American Institute of Mathematical Sciences

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

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

Wen Deng 1,

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.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[1]

Yigui Ou, Yuanwen Liu. A memory gradient method based on the nonmonotone technique. Journal of Industrial & Management Optimization, 2017, 13 (2) : 857-872. doi: 10.3934/jimo.2016050
[2]

Stilianos Louca, Fatihcan M. Atay. Spatially structured networks of pulse-coupled phase oscillators on metric spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3703-3745. doi: 10.3934/dcds.2014.34.3703
[3]

Claudio Meneses. Linear phase space deformations with angular momentum symmetry. Journal of Geometric Mechanics, 2019, 11 (1) : 45-58. doi: 10.3934/jgm.2019003
[4]

Yong-Jung Kim. A generalization of the moment problem to a complex measure space and an approximation technique using backward moments. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 187-207. doi: 10.3934/dcds.2011.30.187
[5]

Pascal Bégout, Jesús Ildefonso Díaz. A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3371-3382. doi: 10.3934/dcds.2014.34.3371
[6]

Fuke Wu, Xuerong Mao, Peter E. Kloeden. Discrete Razumikhin-type technique and stability of the Euler--Maruyama method to stochastic functional differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 885-903. doi: 10.3934/dcds.2013.33.885
[7]

Bin Li, Xiaolong Guo, Xiaodong Zeng, Songyi Dian, Minhua Guo. An optimal pid tuning method for a single-link manipulator based on the control parametrization technique. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1813-1823. doi: 10.3934/dcdss.2020107
[8]

Hartmut Schwetlick, Daniel C. Sutton, Johannes Zimmer. On the $\Gamma$-limit for a non-uniformly bounded sequence of two-phase metric functionals. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 411-426. doi: 10.3934/dcds.2015.35.411
[9]

Alexander V. Rezounenko, Petr Zagalak. Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 819-835. doi: 10.3934/dcds.2013.33.819
[10]

Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123
[11]

Rogério Martins. One-dimensional attractor for a dissipative system with a cylindrical phase space. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 533-547. doi: 10.3934/dcds.2006.14.533
[12]

P. M. Jordan, Louis Fishman. Phase space and path integral approach to wave propagation modeling. Conference Publications, 2001, 2001 (Special) : 199-210. doi: 10.3934/proc.2001.2001.199
[13]

Oskar Weinberger, Peter Ashwin. From coupled networks of systems to networks of states in phase space. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 2021-2041. doi: 10.3934/dcdsb.2018193
[14]

Evgeny L. Korotyaev. Estimates for solutions of KDV on the phase space of periodic distributions in terms of action variables. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 219-225. doi: 10.3934/dcds.2011.30.219
[15]

Nicolas Lerner, Yoshinori Morimoto, Karel Pravda-Starov, Chao-Jiang Xu. Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators. Kinetic & Related Models, 2013, 6 (3) : 625-648. doi: 10.3934/krm.2013.6.625
[16]

Zhendong Luo. A reduced-order SMFVE extrapolation algorithm based on POD technique and CN method for the non-stationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1189-1212. doi: 10.3934/dcdsb.2015.20.1189
[17]

Caiping Liu, Heungwing Lee. Lagrange multiplier rules for approximate solutions in vector optimization. Journal of Industrial & Management Optimization, 2012, 8 (3) : 749-764. doi: 10.3934/jimo.2012.8.749
[18]

Santanu Sarkar. Analysis of Hidden Number Problem with Hidden Multiplier. Advances in Mathematics of Communications, 2017, 11 (4) : 805-811. doi: 10.3934/amc.2017059
[19]

Weinan E, Weiguo Gao. Orbital minimization with localization. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 249-264. doi: 10.3934/dcds.2009.23.249
[20]

Radu Balan, Peter G. Casazza, Christopher Heil and Zeph Landau. Density, overcompleteness, and localization of frames. Electronic Research Announcements, 2006, 12: 71-86.

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences