2019 Fiscal Year Researchstatus Report
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 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

Keywords  Evolution equations / Strict solutions / Wiener process 
Outline of Annual Research Achievements 
We considered a semilinear evolution equation with additive noise of the form dX+AXdt=[F_1(t)+F_2(X)]dt+G(t)dW(t) in a Banach space. Here, we assume that the linear operator A is a sectorial operator generating an analytical semigroup. And, W is a cylindrical Wiener process. By using the semigroup approach and fixed point arguments, under some conditions on the coefficients F_1, F_2, we proved existence of strict solutions to the equation. In addtion, the regularity of the solutions is also obtained.

Current Status of Research Progress 
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
I do not use the Young integral approach but the semigroup approach. The latter approach is effective for the equation considered in the Summary of Research Achievements.

Strategy for Future Research Activity 
I will be in the plan stated in the original proposal. Now I consider a semilinear equation with multiple noise: dX+AXdt=[F_1(t)+F_2(X)]dt+G(t,X)dW(t).
I will try to use the Young integral approach as stated in the original proposal but also the semigroup approach. The final goal is to construct a solution to the equation and then show its regularity.
For the semigroup approach, the variable appearing in stochastic convolutions will be explained as as a multiplication operator. Roughly speaking, any element U in L2 space can be explained as a linear operator from L2 to itself by U(v)=Uv if the product between U and v is still an element of L2. In this way, we may obtain a meaningful stochastic convolution, and therefore a solution to the equation.

Causes of Carryover 
I canceled some business trips due to coronavirus outbreak in this fiscal year. I would like to carry the amount to the next fiscal year. I will buy books, a PC, and make business trips that I could not do in the current fiscal year.
