Linear stability of viscous shock wave for 1-D compressible Navier-Stokes equations with Maxwell’s law

Hu, Yuxi; Wang, Zhao

doi:10.1090/qam/1608

Authors: Yuxi Hu and Zhao Wang
Journal: Quart. Appl. Math. 80 (2022), 221-235
MSC (2020): Primary 35B35, 76N10, 35B40
DOI: https://doi.org/10.1090/qam/1608
Published electronically: January 10, 2022
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Abstract: In this paper, we consider the linear stability of traveling wave solutions for one-dimensional compressible isentropic Navier-Stokes equations with Maxwell’s Law. The global stability of traveling wave solution is established with shock-profile initial data for the linearized system. Anti-derivative and some delicate energy methods are explored to get the desired results. Moreover, the relaxation limit of traveling wave solution is also obtained.

References

J. C. Maxwell, On the dynamical theory of gases, Philos. Trans. Roy. Soc. A. 157 (1867), 49–88.
Y. I. Kanel, One model system of equations of one-dimensional gas motions, J. Diff. Equations 4 (1968), 374–380.
A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, Prikl. Mat. Meh. 41 (1977), no. 2, 282–291 (Russian); English transl., J. Appl. Math. Mech. 41 (1977), no. 2, 273–282. MR 0468593, DOI 10.1016/0021-8928(77)90011-9
Shuichi Kawashima and Akitaka Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101 (1985), no. 1, 97–127. MR 814544, DOI 10.1007/BF01212358
Akitaka Matsumura and Kenji Nishihara, On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math. 2 (1985), no. 1, 17–25. MR 839317, DOI 10.1007/BF03167036
Akitaka Matsumura and Kenji Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas, Comm. Math. Phys. 222 (2001), no. 3, 449–474. MR 1888084, DOI 10.1007/s002200100517
Akitaka Matsumura and Ming Mei, Convergence to travelling fronts of solutions of the $p$-system with viscosity in the presence of a boundary, Arch. Ration. Mech. Anal. 146 (1999), no. 1, 1–22. MR 1682659, DOI 10.1007/s002050050134
Hakho Hong and Feimin Huang, Asymptotic behavior of solutions toward the superposition of contact discontinuity and shock wave for compressible Navier-Stokes equations with free boundary, Acta Math. Sci. Ser. B (Engl. Ed.) 32 (2012), no. 1, 389–412. MR 2921885, DOI 10.1016/S0252-9602(12)60025-3
Song Jiang, On the asymptotic behavior of the motion of a viscous, heat-conducting, one-dimensional real gas, Math. Z. 216 (1994), no. 2, 317–336. MR 1278427, DOI 10.1007/BF02572324
Feimin Huang, Akitaka Matsumura, and Xiaoding Shi, A gas-solid free boundary problem for a compressible viscous gas, SIAM J. Math. Anal. 34 (2003), no. 6, 1331–1355. MR 2000974, DOI 10.1137/S0036141002403730
Yuxi Hu and Na Wang, Global existence versus blow-up results for one dimensional compressible Navier-Stokes equations with Maxwell’s law, Math. Nachr. 292 (2019), no. 4, 826–840. MR 3937620, DOI 10.1002/mana.201700418
Yuxi Hu and Reinhard Racke, Compressible Navier-Stokes equations with revised Maxwell’s law, J. Math. Fluid Mech. 19 (2017), no. 1, 77–90. MR 3607446, DOI 10.1007/s00021-016-0266-5
Tohru Nakamura and Shuichi Kawashima, Viscous shock profile and singular limit for hyperbolic systems with Cattaneo’s law, Kinet. Relat. Models 11 (2018), no. 4, 795–819. MR 3810847, DOI 10.3934/krm.2018032
Akitaka Matsumura and Yang Wang, Asymptotic stability of viscous shock wave for a one-dimensional isentropic model of viscous gas with density dependent viscosity, Methods Appl. Anal. 17 (2010), no. 3, 279–290. MR 2785875, DOI 10.4310/MAA.2010.v17.n3.a3
Tai-Ping Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math. 50 (1997), no. 11, 1113–1182. MR 1470318, DOI 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.3.CO;2-8
Tai-Ping Liu and Yanni Zeng, Shock waves in conservation laws with physical viscosity, Mem. Amer. Math. Soc. 234 (2015), no. 1105, vi+168. MR 3244333, DOI 10.1090/memo/1105
Corrado Mascia and Kevin Zumbrun, Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems, Arch. Ration. Mech. Anal. 172 (2004), no. 1, 93–131. MR 2048568, DOI 10.1007/s00205-003-0293-2
Anders Szepessy and Zhou Ping Xin, Nonlinear stability of viscous shock waves, Arch. Rational Mech. Anal. 122 (1993), no. 1, 53–103. MR 1207241, DOI 10.1007/BF01816555
Alexis F. Vasseur and Lei Yao, Nonlinear stability of viscous shock wave to one-dimensional compressible isentropic Navier-Stokes equations with density dependent viscous coefficient, Commun. Math. Sci. 14 (2016), no. 8, 2215–2228. MR 3576272, DOI 10.4310/CMS.2016.v14.n8.a5
Reinhard Racke and Jürgen Saal, Hyperbolic Navier-Stokes equations I: Local well-posedness, Evol. Equ. Control Theory 1 (2012), no. 1, 195–215. MR 3085225, DOI 10.3934/eect.2012.1.195
Reinhard Racke and Jürgen Saal, Hyperbolic Navier-Stokes equations II: Global existence of small solutions, Evol. Equ. Control Theory 1 (2012), no. 1, 217–234. MR 3085226, DOI 10.3934/eect.2012.1.217
Alexander Schöwe, A quasilinear delayed hyperbolic Navier-Stokes system: global solution, asymptotics and relaxation limit, Methods Appl. Anal. 19 (2012), no. 2, 99–118. MR 3011596, DOI 10.4310/MAA.2012.v19.n2.a1
Wen-An Yong, Newtonian limit of Maxwell fluid flows, Arch. Ration. Mech. Anal. 214 (2014), no. 3, 913–922. MR 3269638, DOI 10.1007/s00205-014-0769-2
G. Maisano, et al., Evidence of anomalous acoustic behavior from Brillouin scattering in supercooled water, Phys. Rev. Lett. 52 (1984), 1025.
M. Pelton, et al., Viscoelastic flows in simple liquids generated by vibrating nanostructures, Phys. Rev. Lett. 111 (2013), 244–502.
F. Sette, et al., Collective dynamics in water by high energy resolution inelastic X-ray scattering, Phys. Rev. Lett. 75 (1995), 850.
D. Chakraborty and J. E. Sader, Constitutive models for linear compressible viscoelastic flows of simple liquids at nanometer length scales, Phys. Fluids 27 (2015), 052002-1–052002-13.