkowski, W. M., On a \(L_p\)-estimate for the linearized compressible Navier-Stokes equations with the Dirichlet boundary conditions, J. Differential Equations, 186, 2, 377-393 (2002) · Zbl 1048.35064
[20] Paicu, M.; Zhang, P.; Zhang, Z., Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density, Comm. Partial Differential Equations, 38, 7, 1208-1234 (2013) · Zbl 1314.35086
[21] Ponce, G.; Racke, R.; Sideris, T. C.; Titi, E. S., Global stability of large solutions to the 3D Navier-Stokes equations, Comm. Math. Phys., 159, 2, 329-341 (1994) · Zbl 0795.35082
[22] Solonnikov, V. A., Estimates for solutions of a non-stationary linearized system of Navier-Stokes equations, Tr. Mat. Inst. Steklova, 70, 213-317 (1964) · Zbl 0163.33803
[23] Solonnikov, V. A., An initial-boundary value problem for a Stokes system that arises in the study of a problem with a free boundary, Tr. Mat. Inst. Steklova, 188, 150-188 (1990), Translated in: Proc. Steklov Inst. Math., in: Boundary value problems of mathematical physics, vol. 14, 1991 (3) 191-239 (Russian) · Zbl 0731.35079
[24] Zadrzyńska, E.; Zaja̧czkowski, W. M., Stability of two-dimensional Navier-Stokes motions in the periodic case, J. Math. Anal. Appl., 423, 2, 956-974 (2015) · Zbl 1308.35182
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.