We treat the 1D shock tube problem, establishing existence of steady solutions of full (nonisentropic) polytropic gas dynamics with arbitrary noncharacteristic data. We present also numerical experiments indicating uniqueness and time-asymptotic stability of such solutions. At the same time, we give an example of an (artificial) equation of state possessing a convex entropy for which there holds nonuniqueness of solutions. This is associated with instability and Hopf bifurcation to time-periodic solutions.
Nous étudions le problème du tube de choc, établissant l’existence et la stabilité de solutions stationnaires des équations de Navier–Stokes non isentropiques pour des données non caractéristiques. Nous présentons aussi des simulations numériques indiquant l’unicité et la stabilité de telles solutions. Dans le même temps, nous donnons un exemple d’équation d’état artificielle possédant une entropie convexe où l’unicité n’a pas lieu. Ce phénomène est associé à une instabilité et une bifurcation de Hopf de solutions périodiques en temps.
Revised:
Accepted:
Published online:
Blake Barker 1; Benjamin Melinand 2; Kevin Zumbrun 3

@article{CML_2023__15__1_0, author = {Blake Barker and Benjamin Melinand and Kevin Zumbrun}, title = {Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible {Navier{\textendash}Stokes} equations}, journal = {Confluentes Mathematici}, pages = {1--26}, publisher = {Institut Camille Jordan}, volume = {15}, year = {2023}, doi = {10.5802/cml.90}, language = {en}, url = {https://cml.centre-mersenne.org/articles/10.5802/cml.90/} }
TY - JOUR AU - Blake Barker AU - Benjamin Melinand AU - Kevin Zumbrun TI - Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier–Stokes equations JO - Confluentes Mathematici PY - 2023 SP - 1 EP - 26 VL - 15 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.90/ DO - 10.5802/cml.90 LA - en ID - CML_2023__15__1_0 ER -
%0 Journal Article %A Blake Barker %A Benjamin Melinand %A Kevin Zumbrun %T Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier–Stokes equations %J Confluentes Mathematici %D 2023 %P 1-26 %V 15 %I Institut Camille Jordan %U https://cml.centre-mersenne.org/articles/10.5802/cml.90/ %R 10.5802/cml.90 %G en %F CML_2023__15__1_0
Blake Barker; Benjamin Melinand; Kevin Zumbrun. Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier–Stokes equations. Confluentes Mathematici, Volume 15 (2023), pp. 1-26. doi : 10.5802/cml.90. https://cml.centre-mersenne.org/articles/10.5802/cml.90/
[1] A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math., Volume 410 (1990), pp. 167-212 | MR | Zbl
[2] Evans function computation, Masters thesis, Brigham Young University, Provo, USA (2009)
[3] Convex entropy, Hopf bifurcation, and viscous and inviscid shock stability, Arch. Ration. Mech. Anal., Volume 217 (2015) no. 1, pp. 309-372 | DOI | MR | Zbl
[4] Evans function computation for the stability of travelling waves, Philos. Trans. R. Soc. Lond., A, Math. Phys. Eng. Sci., Volume 376 (2018) no. 2117, 20170184 | DOI | MR | Zbl
[5] Viscous hyperstabilization of detonation waves in one space dimension, SIAM J. Appl. Math., Volume 75 (2015) no. 3, pp. 885-906 | DOI | MR | Zbl
[6] Viscous hyperstabilization of detonation waves in one space dimension, SIAM J. Appl. Math., Volume 75 (2015) no. 3, pp. 885-906 | DOI | MR | Zbl
[7] Balanced flux formulations for multidimensional Evans-function computations for viscous shocks, Q. Appl. Math., Volume 76 (2018) no. 3, pp. 531-545 | DOI | MR | Zbl
[8] Euler versus Lagrange: the role of coordinates in practical Evans-function computations, SIAM J. Appl. Dyn. Syst., Volume 17 (2018) no. 2, pp. 1766-1785 | DOI | MR | Zbl
[9] A MATLAB-based numerical library for Evans function computation, 2015 (https://github.com/nonlinear-waves/stablab.git)
[10] One-dimensional stability of parallel shock layers in isentropic magnetohydrodynamics, J. Differ. Equations, Volume 249 (2010) no. 9, pp. 2175-2213 | DOI | MR | Zbl
[11] Stability of viscous St. Venant roll waves: from onset to infinite Froude number limit, J. Nonlinear Sci., Volume 27 (2017) no. 1, pp. 285-342 | DOI | MR | Zbl
[12] Existence and stability of viscoelastic shock profiles, Arch. Ration. Mech. Anal., Volume 200 (2011) no. 2, pp. 491-532 | DOI | MR | Zbl
[13] Transverse bifurcation of viscous slow MHD shocks, Physica D, Volume 420 (2021), 132857 | DOI | MR | Zbl
[14] Stability of MHD Shock Waves, Masters thesis, Brigham Young University, Provo, USA (2020)
[15] An introduction to fluid dynamics, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999 | MR | Zbl
