Large-time behavior of discrete velocity Boltzmann equations
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).
scientific article; zbMATH DE number 4039503
scientific article