## 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

### Keywords:

Boltzmann equation; Landau equation; compactness property
Full Text: