Resolvent estimates for a two-dimensional non-self-adjoint operator
Resolvent estimates for a two-dimensional non-self-adjoint operator
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二维非自伴算子的求解估计
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发表时间：
2012
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影响因子：
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通讯作者：
Wen Deng
中科院分区：
文献类型：
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作者：
Wen Deng
We consider a two-dimensional non-self-adjoint differential operator, originated from a stability problem in the two-dimensional Navier-Stokes equation, given by ${mathcal L}_alpha=-Delta+|x|^2+alpha sigma(|x|)partial_ heta$, where $sigma(r)=r^{-2}(1-e^{-r^2})$, $partial_ heta=x_1partial_2-x_2partial_1$ and $alpha$ is a positive parameter tending to $+infty$. We give a complete study of the resolvent of ${mathcal L}_alpha$ along the imaginary axis in the fast rotation limit $alpha o+infty$ and we prove $sup_{lambdain mathbb{R}}|({mathcal L}_alpha-ilambda)^{-1}|_{{mathcal L}( ilde L^2(mathbb{R}^2))}leq Calpha^{-1/3}$, which is an optimal estimate. Our proof is based on a multiplier method, metrics on the phase space and localization techniques.
We consider a two-dimensional non-self-adjoint differential operator, originated from a stability problem in the two-dimensional Navier-Stokes equation, given by ${mathcal L}_alpha=-Delta+|x|^2+alpha sigma(|x|)partial_ heta$, where $sigma(r)=r^{-2}(1-e^{-r^2})$, $partial_ heta=x_1partial_2-x_2partial_1$ and $alpha$ is a positive parameter tending to $+infty$. We give a complete study of the resolvent of ${mathcal L}_alpha$ along the imaginary axis in the fast rotation limit $alpha o+infty$ and we prove $sup_{lambdain mathbb{R}}|({mathcal L}_alpha-ilambda)^{-1}|_{{mathcal L}( ilde L^2(mathbb{R}^2))}leq Calpha^{-1/3}$, which is an optimal estimate. Our proof is based on a multiplier method, metrics on the phase space and localization techniques.