2022 Fiscal Year Annual Research Report
| Project/Area Number |
21F50732
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| Allocation Type | Single-year Grants |
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| Research Institution | Shinshu University |
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Principal Investigator |
謝 賓 信州大学, 学術研究院理学系, 教授 (50510038)
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| Co-Investigator(Kenkyū-buntansha) |
ZHU MIN 信州大学, 理学部, 外国人特別研究員
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| Project Period (FY) |
2021-11-18 – 2024-03-31
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| Keywords | p-Laplacian / delay / BDG inequality / anisotropic / distribution dependent / transformation |
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| Outline of Annual Research Achievements |
We mainly studied stochastic anisotropic stochastic partial differential equations and approximation of path-distribution dependent SDEs driven by alpha-stable noise.The details are following:
(1) We study well-posedness for stochastic anisotropic p-Laplacian reaction-diffusion equation with delay under non-Lipschitz conditions and stochastic anisotropic Navier-Stokes equations with delay. It is worth pointing out that, in contrast to the exist results, the essential difficulty in this work is how to discuss the problem of the uniqueness of solutions for anisotropic SPFDEs under non-Lipschitz condition, other than dealing with the anisotropic one. Under weaker conditions than those imposed in existing literature we show the well-posedness of stochastic partial functional differential equations with anisotropic exponents. Based on the result obtained, we explore the well-posedness of two kinds of typical examples.
(2) We show via interacting particle systems an approximation issue on a class of path-distribution dependent SDEs driven by alpha-stable noise, where the drift is singular. The Zvonkin-type approach based on the existing literature does not work for the case of SDEs with multiplicative noises, although the jump diffusion processes can be estimated. Our work involves primarily two parts. The first part shows the well-posedness of path-distribution dependent stochastic differential equations and the corresponding stochastic interacting particle systems. The second part shows approximations of these path-distribution dependent stochastic differential equations.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
In view of the purpose of the research, we have almost completed the expected
result. Our work involves primarily two parts. In the first part, we adopt the general variational approach to analyze stochastic anisotropic partial functional differential equations in abstract form. Furthermore, we have studied the well-posedness of stochastic anisotropic p-Laplacian reaction-diffusion equations with delay under non-Lipschitz conditions, and stochastic anisotropic Navier-Stokes equations. In the second part, we have studied the convergence rate of EM scheme for path-distribution dependent stochastic differential equations driven by alpha-stable noise, where the drift is singular. Via Zvonkin's transformation, the propagation of chaos and convergence rate of the truncated Euler-Maruyama scheme associated with the interacting particle systems are investigated.
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| Strategy for Future Research Activity |
We will study the asymptotic behavior of stochastic partial differential equations with anisotropic operators. Firstly, we will show the existence and uniqueness of stationary solutions to anisotropic differential equations. Secondly, we will study the moment exponential stability of weak solutions to stochastic anisotropic Navier-Stokes equation under some appropriate conditions. Based on this result, we will show the moment exponential stability of weak solutions to stochastic Navier-Stokes equations with slow diffusion. Finally, we consider almost sure exponential stability of weak solutions to stochastic anisotropic Navier-Stokes equations.
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