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The Boltzmann equation in the whole space. (English) Zbl 1065.35090

The author proves the existence of global \(L_2\) solutions near Maxwellian to the Boltzmann equations in the whole space. The solution \(F\) is sought in the form \(F= \mu +\sqrt{\mu}f\), where \(\mu\) is a Maxwellian and \(f\) is its perturbation which is assumed to have a sufficiently small initial energy. Let us write \(f= \mathbb{P}f +(\mathbb{I}-\mathbb{P})f\), where \(\mathbb{P}\) is the projection onto the five dimensional null-space \(\mathcal{N}\) of the linearized collision operator \(L\). It is known that \(L\) is positive definite with respect to the microscopic part \((\mathbb{I}-\mathbb{P})f\). The construction of the global solution in this paper uses the nonlinear energy method, recently developed by the author, which consists in showing that \(L\) is positive definite for any solution \(f\) with sufficiently small amplitude. This is done by a careful analysis of equations for coefficients of the expansion of \(\mathbb{P}f\) in the basis of \(\mathcal{N}\).

MSC:

35F20 Nonlinear first-order PDEs
82B40 Kinetic theory of gases in equilibrium statistical mechanics
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