Geodesic
The pseudospectrum of the convention-diffusion operator with a variable reaction term
Vestnik rossijskih universitetov. Matematika, Tome 24 (2019) no. 128, pp. 354-367 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper, we study the spectrum of non-self-adjoint convection-diffusion operator with a variable reaction term defined on an unbounded open set $\Omega$ of $\mathbb{R}^n.$ Our idea is to build a family of operators that have the same convection-diffusion-reaction formula, but which will be defined on bounded open sets $\left\{\Omega_{\eta}\right\}_{\eta\in\left]0,1\right[}$ of $\mathbb{R}^n.$ Based on the relationships that link this family to $\Omega,$ we obtain relations between the spectrum and the pseudospectrum. We use the notion of the pseudospectrum to build relationships between convection-diffusion operator and its restrictions to bounded domains. Using these relationships we are able to find the spectrum of our operator in $\mathbb{R}^+.$ Also, the techniques developed to obtain the spectrum allow us to study the properties of the spectrum of this operator when we go to the limit as the reaction term tends to zero. Indeed, we show a spectral localization result for the same convection-diffusion-reaction operator when a perturbation is carried on the reaction term and no longer on the definition domain.
Keywords: differential operator; spectrum; pseudospectrum; convention-diffusion operator. 
@article{VTAMU_2019_24_128_a1,
     author = {H. Guebbai and S. Segni and M. Ghiat and W. Merchela},
     title = {The pseudospectrum of the convention-diffusion operator with a variable reaction term},
     journal = {Vestnik rossijskih universitetov. Matematika},
     pages = {354--367},
     year = {2019},
     volume = {24},
     number = {128},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTAMU_2019_24_128_a1/}
}
TY  - JOUR
AU  - H. Guebbai
AU  - S. Segni
AU  - M. Ghiat
AU  - W. Merchela
TI  - The pseudospectrum of the convention-diffusion operator with a variable reaction term
JO  - Vestnik rossijskih universitetov. Matematika
PY  - 2019
SP  - 354
EP  - 367
VL  - 24
IS  - 128
UR  - http://geodesic.mathdoc.fr/item/VTAMU_2019_24_128_a1/
LA  - ru
ID  - VTAMU_2019_24_128_a1
ER  - 
%0 Journal Article
%A H. Guebbai
%A S. Segni
%A M. Ghiat
%A W. Merchela
%T The pseudospectrum of the convention-diffusion operator with a variable reaction term
%J Vestnik rossijskih universitetov. Matematika
%D 2019
%P 354-367
%V 24
%N 128
%U http://geodesic.mathdoc.fr/item/VTAMU_2019_24_128_a1/
%G ru
%F VTAMU_2019_24_128_a1
H. Guebbai; S. Segni; M. Ghiat; W. Merchela. The pseudospectrum of the convention-diffusion operator with a variable reaction term. Vestnik rossijskih universitetov. Matematika, Tome 24 (2019) no. 128, pp. 354-367. http://geodesic.mathdoc.fr/item/VTAMU_2019_24_128_a1/

[1] H. Guebbai, A. Largillier, “Spectra and Pseudospectra of Convection-Diffusion Operator”, Lobachevskii Journal of Mathematics, 33:1 (2012), 274–283 | DOI | MR | Zbl

[2] E. Shargorodsky, “On the definition of pseudospectra”, Bull. London Math. Soc., 41:2 (2009), 524–534 | DOI | MR | Zbl

[3] S. Roch, B. Silbermann, “C*-algebra techniques in numerical analysis”, J. Operator Theory, 35 (1996), 241–280 | MR | Zbl

[4] L. N. Trefethen, Pseudospectra of matrices, Longman Sci. Tech. Publ., Harlow, 1992 | MR | Zbl

[5] L. N. Trefethen, “Pseudospectra of linear operators”, SIAM Review, 39:3 (1997), 383–406 | DOI | MR | Zbl

[6] E. B. Davies, “Pseudospectra of Differential Operators”, J. Operator Theory, 43:3 (2000), 243–262 | MR | Zbl

[7] E. B. Davies, Spectral Theory and Differential Operators, Cambridge University Press, New York, 1995 | MR | Zbl

[8] L. Boulton, Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra, 1999, arXiv: math/9909179[math.SP] | MR | Zbl

[9] S. C. Reddy, L. N. Trefethen, “Pseudospectra of the convection-diffusion operator”, SIAM J. Appl. Math., 54:1 (1994), 1634–1649 | DOI | MR | Zbl

[10] T. Kato, Perturbation Theory of Linear Operators, Springer-Verlag, Berlin, 1980 | MR | Zbl

[11] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975 | MR | Zbl