# American Institute of Mathematical Sciences

June  2013, 6(2): 317-372. doi: 10.3934/krm.2013.6.317

## Spectral decompositions and $\mathbb{L}^2$-operator norms of toy hypocoercive semi-groups

 1 Institut Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne F-31062 Toulouse Cedex 9, France, France

Received  February 2012 Revised  October 2012 Published  February 2013

For any $a>0$, consider the hypocoercive generators $y∂_x+a∂_y^2-y∂_y$ and $y∂_x-ax∂_y+∂_y^2-y∂_y$, respectively for $(x,y)\in\mathbb{R}/(2\pi\mathbb{Z})\times\mathbb{R}$ and $(x,y)\in\mathbb{R}\times\mathbb{R}$. The goal of the paper is to obtain exactly the $\mathbb{L}^2(\mu_a)$-operator norms of the corresponding Markov semi-group at any time, where $\mu_a$ is the associated invariant measure. The computations are based on the spectral decomposition of the generator and especially on the scalar products of the eigenvectors. The motivation comes from an attempt to find an alternative approach to classical ones developed to obtain hypocoercive bounds for kinetic models.
Citation: Sébastien Gadat, Laurent Miclo. Spectral decompositions and $\mathbb{L}^2$-operator norms of toy hypocoercive semi-groups. Kinetic & Related Models, 2013, 6 (2) : 317-372. doi: 10.3934/krm.2013.6.317
##### References:

show all references

##### References:
 [1] Pierre Monmarché. Hypocoercive relaxation to equilibrium for some kinetic models. Kinetic & Related Models, 2014, 7 (2) : 341-360. doi: 10.3934/krm.2014.7.341 [2] Christian Licht, Thibaut Weller. Approximation of semi-groups in the sense of Trotter and asymptotic mathematical modeling in physics of continuous media. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1709-1741. doi: 10.3934/dcdss.2019114 [3] O. A. Veliev. On the spectrality and spectral expansion of the non-self-adjoint mathieu-hill operator in $L_{2}(-\infty, \infty)$. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1537-1562. doi: 10.3934/cpaa.2020077 [4] Simona Fornaro, Abdelaziz Rhandi. On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5049-5058. doi: 10.3934/dcds.2013.33.5049 [5] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020077 [6] Sergey A. Denisov. The generic behavior of solutions to some evolution equations: Asymptotics and Sobolev norms. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 77-113. doi: 10.3934/dcds.2011.30.77 [7] Peter Scott and Gadde A. Swarup. Regular neighbourhoods and canonical decompositions for groups. Electronic Research Announcements, 2002, 8: 20-28. [8] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025 [9] J. Colliander, M. Keel, Gigliola Staffilani, H. Takaoka, T. Tao. Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 665-686. doi: 10.3934/dcds.2008.21.665 [10] Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure & Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049 [11] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307 [12] Andrea Bondesan, Laurent Boudin, Marc Briant, Bérénice Grec. Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium maxwell distributions. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2549-2573. doi: 10.3934/cpaa.2020112 [13] Annalisa Cesaroni, Matteo Novaga, Enrico Valdinoci. A symmetry result for the Ornstein-Uhlenbeck operator. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2451-2467. doi: 10.3934/dcds.2014.34.2451 [14] Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations & Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017 [15] Daniel Guo, John Drake. A global semi-Lagrangian spectral model for the reformulated shallow water equations. Conference Publications, 2003, 2003 (Special) : 375-385. doi: 10.3934/proc.2003.2003.375 [16] Alin Pogan, Kevin Zumbrun. Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks. Kinetic & Related Models, 2019, 12 (1) : 1-36. doi: 10.3934/krm.2019001 [17] Tadeusz Iwaniec, Gaven Martin, Carlo Sbordone. $L^p$-integrability & weak type $L^{2}$-estimates for the gradient of harmonic mappings of $\mathbb D$. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 145-152. doi: 10.3934/dcdsb.2009.11.145 [18] Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 737-744. doi: 10.3934/dcdss.2016025 [19] Vedran Sohinger. Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3733-3771. doi: 10.3934/dcds.2012.32.3733 [20] Brandon Seward. Krieger's finite generator theorem for actions of countable groups Ⅱ. Journal of Modern Dynamics, 2019, 15: 1-39. doi: 10.3934/jmd.2019012

2019 Impact Factor: 1.311

## Metrics

• HTML views (0)
• Cited by (9)

• on AIMS