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流体力学における非線形発展方程式の適切性と漸近挙動

藤井, 幹大 FUJII, Mikihiro フジイ, ミキヒロ 九州大学

2022.09.22

概要

本稿は,流体力学において現れる種々の非線形発展方程式において,その適切性や漸近挙動を調べた著者の博士課程における研究内容をまとめた論文である.特に,方程式の線形部分に分散性を呈する項を持つ場合や,線形部の消散効果が弱い場合についてそれらの特徴が非線形問題の可解性や解の漸近挙動に与える影響に焦点を当てて解析を行った.本稿で扱う方程式は「2次元準地衡流方程式」,「異方的粘性項をもつ3次元非圧縮性Navier-Stokes方程式」,「3次元非圧縮性Hall項つき磁気流体方程式」,「3次元圧縮性Navier-Stokes-Coriolis方程式」の4つである.

2次元準地衡流方程式の研究は大きく三つに分かれる.第一の研究では消散項を持たず,分散項を持つ場合について考察し,分散係数が大きい場合に長時間解の存在を局所可解性が保証される正則性の枠組みにおいて証明し,さらに分散係数の大きさを無限大とすると対応する線形分散解に漸近することを証明する.第二の研究では優臨界である分数べきLaplacianからなる弱い消散性をもち分散項も有する場合についてその時間大域解をスケール劣臨界,臨界なSobolev空間で構成する.特に,初期値のノルムが大きい場合でも分散係数がそれに応じて大きければ時間大域解が構成できることを示す.第三の研究では分散項がなく優臨界である分数べきLaplacianによる消散項をもつ場合について時間周期的外力を与えた場合に時間周期解が一意存在することを証明する.

次に水平方向にのみ粘性を持つ3次元非圧縮性Navier-Stokes方程式の解の時刻無限大における挙動を調べる.この方程式は空間鉛直方向の変数に関する粘性がないという意味で通常のNavier-Stokes方程式より弱い消散性をもつ.本研究では速度場の水平成分は二次元の熱核と同様の減衰率をもち,鉛直成分は3次元の熱核と同様の減衰率をもつことを証明する.さらに時刻無限大における解の漸近展開を行う.

Hall項付き磁気流体方程式は2階の微分をもつ非線形項を保つため,その消散項は通常のLaplacianであるが相対的に消散が弱い準線形の方程式である.まずは,零磁場周りの解を考察しその時間大域的可解性を保証する関数空間のクラスと初期値に対する小ささ条件を改良する.つぎに定磁場周りの摂動問題を考察する.この摂動の方程式では分散性を呈する項が線形部分に現れる.この分散性を制御するためにFourier-Besov空間やStrichartz評価を用いて解析を行い,時間大域的適切性を様々な枠組みで証明する.

最後に圧縮性Navier-Stokes-Coriolis方程式の回転と音波による分散性に着目し,長時間解をスケール臨界である斉次Besov空間で構築する.本方程式は固有周波数を与える特性方程式が複雑な4次方程式であるため線形解をFourier積分を用いて陽に表示することは得策ではない.そこで非粘性の場合の分散評価と,周波数帯を三段階に分けた時空間ノルムに関するエネルギー法を構築することで困難点を解消し長時間解の存在を証明する.

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