Averages over spheres for kinetic transport equations; hyperbolic Sobolev spaces and Strichartz inequalities.
The authors consider averages over spheres \[ \rho(t,x) := \int_{S^{d-1}} f(t,x,v)\,d\sigma(v) \] of solutions \(f : \mathbb R \times \mathbb R^d \times S^{d-1} \to \mathbb R\) of the transport equation \[ \partial_t f + v \cdot \nabla_x f = g. \] As in the case of averaging over balls, it turns out that the averaged quantity is more regular than \(f\) itself [see, e.g., \textit{F. Golse}, \textit{P. L. Lions}, \textit{B. Perthame} and \textit{R. Sentis}, J. Funct. Anal. 76, 110--125 (1988; Zbl 0652.47031) and \textit{F. Bouchut}, in: Kinetic equations and asymptotic theory, Series in Appl. Math. 4, Elsevier (2000; Zbl 0979.82048)]. In particular, for \(d \geq 3\) it is shown that \[ \left\| (1+| \tau| + | \xi| )^{1/2} \widehat\rho(\tau,\xi)\right\|_{L^2(\mathbb R^{1+d})} \leq C(d) \left( \left\| f \right\|_{L^2(\mathbb R^{1+d}\times S^{d-1})} + \left\| g \right\|_{L^2(\mathbb R^{1+d}\times S^{d-1})} \right), \] which is exactly the same smoothing effect as in the case of averages over balls. However, in the special case \(d=2\) the estimate turns out to be worse ``by a quarter of derivative'': \[ \left\| (1+| \tau| + | \xi| )^{1/4} \widehat\rho(\tau,\xi)\right\|_{L^2(\mathbb R^{1+2})} \leq C \left( \left\| f \right\|_{L^2(\mathbb R^{1+2}\times S^1)} + \left\| g \right\|_{L^2(\mathbb R^{1+2}\times S^1)} \right). \] Nevertheless, a careful analysis shows that, far from the light cone \(| \tau| = | \xi| \) the quarter of derivative is somehow recovered. More exactly, the following hyperbolic Sobolev space estimate holds true: \[ \left\| \big(1+\big| | \tau| - | \xi| \big| \big)^{1/4} \big( 1 + | \tau| + | \xi| \big)^{1/4} \widehat\rho(\tau,\xi)\right\|_{L^2(\mathbb R^{1+2})} \] \[ \leq C \left( \left\| f \right\|_{L^2(\mathbb R^{1+2}\times S^1)} + \left\| g \right\|_{L^2(\mathbb R^{1+2}\times S^1)} \right). \] Eventually, this allows the authors to prove the Strichartz-like estimate \[ \left\| \rho \right\|_{L^3(\mathbb R^{1+2})} \leq C \left( \left\| f \right\|_{L^2(\mathbb R^{1+2}\times S^1)} + \left\| g \right\|_{L^2(\mathbb R^{1+2}\times S^1)} \right) \] [see \textit{M. Keel} and \textit{T. Tao}, Am. J. Math. 120, 955--980 (1998; Zbl 0922.35028)].
scientific article; zbMATH DE number 1770835
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