January  2013, 12(1): 547-596. doi: 10.3934/cpaa.2013.12.547

Resolvent estimates for a two-dimensional non-self-adjoint operator

1. 

Institut de Mathématiques de Jussieu, Université Pierre-et-Marie-Curie (Paris 6), 4 place Jussieu, 75005 Paris, France

Received  May 2011 Revised  January 2012 Published  September 2012

We consider a two-dimensional non-self-adjoint differential operator, originated from a stability problem in the two-dimensional Navier-Stokes equation, given by ${\mathcal L}_\alpha=-\Delta+|x|^2+\alpha \sigma(|x|)\partial_\theta$, where $\sigma(r)=r^{-2}(1-e^{-r^2})$, $\partial_\theta=x_1\partial_2-x_2\partial_1$ 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 $\alpha\to+\infty$ and we prove $\sup_{\lambda\in \mathbb{R}}\|({\mathcal L}_\alpha-i\lambda)^{-1}\|_{{\mathcal L}(\tilde L^2(\mathbb{R}^2))}\leq C\alpha^{-1/3}$, which is an optimal estimate. Our proof is based on a multiplier method, metrics on the phase space and localization techniques.
Citation: Wen Deng. Resolvent estimates for a two-dimensional non-self-adjoint operator. Communications on Pure & Applied Analysis, 2013, 12 (1) : 547-596. doi: 10.3934/cpaa.2013.12.547
References:
[1]

N. Dencker, J. Sjöstrand and M. Zworski, Pseudospectra of semiclassical (pseudo-) differential operators,, Comm. Pure Appl. Math., 57 (2004), 384. doi: 10.1002/cpa.20004.

[2]

I. Gallagher, T. Gallay and F. Nier, Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator,, Int. Math. Res. Not. IMRN, 12 (2009), 2147. doi: 10.1002/cpa.20004.

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T. Gallay and C. E. Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on $R^2$,, Arch. Ration. Mech. Anal., 163 (2002), 209. doi: 10.1007/s002050200200.

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T. Gallay and C. E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation,, Comm. Math. Phys., 225 (2005), 97. doi: 10.1007/s00220-004-1254-9.

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L. Hörmander, "The Analysis of Linear Partial Differential Operators. I,", Springer-Verlag, (1983). doi: 10.1007/978-3-642-96750-4.

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L. Hörmander, "The Analysis of Linear Partial Differential Operators. III,", Springer-Verlag, (1985).

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T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1995).

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N. Lerner, "Metrics on the Phase Space and Non-selfadjoint Pseudo-differential Operators,", Birkh\, (2010). doi: 10.1007/978-3-7643-8510-1.

[9]

K. Pravda-Starov, A general result about the pseudo-spectrum of Schrödinger operators,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), 471. doi: 10.1098/rspa.2003.1194.

[10]

L. N. Trefethen, Pseudospectra of linear operators,, SIAM Rev., 39 (1997), 383. doi: 10.1137/S0036144595295284.

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L. N. Trefethen and M. Embree, "Spectra and Pseudospectra,", Princeton University Press, (2005).

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C. Villani, Hypocoercive diffusion operators,, International Congress of Mathematicians. Vol. {III} (2006), (2006), 473.

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C. Villani, Hypocoercivity,, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00567-5.

show all references

References:
[1]

N. Dencker, J. Sjöstrand and M. Zworski, Pseudospectra of semiclassical (pseudo-) differential operators,, Comm. Pure Appl. Math., 57 (2004), 384. doi: 10.1002/cpa.20004.

[2]

I. Gallagher, T. Gallay and F. Nier, Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator,, Int. Math. Res. Not. IMRN, 12 (2009), 2147. doi: 10.1002/cpa.20004.

[3]

T. Gallay and C. E. Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on $R^2$,, Arch. Ration. Mech. Anal., 163 (2002), 209. doi: 10.1007/s002050200200.

[4]

T. Gallay and C. E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation,, Comm. Math. Phys., 225 (2005), 97. doi: 10.1007/s00220-004-1254-9.

[5]

L. Hörmander, "The Analysis of Linear Partial Differential Operators. I,", Springer-Verlag, (1983). doi: 10.1007/978-3-642-96750-4.

[6]

L. Hörmander, "The Analysis of Linear Partial Differential Operators. III,", Springer-Verlag, (1985).

[7]

T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1995).

[8]

N. Lerner, "Metrics on the Phase Space and Non-selfadjoint Pseudo-differential Operators,", Birkh\, (2010). doi: 10.1007/978-3-7643-8510-1.

[9]

K. Pravda-Starov, A general result about the pseudo-spectrum of Schrödinger operators,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), 471. doi: 10.1098/rspa.2003.1194.

[10]

L. N. Trefethen, Pseudospectra of linear operators,, SIAM Rev., 39 (1997), 383. doi: 10.1137/S0036144595295284.

[11]

L. N. Trefethen and M. Embree, "Spectra and Pseudospectra,", Princeton University Press, (2005).

[12]

C. Villani, Hypocoercive diffusion operators,, International Congress of Mathematicians. Vol. {III} (2006), (2006), 473.

[13]

C. Villani, Hypocoercivity,, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00567-5.

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