Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II
The main result of these two papers is a new estimate for the gain part of the Boltzmann collision operator. Under reasonable assumptions on the collision kernel, it is shown that \[ \bigl |Q^+ (f,g) \bigr |_{H^{N - 1 \over 2}} \leq C |f |_{L^1} |g |_{ L^2} \tag{1} \] if \(g \in L^2 (\mathbb{R}^N_v)\), \(f \in L^1 (\mathbb{R}^N_v)\) \((f\) and \(g\) can be exchanged). The proof given in part I of the paper is based on Fourier integral operators. We mention that a more direct proof based on Radon transformations has more recently been given by \textit{B. Wennberg} [Commun. Partial Differ. Equations 19, No. 11-12, 2057-2074 (1994; Zbl 0818.35128).] The author discusses the implications of this ``compactness estimate'' for the existence theory of renormalized solutions of the Boltzmann equation. The main improvement over the original proof is that if \(f^n \to f\) weakly in \(L^1 (\mathbb{R}^{2N} \times (0,T))\) (where \((f^n)\) is a sequence of renormalized solutions of the Boltzmann equation), then \(Q^+ (f^n, f^n)\) converges in measure on \(\mathbb{R}^N_x \times \{|v |< R \} \times (0,T)\) to \(Q^+ (f,f) \) \(\forall R\), \(T \in (0, \infty)\). Another application of the estimate (1) is a new proof of strong convergence to equilibrium for renormalized solutions of the Boltzmann equation in a box with periodic boundary conditions. In part II of this work, it is shown that a subsequence of the \(f^n\) will converge strongly in \(C([0,T]; L^1 (\mathbb{R}^{2N}))\) to a renormalized solution \(f\) if strong \(L^1\)-convergence holds for the initial values. Then a class of new inequalities (referred to as ``dissipation inequalities'') satisfied by smooth solutions of the Boltzmann equation is derived, and it is shown that a renormalized solution satisfying these inequalities exists. As the final application of these inequalities, it is shown that if a strong solution of the Boltzmann equation exists, then every renormalized solution satisfying such inequalities will coincide with this strong solution. [Part III, cf. ibid. 34, No. 3, 539-584 (1994)].
scientific article; zbMATH DE number 701493