Well-posedness of spatially homogeneous Boltzmann equation with full-range interaction. (English) Zbl 1253.35093
Summary: In the present work, we give the coercivity estimates for the Boltzmann collision operator \(Q(\cdot , \cdot )\) without angular cut-off to clarify in which case the functional \({\langle -Q(g, f), f\rangle}\) will become truly sub-elliptic. Based on this observation and commutator estimates in [R. Alexandre et al., Arch. Ration. Mech. Anal. 198, No. 1, 39–123 (2010; Zbl 1257.76099)], the upper bound estimates for the collision operator in [Y. Chen and L. He, Arch. Ration. Mech. Anal. 201, No. 2, 501–548 (2011; Zbl 1318.76018)] and the stability results in [L. Desvillettes and C. Mouhot, Arch. Ration. Mech. Anal. 193, No. 2, 227–253 (2009; Zbl 1169.76054)], in the function space \({L^1_q\cap H^N}\), we establish global well-posedness or local well-posedness for the spatially homogeneous Boltzmann equation with full-range interaction (covering most of physical collision kernels).
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