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Large-time behavior of discrete velocity Boltzmann equations. (English) Zbl 0637.76070
The asymptotic behaviour of equations representing one-dimensional motions in a gas with a discrete set of velocities is studied. This fictitious gas is described by the following system of equations (discrete velocity Boltzmann equations): $u_{i,t}+c_ iu_{i,x}=\sum^{n}_{j,k=1}a_ i^{jk}u_ ju_ k\equiv F_ i(u_ 1,..,u_ n),\quad i=1,...,n,$ where $$u_ i(x,t)$$ is the density of particles with speeds $$c_ i$$, $$a_ i^{jk}$$ are the interaction coefficients (they are constant satisfying conditions for mass and momentum conservation and an entropy condition; $$a_ i^{jk}=a_ i^{kj})$$. The initial condition $$u_ i(x,0)=u_{0i}(x)$$, $$i=1,..,n$$, with $$u_{0i}\geq 0$$ in $$L^ 1\cap L^{\infty}$$ is prescribed. The solution $$u_ i(x,t)$$ is in class $$C(0,T;L^ p({\mathbb{R}}))$$ for $$1\leq p\leq \infty$$, with $$T>0$$ arbitrary. $$u_ i(x,t)$$ as $$t\to \infty$$ have the form $$u_ i^{\infty}(x-c_ it)$$ it corresponds to a state in which the interaction is negligible. 3 theorems are proved $$(L^ 1$$-asymptotics; boundedness and $$L^ p$$- asymptotics; uniform asymptotics).
Reviewer: I.Grosu

##### MSC:
 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 82B40 Kinetic theory of gases in equilibrium statistical mechanics
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