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A kinetic equation with kinetic entropy functions for scalar conservation laws. (English) Zbl 0729.76070
The authors consider nonlinear kinetic equation of the type $[\partial_ t+a(v)\cdot \partial_ x]f_{\epsilon}(x,v,t) = \frac{1}{\epsilon}[\chi_{u_{\epsilon}(x,t)}(v)- f_{\epsilon}(x,v,t)],$ where $$x\in {\mathbb{R}}^ d$$, $$v\in {\mathbb{R}}$$, $$t\in {\mathbb{R}}_+$$, $$u_{\epsilon}(x,t)=\int f_{\epsilon}(x,v,t)dv$$ and
sgn u if $$(u-v)=0$$ $$\chi_ u(v)= 0$$ otherwise.
This model resembles the well-known BGK model. It is proved that the initial value problem is uniformly in $$\epsilon$$ well-posed (i.e. bounds and continuous dependence on the data hold uniformly with respect to $$\epsilon$$). Moreover, the kinetic equation possesses a family of kinetic entropy functions, which translate as $$\epsilon\to 0$$ to Krushkov-type entropy inequalities for the corresponding multidimensional conservation law $\partial_ tu(x,t)+\sum^{d}_{i=1}\partial_ x[A_ i(u(x,t))]=0,$ where $a(v)\cdot \partial_ x\equiv \sum^{d}_{i=1}a_ i(v)\partial_{x_ i},\text{ and } a_ i(\cdot)=A_ i'(\cdot).$ Finally, bounded variation arguments in the multidimensional case and a compensated compactness argument in the one- dimensional case are used to prove that the local density $$u_{\epsilon}$$ of $$f_{\epsilon}$$ converges strongly with $$\epsilon\searrow 0$$ to the unique entropy solution of the conservation law.

##### MSC:
 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 82C40 Kinetic theory of gases in time-dependent statistical mechanics 35L65 Hyperbolic conservation laws
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##### References:
  Boltzmann, L.: Vorlesungen uber Gas theorie, Liepzig 1886 · JFM 18.1033.01  Y. Brenier, Y.: Averaged multivaried solutions for scalar conservation laws. SIAM J. Numer. Anal.21, 1013–1037 (1986) · Zbl 0565.65054 · doi:10.1137/0721063  Caflisch, R.: The fluid dynamic limit of the nonlinear Boltzmann equation. Commun. Pure Appl. Math.33, 651–666 (1980) · Zbl 0435.76049 · doi:10.1002/cpa.3160330506  Crandall, M., Majda, A.: Monotone difference approximations for scalar conservation laws. Math. Comp.34, 1–21 (1980) · Zbl 0423.65052 · doi:10.1090/S0025-5718-1980-0551288-3  Crandall, M., Tartar, L.: Some relations between non-expansive and order preserving mappings. Proc. Am. Math. Soc.78 (3) 385–390 (1980) · Zbl 0449.47059 · doi:10.1090/S0002-9939-1980-0553381-X  DiPerna, R., Lions, P. L. On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math. (1989) · Zbl 0698.45010  Giga, Y., Miyakawa, T.: A kinetic construction of global solutions of first order quasilinear equations. Duke Math. J.50, 505–515 (1983) · Zbl 0519.35053 · doi:10.1215/S0012-7094-83-05022-6  Krushkov, S. N.: Frist order quasilinear equations in several independent variables. Math. USSR Sb.10, 217–243 (1970) · Zbl 0215.16203 · doi:10.1070/SM1970v010n02ABEH002156  Lax, P. D.: Hyperbolic systems of conservation laws and the mathematical theory of shock waves. SIAM Regional Conference Series in Applied Mathematics, vol. 11 · Zbl 0268.35062  Murat, F.: Compacité per compensation. Ann. Scuola Norm. Sup. Disa Sci. Math.5, 489–507 (1978) and8, 69–102 (1981) · Zbl 0399.46022  Perthame, B.: Global existence of solutions to the BGK model of Boltzmann equations. J. Diff. Eq.81, 191–205 (1989) · Zbl 0694.35134 · doi:10.1016/0022-0396(89)90173-3  Tadmor, E.: Semi-discrete approximations to nonlinear systems of conservation laws; consistency andL stability imply convergence. ICASE Report No. 88-41.  Tartar, L.: Compensated compactness and applications to partial differential equations. In: Research Notes in Mathematics, vol. 39, Nonlinear Analysis and Mechanics, Heriot-Watt Sympos., vol. 4. Knopps, R. J. (ed.) pp. 136–211. Boston, London: Pittman Press 1975
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