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2014 年度 実績報告書

New Frontiers in Kinetic Equation Theory

研究課題

研究課題/領域番号 26887008
研究機関埼玉大学

研究代表者

BEZ NEAL  埼玉大学, 研究機構研究企画推進室, 准教授 (30729843)

研究期間 (年度) 2014-08-29 – 2016-03-31
キーワードSmoothing estimates / Dispersive equations
研究実績の概要

Progress has been achieved on numerous fronts, primarily by pioneering certain techniques from harmonic analysis and geometric analysis to problems in partial differential equations.

The Funk-Hecke theorem, in particular, has proved to be particularly effective. This is a result from classical harmonic analysis and has been used to make several breakthroughs on this grant. For example, a number of new results have been obtained in the context of Kato-smoothing estimates and related trace estimates, especially regarding the identification of optimal constants and extremal input functions in these estimates. Such estimates are well-known to have numerous consequences in the theory of dispersive and wave-like partial differential equations and it is anticipated that the new results obtained on this grant will find several applications in these directions.

現在までの達成度 (区分)
現在までの達成度 (区分)

1: 当初の計画以上に進展している

理由

The original plan was to apply powerful techniques from harmonic and geometric analysis to develop the theory of kinetic equations. In addition to making progress on this goal, such techniques have also been used to solve a number of problems in closely related topics, including smoothing estimates for dispersive and wave-like partial differential equations.

今後の研究の推進方策

In the next year of this grant, the techniques which have been utilised in the research achievements so far will be adopted to push forward the rigorous mathematical theory of the kinetic transport equation. More specifically, the Funk-Hecke theorem will be used to understand the smoothing effect of the velocity averages for the solution of the kinetic transport equation in the case of square-integrable initial data. To go beyond this and to provide a fuller theory in more general Lebesgue spaces, other techniques from harmonic analysis will be adopted.

  • 研究成果

    (5件)

すべて 2015 2014

すべて 雑誌論文 (2件) (うち国際共著 2件、 査読あり 2件、 謝辞記載あり 2件) 学会発表 (3件) (うち招待講演 3件)

  • [雑誌論文] Optimal forward and reverse estimates of Morawetz and Kato-Yajima type with angular smoothing index2015

    • 著者名/発表者名
      Neal Bez, Mitsuru Sugimoto
    • 雑誌名

      Journal of Fourier Analysis and Applications

      巻: 21 ページ: 318-341

    • DOI

      10.1007/s00041-014-9371-0

    • 査読あり / 国際共著 / 謝辞記載あり
  • [雑誌論文] Applications of the Funk-Hecke theorem to smoothing and trace estimates2015

    • 著者名/発表者名
      Neal Bez, Hiroki Saito, Mitsuru Sugimoto
    • 雑誌名

      Advances in Mathematics

      巻: 285 ページ: 1767-1795

    • DOI

      10.1016/j.aim.2015.08.025

    • 査読あり / 国際共著 / 謝辞記載あり
  • [学会発表] Some inequalities from geometric and harmonic analysis via induction-on-scales2015

    • 著者名/発表者名
      Neal Bez
    • 学会等名
      Spring Meeting of Mathematical Society of Japan
    • 発表場所
      明治大学駿河台キャンパス (東京都千代田区)
    • 年月日
      2015-03-22 – 2015-03-22
    • 招待講演
  • [学会発表] Lectures on the linear and multilinear restriction conjectures2014

    • 著者名/発表者名
      Neal Bez
    • 学会等名
      Harmonic Analysis Workshop
    • 発表場所
      ホテルヒルズサンピア山形 (山形県山形市)
    • 年月日
      2014-12-25 – 2014-12-27
    • 招待講演
  • [学会発表] Multilinear Radon-like transforms2014

    • 著者名/発表者名
      Neal Bez
    • 学会等名
      Differential Equations Workshop
    • 発表場所
      京都大学吉田キャンパス (京都府京都市)
    • 年月日
      2014-12-20 – 2014-12-20
    • 招待講演

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公開日: 2017-01-06  

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