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

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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

Abstract
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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.

Keywords: Multiplier method, metric on the phase space, localization technique..

Mathematics Subject Classification: Primary: 47A10; Secondary: 35Q3.

Citation: Wen Deng. Resolvent estimates for a two-dimensional non-self-adjoint operator. Communications on Pure and 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), 384-415. 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), 2147-2199. 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), 209-258. 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), 97-129. doi: 10.1007/s00220-004-1254-9.
[5]	L. Hörmander, "The Analysis of Linear Partial Differential Operators. I," Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-96750-4.
[6]	L. Hörmander, "The Analysis of Linear Partial Differential Operators. III," Springer-Verlag, Berlin, 1985.
[7]	T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, Berlin, 1995.
[8]	N. Lerner, "Metrics on the Phase Space and Non-selfadjoint Pseudo-differential Operators," Birkhäuser Verlag, Basel, 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), 471-477. doi: 10.1098/rspa.2003.1194.
[10]	L. N. Trefethen, Pseudospectra of linear operators, SIAM Rev., 39 (1997), 383-406. doi: 10.1137/S0036144595295284.
[11]	L. N. Trefethen and M. Embree, "Spectra and Pseudospectra," Princeton University Press, Princeton, NJ, 2005.
[12]	C. Villani, Hypocoercive diffusion operators, International Congress of Mathematicians. Vol. III (2006), 473-498.
[13]	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), 384-415. 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), 2147-2199. 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), 209-258. 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), 97-129. doi: 10.1007/s00220-004-1254-9.
[5]	L. Hörmander, "The Analysis of Linear Partial Differential Operators. I," Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-96750-4.
[6]	L. Hörmander, "The Analysis of Linear Partial Differential Operators. III," Springer-Verlag, Berlin, 1985.
[7]	T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, Berlin, 1995.
[8]	N. Lerner, "Metrics on the Phase Space and Non-selfadjoint Pseudo-differential Operators," Birkhäuser Verlag, Basel, 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), 471-477. doi: 10.1098/rspa.2003.1194.
[10]	L. N. Trefethen, Pseudospectra of linear operators, SIAM Rev., 39 (1997), 383-406. doi: 10.1137/S0036144595295284.
[11]	L. N. Trefethen and M. Embree, "Spectra and Pseudospectra," Princeton University Press, Princeton, NJ, 2005.
[12]	C. Villani, Hypocoercive diffusion operators, International Congress of Mathematicians. Vol. III (2006), 473-498.
[13]	C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00567-5.

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