Regularity for doubly nonlinear degenerate and singular parabolic equations and a geometric flow

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K03375
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Kumamoto University
• Principal Investigator: Misawa Masashi 熊本大学, 大学院先端科学研究部(理), 教授 (40242672)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: ソボレフ流 / ソボレフ不等式 / 山辺問題とその熱流 / 二重非線形退化特異放物型方程式 / 正則性 / 正値性伝播

Outline of Final Research Achievements

We study a doubly nonlinear degenerate and singular parabolic equation. We consider the so-called p-Sobolev flow, which contains the geometric flow named Yamabe flow in differential geometry. We proved the global existence of a regular solution for the Cauchy-zero Dirichlet problem of the p-Sobolev flow, under the condition that the initial-boundary datum is non-negative, bounded and belongs to the energy class. The regular solutions is the weak solution, satisfying the regularity that the solution itself and its gradient are continuous in time-space. We also a prior regularity estimates for a weak solution of the p-Sobolev flow. The key of the proof is the so-called expansion positivity of a weak solution to the doubly nonlinear degenerate and singular parabolic equation describing the p-Sobolev flow, based on some local energy estimates.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

多孔媒質型方程式およびpラプラス方程式を含む二重非線形退化特異放物型方程式の弱解の構成と弱解の正値性および空間一階導関数の時空連続性を証明した. 線形放物型方程式を含む非線形退化特異放物型に対する正則性理論に貢献できた. また, 幾何学に現れる山辺問題の熱流の初期値零境界値問題に対して, 定義域および初期値についてより一般的な条件のもと, 解析的な評価によって, 正則解の大域存在を証明した. 幾何学的発展方程式の解の構成および正則性に寄与した.