[16] Über Abbildung von Mannigfaltigkeiten, Math. Ann., Volume 71 (1911) no. 1, pp. 97-115 | DOI | Zbl
[17] Brouwer degree, the Core of Nonlinear Analysis, Progress in Nonlinear Differential Equations and Their Applications, 95, Birkhäuser, Basel, 2021 | DOI
[18] Numerical solution of Orr–Sommerfeld-type equations, J. Comput. Phys., Volume 37 (1980) no. 1, pp. 133-139 | DOI | MR | Zbl
[19] Stability of combustion waves in a simplified gas–solid combustion model in porous media, Philos. Trans. R. Soc. Lond., A, Math. Phys. Eng. Sci., Volume 376 (2018) no. 2117, 20170185 | DOI | MR | Zbl
[20] The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math., Volume 51 (1998) no. 7, pp. 797-855 | DOI | MR | Zbl
[21] Spectral Analysis of Fronts in a Marangoni-Driven Thin Liquid Film Flow Down a Slope, SIAM J. Appl. Math., Volume 80 (2020) no. 1, pp. 95-118 | DOI | MR | Zbl
[22] Differential topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1994 (corrected reprint of the 1976 original) | MR | Zbl
[23] Spectral stability of ideal-gas shock layers, Arch. Ration. Mech. Anal., Volume 194 (2009) no. 3, pp. 1029-1079 | DOI | MR | Zbl
[24] Multidimensional stability of large-amplitude Navier–Stokes shocks, Arch. Ration. Mech. Anal., Volume 226 (2017) no. 3, pp. 923-973 | DOI | MR | Zbl
[25] An efficient shooting algorithm for Evans function calculations in large systems, Phys. D, Volume 220 (2006) no. 2, pp. 116-126 | DOI | MR | Zbl
[26] On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws, Tôhoku Math. J. (2), Volume 40 (1988) no. 3, pp. 449-464 | DOI | MR | Zbl
[27] A BVP solver that controls residual and error, J. Numer. Anal. Ind. Appl. Math., Volume 3 (2008) no. 1-2, pp. 27-41 | MR | Zbl
[28] Compressible Navier–Stokes equations in a bounded domain with inflow boundary condition, SIAM J. Math. Anal., Volume 28 (1997) no. 1, pp. 94-108 | DOI | MR | Zbl
[29] Hyperbolic systems of conservation laws and the mathematical theory of shock waves, CBMS-NSF Regional Conference Series in Applied Mathematics, 11, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973 | DOI | MR | Zbl
[30] Mathematical topics in fluid mechanics. Vol. 2 Compressible models, Oxford Lecture Series in Mathematics and its Applications; Oxford Science Publications, 10, The Clarendon Press; Oxford University Press, New York, 1998 | MR | Zbl
[31] Stability of Planar Detonations in the Reactive Navier-Stokes Equations, Ph. D. Thesis, Brigham Young University, Provo, USA (2017) | MR
[32] Existence and stability of steady compressible Navier–Stokes solutions on a finite interval with noncharacteristic boundary conditions, Physica D, Volume 394 (2019), pp. 16-25 | DOI | MR | Zbl
[33] Existence and behavior of steady solutions on an interval for general hyperbolic-parabolic systems of conservation laws (2021) | arXiv
[34] Topology from the differentiable viewpoint, The University Press of Virginia, Charlottesville, Va., 1965 (based on notes by David W. Weaver) | MR | Zbl
[35] Elements of combinatorial and differential topology, Graduate Studies in Mathematics, 74, American Mathematical Society, Providence, RI, 2006 (translated from the 2004 Russian original by Olga Sipacheva) | DOI | MR | Zbl
[36] Inviscid boundary conditions and stability of viscous boundary layers, Asymptot. Anal., Volume 26 (2001) no. 3-4, pp. 285-306 | MR | Zbl
[37] Boundary layer stability in real vanishing viscosity limit, Comm. Math. Phys., Volume 221 (2001) no. 2, pp. 267-292 | DOI | MR | Zbl
[38] The MATLAB ODE Suite, SIAM Journal on Scientific Computing, Volume 18 (1997) no. 1, pp. 1-22 | DOI | MR | Zbl
[39] Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions, Comm. Math. Phys., Volume 302 (2011) no. 1, pp. 1-51 | DOI | MR | Zbl
[40] Multidimensional stability of planar viscous shock waves, Advances in the theory of shock waves (Progress in Nonlinear Differential Equations and their Applications), Volume 47, Birkhäuser Boston, Boston, MA, 2001, pp. 307-516 | DOI | MR | Zbl
[41] Stability of noncharacteristic boundary layers in the standing-shock limit, Trans. Amer. Math. Soc., Volume 362 (2010) no. 12, pp. 6397-6424 | DOI | MR | Zbl
[42] Viscous and inviscid stability of multidimensional planar shock fronts, Indiana Univ. Math. J., Volume 48 (1999) no. 3, pp. 937-992 | DOI | MR | Zbl
[43] Errata to: “Pointwise semigroup methods, and stability of viscous shock waves” [Indiana Univ. Math. J. (47) (1998), no. 3, 741–871; MR1665788 (99m:35157)], Indiana Univ. Math. J., Volume 51 (2002) no. 4, pp. 1017-1021 | DOI | MR
Cited by Sources: