##
**Global asymptotics toward the rarefaction waves for solutions to the Cauchy problem of the scalar conservation law with nonlinear viscosity.**
*(English)*
Zbl 1446.35062

Summary: In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem for the scalar viscous conservation law where the far field states are prescribed. Especially, we deal with the case when the viscosity is of non-Newtonian type, including a pseudo-plastic case. When the corresponding Riemann problem for the hyperbolic part admits a Riemann solution which consists of single rarefaction wave, under a condition on nonlinearity of the viscosity, it is proved that the solution of the Cauchy problem tends toward the rarefaction wave as time goes to infinity, without any smallness conditions.

### MSC:

35K59 | Quasilinear parabolic equations |

35K55 | Nonlinear parabolic equations |

35B40 | Asymptotic behavior of solutions to PDEs |

35L65 | Hyperbolic conservation laws |

PDF
BibTeX
XML
Cite

\textit{A. Matsumura} and \textit{N. Yoshida}, Osaka J. Math. 57, No. 1, 187--205 (2020; Zbl 1446.35062)

### References:

[1] | G.I. Barenblatt: On the motion of suspended particles in a turbulent flow taking up a half-space or a plane open channel of finite depth, Prikl. Mat. Meh., 19 (1955), 61-88 (in Russian). |

[2] | J.A. Carrillo and G. Toscani: Asymptotic \(L^1\)-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J, 49 (2000), 113-142. · Zbl 0963.35098 |

[3] | R.P. Chhabra: Bubbles, drops and particles in non-Newtonian Fluids, CRC, Boca Raton, FL, 2006. |

[4] | R.P. Chhabra: Non-Newtonian Fluids: An Introduction, available at http://www.physics.iitm.ac.in/ compflu/Lect-notes/chhabra.pdf. · Zbl 1298.76014 |

[5] | R.P. Chhabra and J.F. Richardson: Non-Newtonian flow and applied rheology, 2nd edition, Butterworth-Heinemann, Oxford, 2008. |

[6] | A. de Waele: Viscometry and plastometry, J. Oil Colour Chem. Assoc., 6 (1923), 33-69. |

[7] | Q. Du and M.D. Gunzburger: Analysis of a Ladyzhenskaya model for incompressible viscous flow, J. Math. Anal. Appl, 155 (1991), 21-45. · Zbl 0712.76039 |

[8] | M.E. Gurtin and R.C. MacCamy: On the diffusion of biological populations, Math. Biosci. 33 (1979), 35-49. · Zbl 0362.92007 |

[9] | I. Hashimoto and A. Matsumura: Large time behavior of solutions to an initial boundary value problem on the half space for scalar viscous conservation law, Methods Appl. Anal. 14 (2007), 45-59. · Zbl 1149.35057 |

[10] | Y. Hattori and K. Nishihara: A note on the stability of rarefaction wave of the Burgers equation, Japan J. Indust. Appl. Math. 8 (1991), 85-96. · Zbl 0725.35013 |

[11] | F. Huang, R. Pan and Z. Wang: \(L^1\) Convergence to the Barenblatt solution for compressible Euler equations with damping, Arch. Ration. Mech. Anal. 200 (2011), 665-689. · Zbl 1229.35196 |

[12] | A.M. Il’in, A.S. Kalašnikov and O.A. Oleĭnik, Second-order linear equations of parabolic type, Uspekhi Math. Nauk SSSR 17 (1962), 3-146 (in Russian), Russian Math. Surveys 17 (1962), 1-143 (in English). |

[13] | A.M. Il’in and O.A. Oleĭnik: Asymptotic behavior of the solutions of the Cauchy problem for some quasi-linear equations for large values of the time, Mat. Sb. 51 (1960), 191-216 (in Russian). |

[14] | P. Jahangiri, R. Streblow and D. Müller: Simulation of Non-Newtonian Fluids using Modelica: in Proceedings of the 9th International Modelica Conference September 3-5, Munich, Germany 57-62, 2012. |

[15] | S. Kamin: Source-type solutions for equations of nonstationary filtration, J. Math. Anal. Appl. 64 (1978), 263-276. · Zbl 0387.76083 |

[16] | Ya.I. Kanel’: A model system of equations for the one-dimensional motion of a gas, Differencial’nya Uravnenija 4 (1968), 721-734 (in Russian). · Zbl 0235.35023 |

[17] | T. Kato: Linear evolution equations of “hyperbolic” type, J. Fac. Sci. Univ. Tokyo Sect. I, 17 (1970), 241-258. · Zbl 0222.47011 |

[18] | T. Kato: Linear evolution equations of “hyperbolic” type, II, J. Math. Soc. Japan 19 (1973), 648-666. · Zbl 0262.34048 |

[19] | S. Kawashima and A. Matsumura: Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Comm. Pure Appl. Math. 47 (1994), 1547-1569. · Zbl 0820.73030 |

[20] | O.A. Ladyženskaja, New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them; in Boundary Value Problems of Mathematical Physics V, AMS, Providence, RI, 1970. |

[21] | O.A. Ladženskaja, V.A. Solonnikov and N.N. Ural’ceva: Linear and quasilinear equations of parabolic type (Russian), Translated from the Russian by S. Smith., Transl. Math. Monogr. 23, AMS, Providence, RI, 1968. |

[22] | P.D. Lax: Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957), 537-566. · Zbl 0081.08803 |

