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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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