Asymptotic analysis of Burgers-type equations with spatial anisotropy in higher dimensions

2022 Fiscal Year Final Research Report

• Project/Area Number: 20K22303
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Shinshu University
• Principal Investigator: Fukuda Ikki 信州大学, 学術研究院工学系, 講師 (60882214)
• Project Period (FY): 2020-09-11 – 2023-03-31
• Keywords: Burgers型方程式 / 一般化KPB方程式 / 一般化ZKB方程式 / 散逸・分散型方程式 / 解の漸近挙動 / 時間減衰評価の最良性 / 空間異方性

Outline of Final Research Achievements

In this study, we mainly considered the large time behavior and decay estimates for solutions to the generalized KP-Burgers equation and the generalized Zakharov-Kuznetsov-Burgers equation, in two dimensions. In particular, in two dimensions, the interaction between dispersion term and dissipation term induced by the spatial anisotropy of the equation has an essential effect on the structure of the solution. As a result, we showed that a unique decay rate appears in the decay estimates of the solution. Moreover, we derived an approximation formula for the solution and used it to prove the optimality for the decay estimates. In addition, as a preparation for these analyses, we also analyzed some one-dimensional Burgers equations with dispersion term and clarified the effect of the shape of the dispersion term on the large time behavior of the solution.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

本研究では, 分散項付きのBurgers型方程式を扱ったが, それらはいずれも非線形波を記述する方程式であり, その理論の整備は数学としても現象の解析としても重要である. 今回, 一般化KP-Burgers方程式と一般化Zakharov-Kuznetsov-Burgers方程式, 即ち空間異方性のあるBurgers型方程式の研究では, 分散型方程式と放物型方程式の両者の手法を組み合わせて解析したことで, 既存の評価と全く異なるものが得られることを見出した. これは, 散逸・分散型方程式に対する解の長時間挙動の理論の深化に繋がったと考えられ, 今後のこの分野のさらなる発展への貢献が期待できる.