[23] | H.W. Liepmann and A. Roshko: Elements of Gas Dynamics, John Wiley & Sons, Inc., New York, 1957. · Zbl 0078.39901 |

[24] | J.L. Lions: Quelques mèthodes de rèsolution des problèmes aux limites non linèaires, Dunod Gauthier-Villars, Paris, 1969 (in French). |

[25] | T.-P. Liu, A. Matsumura and K. Nishihara: Behaviors of solutions for the Burgers equation with boundary corresponding to rarefaction waves, SIAM J. Math. Anal. 29 (1998), 293-308. · Zbl 0916.35103 |

[26] | J. Málek: Some frequently used models for non-Newtonian fluids, available at http://www.karlin.mff.cuni.cz/ malek/new/images/Lecture4.pdf. |

[27] | J. Málek, D. Pražák and M. Steinhauer: On the existence and regularity of solutions for degenerate power-law fluids, Differential Integral Equations 19 (2006), 449-462. · Zbl 1200.76020 |

[28] | A. Matsumura and M. Mei: Nonlinear stability of viscous shock profile for a non-convex system of viscoelasticity, Osaka J. Math. 34 (1997), 589-603. · Zbl 0945.74539 |

[29] | A. Matsumura and K. Nishihara: Asymptotic toward the rarefaction wave of solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math. 3 (1986), 1-13. · Zbl 0612.76086 |

[30] | A. Matsumura and K. Nishihara: Asymptotics toward the rarefaction wave of the solutions of Burgers’ equation with nonlinear degenerate viscosity, Nonlinear Anal. 23 (1994), 605-614. · Zbl 0811.35127 |

[31] | A. Matsumura and K. Nishihara: Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Comm. Math. Phys. 165 (1994), 83-96. · Zbl 0811.35080 |

[32] | A. Matsumura and N. Yoshida: Asymptotic behavior of solutions to the Cauchy problem for the scalar viscous conservation law with partially linearly degenerate flux, SIAM J. Math. Anal. 44 (2012), 2526-2544. · Zbl 1255.35048 |

[33] | T. Nagai and M. Mimura: Some nonlinear degenerate diffusion equations related to population dynamics, J. Math. Soc. Japan 35 (1983), 539-562. · Zbl 0535.92019 |

[34] | T. Nagai and M. Mimura: Asymptotic behavior for a nonlinear degenerate diffusion equation in population dynamics, SIAM J. Appl. Math. 43 (1983), 449-464. · Zbl 0554.35060 |

[35] | W. Ostwald: Über die Geschwindigkeitsfunktion der Viskositat disperser Systeme, I. Colloid Polym. Sci. 36 (1925), 99-117 (in German). |

[36] | R.E. Pattle: Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quart. J. Mech. Appl. Math. 12 (1959), 407-409. · Zbl 0119.30505 |

[37] | J. Smoller: Shock Waves and Reaction-diffusion Equations, Springer-Verlag, New York, 1983. · Zbl 0508.35002 |

[38] | T. Sochi: Pore-Scale Modeling of Non-Newtonian Flow in Porous Media, PhD thesis, Imperial College London, 2007. · Zbl 1356.76363 |

[39] | J.L. Vázquez: Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type, Oxford Math. and Appl. 33, Oxford University Press, Oxford, 2006. |

[40] | J.L. Vázquez: The Porous Medium Equation: Mathematical Theory, Oxford Math. Monogr., Oxford: Clarendon Press, New York, 2007. · Zbl 1107.35003 |

[41] | N. Yoshida: Decay properties of solutions toward a multiwave pattern for the scalar viscous conservation law with partially linearly degenerate flux, Nonlinear Anal. 96 (2014), 189-210. · Zbl 1284.35082 |

[42] | N. Yoshida: Decay properties of solutions to the Cauchy problem for the scalar conservation law with nonlinearly degenerate viscosity, Nonlinear Anal. 128 (2015), 48-76. · Zbl 1329.35070 |

[43] | N. Yoshida: Large time behavior of solutions toward a multiwave pattern for the Cauchy problem of the scalar conservation law with degenerate flux and viscosity: in Sūrikaisekikenkyūsho Kōkyūroku “Mathematical Analysis in Fluid and Gas Dynamics”, 1947, 205-222, 2015. |

[44] | N. Yoshida: Asymptotic behavior of solutions toward a multiwave pattern for the scalar conservation law with the Ostwald-de Waele-type viscosity, SIAM J. Math. Anal. 49 (2017), 2009-2036. · Zbl 1377.35152 |

[45] | N. Yoshida: Decay properties of solutions toward a multiwave pattern to the Cauchy problem for the scalar conservation law with degenerate flux and viscosity, J. Differential Equations 263 (2017), 7513-7558. · Zbl 1383.35125 |

[46] | N. Yoshida: Asymptotic behavior of solutions toward the viscous shock waves to the Cauchy problem for the scalar conservation law with nonlinear flux and viscosity, SIAM J. Math. Anal. 50 (2018), 891-932. · Zbl 1387.35349 |

[47] | N. Yoshida: Asymptotic behavior of solutions toward a multiwave pattern to the Cauchy problem for the dissipative wave equation with partially linearly degenerate flux, appear in Funkcialaj Ekvacioj. |

[48] | Ya.B. Zel’dovič and A.S. Kompaneec: On the theory of propagation of heat with the heat conductivity depending upon the temperature: in Collection in honor of the seventieth birthday of academician A.F. Ioffe, Izdat. Akad. Nauk SSSR, 61-71, 1950 (in Russian). |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.