Space-time behavior of the compressible Navier-Stokes equations with hyperbolic heat conduction. (English) Zbl 1531.76076
Summary: We consider compressible Navier-Stokes equations with hyperbolic heat conduction in \(\mathbb{R}^3\). A space-time description of the classical solution is given when the initial perturbation is suitable small. The result implies that all of the unknowns obey the generalized Huygens’ principle as the classical compressible Navier-Stokes equations in [T.-P. Liu and W. Wang, Commun. Math. Phys. 196, No. 1, 145–173 (1998; Zbl 0912.35122)]. Additionally, we show that the decay of the flux \(\mathbf{q}\) is faster than the other unknowns.
©2023 American Institute of Physics
©2023 American Institute of Physics
MSC:
| 76N06 | Compressible Navier-Stokes equations |
| 80A19 | Diffusive and convective heat and mass transfer, heat flow |
Citations:
Zbl 0912.35122References:
| [1] | Liu, T. P.; Wang, W. K., The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Commun. Math. Phys., 196, 145-173 (1998) · Zbl 0912.35122 · doi:10.1007/s002200050418 |
| [2] | Hu, Y.; Racke, R., Compressible Navier-Stokes equations with hyperbolic heat conduction, J. Hyperbolic Differ. Equations, 13, 2, 233-247 (2016) · Zbl 1398.35171 · doi:10.1142/s0219891616500077 |
| [3] | Jun Choe, H.; Kim, H., Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Differ. Equations, 190, 504-523 (2003) · Zbl 1022.35037 · doi:10.1016/s0022-0396(03)00015-9 |
| [4] | Feireisl, E.; Novotny, A.; Petzeltová, H., On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3, 358-392 (2001) · Zbl 0997.35043 · doi:10.1007/pl00000976 |
| [5] | Fernández Sare, H. D.; Racke, R., On the stability of damped timoshenko systems: Cattaneo versus Fourier law, Arch. Ration. Mech. Anal., 194, 221-251 (2009) · Zbl 1251.74011 · doi:10.1007/s00205-009-0220-2 |
| [6] | Hoff, D., Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differ. Equations, 120, 215-254 (1995) · Zbl 0836.35120 · doi:10.1006/jdeq.1995.1111 |
| [7] | Huang, X. D.; Li, J.; Xin, Z. P., Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Commun. Pure Appl. Math., 65, 549-585 (2012) · Zbl 1234.35181 · doi:10.1002/cpa.21382 |
| [8] | Jiang, S.; Racke, R., Evolution Equations in Thermoelasticity. Monographs Surveys Pure and Applied Mathematics Vol. 112 (2000), Chapman & Hall/CRC: Chapman & Hall/CRC, Boca Raton · Zbl 0968.35003 |
| [9] | Kawashima, S., Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics (1983), Kyoto University |
| [10] | Kobayashi, T.; Shibata, Y., Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in \(\mathbb{R}^3\), Commun. Math. Phys., 200, 621-659 (1999) · Zbl 0921.35092 · doi:10.1007/s002200050543 |
| [11] | Lions, P. L., Mathematical Topics in Fluid Mechanics: Compressible Models, Vol. II (1998), Clarendon Press: Clarendon Press, Oxford · Zbl 0908.76004 |
| [12] | Duan, R. J.; Ukai, S.; Yang, T.; Zhao, H. J., Optimal convergence rates for the compressible Navier-Stokes equations with potential forces, Math. Models Methods Appl. Sci., 17, 737-758 (2007) · Zbl 1122.35093 · doi:10.1142/s021820250700208x |
| [13] | Duan, R.; Liu, H.; Ukai, S.; Yang, T., Optimal L^p-L^q convergence rates for the compressible Navier-Stokes equations with potential force, J. Differ. Equations, 238, 220-233 (2007) · Zbl 1121.35096 · doi:10.1016/j.jde.2007.03.008 |
| [14] | 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, 55, 337-342 (1979) · Zbl 0447.76053 · doi:10.3792/pjaa.55.337 |
| [15] | Matsumura, A.; Nishida, T., The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20, 67-104 (1980) · Zbl 0429.76040 · doi:10.1215/kjm/1250522322 |
| [16] | Ponce, G., Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9, 5, 399-418 (1985) · Zbl 0576.35023 · doi:10.1016/0362-546x(85)90001-x |
| [17] | Li, H. L.; Zhang, T., Large time behavior of isentropic compressible Navier-Stokes system in \(\mathbb{R}^3\), Math. Methods Appl. Sci., 34, 6, 670-682 (2011) · Zbl 1214.35047 · doi:10.1002/mma.1391 |
