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Asymptotic behavior of solutions to nonlinear hyperbolic and dispersive equations with weakly dissipative structure

西井, 良徳 大阪大学

2022.03.24

概要

本論文は,半線形波動方程式及び非線形Schrödinger方程式の初期値問題に対する解の長時間挙動について研究したものである.

双曲型方程式には平滑化効果がないため,非線形双曲型方程式の初期値問題を考える際,初期値が小さく,滑らかで,空間遠方で十分速く減衰している場合でも一般には有限時刻で解に特異性が生じる.そのため,非線形項が解の特異性の発生や,長時間挙動にどのように影響するかが興味の対象となる.1986年にKlainermanとChristodoulouによって導入された零条件(nullcondition)は,空間3次元における準線形波動方程式の小振幅解の時間大域的存在と漸近自由性を保証する非線形項の構造条件であり,非線形双曲型方程式における最重要概念の1つとして広く知られている.2000年代以降,Lindblad, Rodnianski, Alinhac, Agemi等により零条件よりも弱い構造条件の研究が進められている.Kubo(2007), Hoshiga(2008), Katayama-Murotani-Sunagawa(2012), Katayama Matsumura-Sunagawa(2015)等により導入されたAgemi型構造条件はその1つで,零条件と消散構造を統合するような構造条件である.また,分散型方程式の代表格である非線形Schrödinger方程式において,Agemi型構造条件と対応する条件がLi-Sunagawa(2016), Sagawa-Sunagawa(2016), Sakoda-Sunagawa(2020), Katayama-Sakoda(2021)等により指摘されている.

本論文では,非線形項がAgemi型構造条件を満たす半線形波動方程式及びAgemi型構造条件と対応する消散構造を伴う非線形Schrödinger方程式に対して,これまであまり結果が知られていなかった消散構造が部分的に退化している状況下での解の漸近挙動について考察する.

以下,本論文の構成および各章の概要について述べる.本論文は全三章からなる.

第一章では問題の背景,先行研究の要約及び本論文の構成を述べる.

第二章では,2次元Euclid空間上で3次の非線形項を伴う半線形波動方程式の考察を行う.前半では方程式が単独である場合を取り扱う.学位申請者は砂川秀明氏,寺下拓貴氏との共同研究により,Agemi型構造条件を仮定し,零条件を仮定しない場合には解のエネルギーノルムが時間減衰することを示し,その減衰率の上からの評価を与えた.後半ではAgemi型構造条件を満たす斉3次の非線形項を伴う或る2成分半線形波動方程式系を考察する.学位申請者はこの連立系について,単独の場合とは対照的に,成分ごとの初期値にある種の大小関係がある場合には解の各成分が共に非自明な自由解に漸近し,特に系の全エネルギーは減衰しないことを示した.証明は,Kleinermanにより創始された可換ベクトル場法を用いて半線形波動方程式を簡約化されたある種の常微分方程式へと変換し,その解の挙動を考察することにより行う.

第三章では,1次元Euclid空間上で斉3次の非線形項を伴うSchrödinger方程式について考察する.学位申請者は李春花氏,佐川侑司氏,砂川秀明氏との共同研究により,弱い消散構造の下で単独微分型Schrödinger方程式の解のL^2ノルムが時間減衰することを示した.さらに,弱い消散構造を伴う非線形Schrödinger方程式の2成分連立系で,解が時間無限大で自由解に漸近するが,非線形項が短距離型の場合には起こりえない散乱状態への制限が生じる例を構成した.証明には,Hayashi-Naumkin(1998)で用いられたSchrödinger自由発展群のMDFM分解により,問題をある種の常微分方程式へと変換し,その解の挙動を考察する手法を用いる.

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