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A kinetic approach to general first order quasilinear equations. (English) Zbl 0568.35063

The authors prove the existence of global weak solutions of the Cauchy problem for the first order quasilinear equation of conservation type with variable coefficients: \(u_ t+\sum^{n}_{i=1}A^ i(x,u)_{x_ i}+B(x,u)=0,\) \(x\in R^ n\), \(t>0\), \(u(x,0)=u_ 0(x).\) The vanishing viscosity method has been used by other authors to construct so-called entropy solutions of the problem. The method developed in this paper uses nonlinear semigroup theory to construct entropy solutions. Approximate solutions are developed from an auxiliary linear equation. The linear equation is viewed as the evolution in time of the microscopic state of gases. Stability properties of the approximate solutions are derived and their convergence to entropy solutions is established. The results are applicable in describing the macroscopic thermo-fluid dynamics of gases.
Reviewer: G.F.Webb

MSC:

35L60 First-order nonlinear hyperbolic equations
47H20 Semigroups of nonlinear operators
35D05 Existence of generalized solutions of PDE (MSC2000)
76N15 Gas dynamics (general theory)

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[1] H. Brézis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Functional Analysis 9 (1972), 63 – 74. · Zbl 0231.47036
[2] M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265 – 298. · Zbl 0226.47038 · doi:10.2307/2373376
[3] Michael G. Crandall, The semigroup approach to first order quasilinear equations in several space variables, Israel J. Math. 12 (1972), 108 – 132. · Zbl 0246.35018 · doi:10.1007/BF02764657
[4] Michael G. Crandall and Andrew Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), no. 149, 1 – 21. · Zbl 0423.65052
[5] Yoshikazu Giga and Tetsuro Miyakawa, A kinetic construction of global solutions of first order quasilinear equations, Duke Math. J. 50 (1983), no. 2, 505 – 515. · Zbl 0519.35053 · doi:10.1215/S0012-7094-83-05022-6
[6] Enrico Giusti, Minimal surfaces and functions of bounded variation, Department of Pure Mathematics, Australian National University, Canberra, 1977. With notes by Graham H. Williams; Notes on Pure Mathematics, 10. · Zbl 0402.49033
[7] Amiram Harten, Peter D. Lax, and Bram van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25 (1983), no. 1, 35 – 61. · Zbl 0565.65051 · doi:10.1137/1025002
[8] Y. Kobayashi, The application of the product formula for semigroups to first order quasilinear equations, Hiroshima Math. J. (to appear).
[9] K. Kobayasi and S. Oharu, On nonlinear evolution operators associated with certain nonlinear equations of evolution, Mathematical analysis on structures in nonlinear phenomena (Tokyo, 1978), Lecture Notes Numer. Appl. Anal., vol. 2, Kinokuniya Book Store, Tokyo, 1980, pp. 139 – 210. · Zbl 0474.35026
[10] S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR-Sb. 10 (1970), 217-243. · Zbl 0215.16203
[11] Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. · Zbl 0268.35062
[12] Wladimir Mazja, Einbettungssätze für Sobolewsche Räume. Teil 2, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 28, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980 (German). With English, French and Russian summaries. · Zbl 0438.46021
[13] I. Miyadera and Y. Kobayashi, Convergence and approximation of nonlinear semigroups, Functional Analysis and Numerical Analysis, Japan-France Seminar, Tokyo and Kyoto , 1976.
[14] Shinnosuke Ôharu and Tadayasu Takahashi, A convergence theorem of nonlinear semigroups and its application to first order quasilinear equations, J. Math. Soc. Japan 26 (1974), 124 – 160. · Zbl 0265.47052 · doi:10.2969/jmsj/02610124
[15] O. A. Oleĭnik, Discontinuous solutions of non-linear differential equations, Amer. Math. Soc. Transl. (2) 26 (1963), 95 – 172. · Zbl 0131.31803
[16] Yann Brenier, Une application de la symétrisation de Steiner aux équations hyperboliques: la méthode de transport et écroulement, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 11, 563 – 566 (French, with English summary). · Zbl 0459.35006
[17] -, Résolution d’équations d’évolution quasilinéaries en dimension \( N\) d’espace à l’aide d’équations linéaires en dimension \( N + 1\), J. Differential Equations 49 (1983). · Zbl 0549.35055
[18] Tetsuro Miyakawa, A kinetic approximation of entropy solutions of first order quasilinear equations, Recent topics in nonlinear PDE (Hiroshima, 1983) North-Holland Math. Stud., vol. 98, North-Holland, Amsterdam, 1984, pp. 93 – 105. · Zbl 0567.35014 · doi:10.1016/S0304-0208(08)71494-4
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