Global in time estimates for the spatially homogeneous Landau equation with soft potentials. (English) Zbl 1296.35112
Summary: This paper deals with some global in time a priori estimates of the spatially homogeneous Landau equation for soft potentials \(\gamma\in [-2,0)\). For the first result, we obtain the estimate of weak solutions in \(L_t^\alpha\) \(L_v^{3-\varepsilon}\) for \(\alpha=\frac{2(3-\varepsilon)}{3(2-\varepsilon)}\) and \(0<\varepsilon<1\), which is an improvement over estimates by N. Fournier and H. Guérin [J. Funct. Anal. 256, No. 8, 2542–2560 (2009; Zbl 1165.35467)]. For the second result, we have the estimate of weak solutions in \(L_t^\infty\) \(L_v^p\), \(p>1\), which extends part of results by Fournier and Guerin [loc. cit.] and R. Alexandre et al. [“Some a priori estimates for the homogeneous Landau equation with soft potentials” (2013), arXiv:1302.1814]. As an application, we deduce some global well-posedness results for \(\gamma\in[-2,0)\). Our estimates include the case \(\gamma=-2\), which is the key point in this paper.
MSC:
| 35Q20 | Boltzmann equations |
| 35B45 | A priori estimates in context of PDEs |
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
| 35D30 | Weak solutions to PDEs |
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
Citations:
Zbl 1165.35467References:
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