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On Boltzmann and Landau equations. (English) Zbl 0809.35137

The paper studies some compactness properties of the solutions of the kinetic equation \[ {{\partial f} \over {\partial t}}+ v \cdot \nabla_ x f= Q(f,f), \qquad x\in \mathbb{R}^ N, \quad v\in \mathbb{R}^ N, \quad t\geq 0,\tag{1} \] where \(N\geq 1\), \(f\) is a nonnegative function and \(Q(f,f)\) is a nonlocal, quadratic operator. The unknown function \(f\) corresponds at each time \(t\) to the density of particles at the point \(x\) with velocity \(v\). The operator \(Q\) is the so-called collision operator.
Different structure of \(Q\) yields to Boltzmann or Landau equation. In both cases, if reasonable assumptions are made on \(Q\), then a sequence of solutions \((f^ n)_ n\) corresponding to a sequence of initial conditions \((f_ 0^ n)_ n\) is proved to be relatively compact in suitable \(L^ r\)-spaces.
Reviewer: S.Totaro (Firenze)

MSC:

35Q72 Other PDE from mechanics (MSC2000)
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
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