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A solution formula and the R-boundedness for the generalized Stokes resolvent problem in an infinite layer with Neumann boundary condition

大石, 健太 名古屋大学

2021.06.16

概要

本論文では、非圧縮粘性流体の流れを記述するNavier-Stokes方程式の数学解析のうちでも,正の時刻において領域が未知となる自由境界問題の解析を目的とする.即ち、初期時刻での領域と流体の速度を既知として,正の時刻における領域、流体の速度及び圧力を求める.ここでは、表面張力を考慮せず、また,初期時刻の領域としてlayer,解の枠組みとして時間Lp空間L。を考える、海洋をモデルとして下面の境界を固定した問題では,Saito(2018)により時間大域的適切性まで得られている、それに対し,本論文では上面と下面を共に自由境界とした問題を考察し、後述のレゾルベント問題に対する.2-有界性を通して時間局所適切性を示した。

自由境界問題は領域が未知であるため、まずEuler座標からLagrange座標に変換することで固定領域上の方程式に帰着し、線形問題の解析と非線形項の評価から不動点定理を用いることで非線型方程式の解を構成する.ここで,解の存在区間を小さく取ることで、上記の変換の単射性が崩れない,即ち流体が重なり合わないようにできる、しかし,その非線形方程式は準線形であるため、線形問題の解析として、解析半群の時間各点評価では不十分でLp-Lg最大正則性が必要となる.

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