Building a Theory of Regular Structures for NonAutonomous and QuasiLinear Rough Evolution Equations, and Applying the Theory to Forest Kinematic Ecosystems
Project/Area Number 
19K14555

Research Category 
GrantinAid for EarlyCareer Scientists

Allocation Type  Multiyear Fund 
Review Section 
Basic Section 12010:Basic analysisrelated

Research Institution  Kyushu University 
Principal Investigator 
タ・ビィエ トン 九州大学, 農学研究院, 准教授 (30771109)

Project Period (FY) 
20190401 – 20230331

Project Status 
Granted (Fiscal Year 2020)

Budget Amount *help 
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2022: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)

Keywords  Evolution equations / Strict solutions / Mild solutions / Wiener process / Forest kinematic model / Evolution Equations / Forest Ecosystems / Coat Patterns 
Outline of Research at the Start 
Many phenomena including noise can be described by rough evolution equations (REEs). Once these equations are solved, it will give new understanding on the phenomena.
This project will construct a new theory to solve a wide class of REEs. The theory will be then used to study the effect of noise on forest kinematic ecosystem and on the coat pattern of animals.

Outline of Annual Research Achievements 
A) We continued to study this evolution equation dX+AXdt=[F_1(t)+F_2(X)]dt+G(t)dW(t). The results include: 1) Existence of both mild solutions and strict solutions; 2) Regularities of these solutions. Roughly speaking, we showed that A^pX is (gamma) Holder continuous functions. To obtain these results, we used the semigroup methods. The results now are published in Communications on Pure and Applied Analysis. B) We constructed a forest kinematic model. We are going to apply the results in A) to study this model.

Current Status of Research Progress 
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
The coefficients in the forest model are very nonregular. The existence of solutions may be obtained but it is difficult to show the behavior of solutions.

Strategy for Future Research Activity 
A) We will prove the existence of solutions to the forest kinematic model that we have constructed. To do this, we will approximate the solutions by solutions of a more "regular" system. Then, we will study the behavior of solutions and make numerical simulations.
B) We will also consider a semilinear evolution equation with multiple noise: dX+AXdt=[F_1(t)+F_2(X)]dt+G(t,X)dW(t). The aim is to construct a solution to this kind of equations.

Report
(2 results)
Research Products
(1 results)