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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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