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Math. Anal."],"published-print":{"date-parts":[[2024,2,29]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>In the paper, we develop an [Formula: see text] approach to construct global solutions to the Cauchy problem on the non-cutoff Boltzmann equation near equilibrium in [Formula: see text]. In particular, only smallness of [Formula: see text] with [Formula: see text] is imposed on initial data [Formula: see text], where [Formula: see text] 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 [Formula: see text] 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\u2013847] and [R. Alexandre et al., J. Funct. Anal., 262 (2012), pp. 915\u20131010], 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 [Formula: see text] norm corresponding to the Weiner space as studied in [R. J. Duan et al., Comm. Pure Appl. Math., 74 (2021), pp. 932\u20131020]. In contrast, for the latter, the extra interplay with the [Formula: see text] 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 [Formula: see text] norm helps gain an almost optimal decay rate [Formula: see text] of the [Formula: see text] norm via the time-weighted energy estimates in the spirit of the idea of [S. Kawashima, J. Hyperbolic Differ. Equ., 1 (2004), pp. 581\u2013603] and, in turn, this is necessarily used for establishing the global existence.<\/jats:p>","DOI":"10.1137\/22m1533232","type":"journal-article","created":{"date-parts":[[2024,1,16]],"date-time":"2024-01-16T04:29:02Z","timestamp":1705379342000},"page":"762-800","source":"Crossref","is-referenced-by-count":9,"title":["An \\(\\boldsymbol{L^1_{k}\\cap L^{p}_{k}}\\) Approach for the Non-Cutoff Boltzmann Equation in \\(\\boldsymbol{\\mathbb{R}^3}\\)"],"prefix":"10.1137","volume":"56","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8821-8829","authenticated-orcid":true,"given":"Renjun","family":"Duan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong, People\u2019s Republic of China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8509-0484","authenticated-orcid":true,"given":"Shota","family":"Sakamoto","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1710-8467","authenticated-orcid":true,"given":"Yoshihiro","family":"Ueda","sequence":"additional","affiliation":[{"name":"Graduate School of Maritime Sciences, Kobe University, Kobe, 658-0022, Japan."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,1,16]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-011-0432-0"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-011-1242-9"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2011.10.007"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.3934\/krm.2013.6.1011"},{"key":"ref5","unstructured":"R. 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