AN Lk1 ∧ Lkp APPROACH FOR THE NON-CUTOFF BOLTZMANN EQUATION IN R3+

In the paper, we develop an L-k(1) boolean AND L-k(p) approach to construct global solutions to the Cauchy problem on the non-cutoff Boltzmann equation near equilibrium in R-3. In particular, only smallness of IITxf0IIL1\capLp(R3k;L2(R3v)) with 3/2 < p \leq oo is imposed on initial data f0(x, v), where Txf0(k, v) is the Fourier transform in space variable. This provides the first result on the global existence of such low-regularity solutions without relying on Sobolev embedding H2(R3x) \subset L\infty(R3x) in the case of the whole space. Different from the use of sufficiently smooth Sobolev spaces in the classical results [P. T. Gressman and R. M. Strain, J. Amer. Math. Soc., 24 (2011), pp. 771--847] and [R. Alexandre et al., J. Funct. Anal., 262 (2012), pp. 915-1010], there is a crucial difference between the torus case and the whole space case for low-regularity solutions under consideration. In fact, for the former, it is enough to take the only L1k norm corresponding to the Weiner space as studied in [R. J. Duan et al., Comm. Pure Appl. Math., 74 (2021), pp. 932-1020]. In contrast, for the latter, the extra interplay with the Lpk norm plays a vital role in controlling the nonlinear collision term due to the degenerate dissipation of the macroscopic component. Indeed, the propagation of the Lpk norm helps gain an almost optimal decay rate (1+t) - 32 (1 - 1p )+ of the L1k norm via the time-weighted energy estimates in the spirit of the idea of [S. Kawashima, J. Hyperbolic Differ. Equ., 1 (2004), pp. 581--603] and, in turn, this is necessarily used for establishing the global existence.