| [18] | Wang, W. J., Optimal convergence rates for the strong solutions to the compressible Navier-Stokes equations with potential force, Nonlinear Anal., 34, 363-378 (2017) · Zbl 1354.35072 · doi:10.1016/j.nonrwa.2016.09.005 |
| [19] | Danchin, R.; Xu, J., Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical L^p framework, Arch. Ration. Mech. Anal., 224, 1, 53-90 (2017) · Zbl 1366.35126 · doi:10.1007/s00205-016-1067-y |
| [20] | Liu, T. P.; Zeng, Y. N., Large Time Behavior of Solutions for General Quasilinear Hyperbolic-Parabolic Systems of Conservation Laws, 125 (1997), American Mathematical Society · Zbl 0884.35073 |
| [21] | Hoff, D.; Zumbrun, K., Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44, 2, 603-676 (1995) · Zbl 0842.35076 · doi:10.1512/iumj.1995.44.2003 |
| [22] | Liu, T. P.; Noh, S. E., Wave propagation for the compressible Navier-Stokes equations, J. Hyperbolic Differ. Equations, 12, 385-445 (2015) · Zbl 1327.76112 · doi:10.1142/s0219891615500113 |
| [23] | Deng, S. J.; Yu, S. H., Green’s function and pointwise convergence for compressible Navier-Stokes equations, Q. Appl. Math., 75, 3, 433-503 (2017) · Zbl 1360.35141 · doi:10.1090/qam/1461 |
| [24] | Du, L.; Wu, Z., Solving the non-isentropic Navier-Stokes equations in odd space dimensions: The Green function method, J. Math. Phys., 58, 10, 101506 (2017) · Zbl 1374.76194 · doi:10.1063/1.5005915 |
| [25] | Wang, W.; Yang, T., The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differ. Equations, 173, 410-450 (2001) · Zbl 0997.35039 · doi:10.1006/jdeq.2000.3937 |
| [26] | Wang, W.; Wu, Z., Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions, J. Differ. Equations, 248, 7, 1617-1636 (2010) · Zbl 1185.35177 · doi:10.1016/j.jde.2010.01.003 |
| [27] | Wu, Z.; Wang, W., Pointwise estimates for bipolar compressible Navier-Stokes-Poisson system in dimension three, Arch. Ration. Mech. Anal., 226, 2, 587-638 (2017) · Zbl 1373.35258 · doi:10.1007/s00205-017-1140-1 |
| [28] | Duan, R., Green’s function and large time behavior of the Navier-Stokes-Maxwell system, Anal. Appl., 10, 133-197 (2012) · Zbl 1241.35145 · doi:10.1142/s0219530512500078 |
| [29] | Deng, S. J.; Yang, X. F., Pointwise structure of a radiation hydrodynamic model in one-dimension, Math. Methods Appl. Sci., 43, 6, 3432-3456 (2020) · Zbl 1440.76135 · doi:10.1002/mma.6130 |
| [30] | Hu, Y.; Racke, R., Formation of singularities in one-dimensional thermoelasticity with second sound, Q. Appl. Math, 72, 311-321 (2014) · Zbl 1295.35311 · doi:10.1090/s0033-569x-2014-01336-2 |
| [31] | Qin, Y.; Li, H., Global existence and exponential stability of solutions to the quasilinear thermo-diffusion equations with second sound, Acta Math. Sci., 34, 3, 759-778 (2014) · Zbl 1313.35024 · doi:10.1016/s0252-9602(14)60047-3 |
| [32] | Racke, R., Thermoelasticity with second sound-exponential stability in linear and non-linear 1d, Math. Methods Appl. Sci., 25, 409-441 (2002) · Zbl 1008.74027 · doi:10.1002/mma.298 |
| [33] | Racke, R.; Dafermos, C. M.; Pokorný, M., Thermoelasticity, Evolutionary Equations: Handbook of Differential Equations, 5, 315-420 (2009), Elsevier · Zbl 1179.35004 |
| [34] | Wu, Z.; Zhou, W.; Li, Y., Decay of the compressible Navier-Stokes equations with hyperbolic heat conduction, Commun. Math. Anal. Appl., 2, 115-141 (2023) · Zbl 1538.35072 · doi:10.4208/cmaa.2022-0022 |
| [35] | Guo, Y.; Wang, Y., Decay of dissipative equations and negative Sobolev spaces, Commun. Partial Differ. Equations, 37, 2165-2208 (2012) · Zbl 1258.35157 · doi:10.1080/03605302.2012.696296 |
| [36] | Li, D. L., The Green’s function of the Navier-Stokes equations for gas dynamics in \(\mathbb{R}^3\), Commun. Math. Phys., 257, 579-619 (2005) · Zbl 1075.76053 · doi:10.1007/s00220-005-1351-4 |
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.
