A linearized Kuramoto–Sivashinsky PDE via an imaginary-Brownian-time-Brownian-angle process

Hassan Allouba

doi:10.1016/S1631-073X(03)00060-8

Partial Differential Equations/Probability Theory

A linearized Kuramoto–Sivashinsky PDE via an imaginary-Brownian-time-Brownian-angle process

Présenté par : Philippe G. Ciarlet

Hassan Allouba ¹

¹ Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA

Comptes Rendus. Mathématique, Volume 336 (2003) no. 4, pp. 309-314.

Résumé
Abstract

Nous introduisons un nouveau processus de temps brownien imaginaire et d'angle brownien, qu'on appelle aussi le processus de Kuramoto–Sivashinsky linéaire (PKSL). En étendant nos techniques développées dans deux articles récents sur la relation entre des processus de temps brownien et des EDPs du quatrième ordre, nous donnons une solution explicite à une version linéaire d'EDP d-dimensionnel de Kuramoto–Sivashinsky : $u_{t} = - \frac{1}{8} Δ^{2} u - \frac{1}{2} Δ u - \frac{1}{2} u$ . La solution est donnée par une fonctionnelle associée à notre PKSL.

We introduce a new imaginary-Brownian-time-Brownian-angle process, which we also call the linear-Kuramoto–Sivashinsky process (LKSP). Building on our techniques in two recent articles involving the connection of Brownian-time processes to fourth order PDEs, we give an explicit solution to a linearized Kuramoto–Sivashinsky PDE in d-dimensional space: $u_{t} = - \frac{1}{8} Δ^{2} u - \frac{1}{2} Δ u - \frac{1}{2} u$ . The solution is given in terms of a functional of our LKSP.

Reçu le : 2002-05-24
Accepté le : 2003-01-23
Publié le : 2003-02-15

DOI : 10.1016/S1631-073X(03)00060-8

Affiliations des auteurs :

Hassan Allouba ¹

¹ Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA

@article{CRMATH_2003__336_4_309_0,
     author = {Hassan Allouba},
     title = {A linearized {Kuramoto{\textendash}Sivashinsky} {PDE} via an {imaginary-Brownian-time-Brownian-angle} process},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {309--314},
     publisher = {Elsevier},
     volume = {336},
     number = {4},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00060-8},
     language = {en},
}

TY  - JOUR
AU  - Hassan Allouba
TI  - A linearized Kuramoto–Sivashinsky PDE via an imaginary-Brownian-time-Brownian-angle process
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 309
EP  - 314
VL  - 336
IS  - 4
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00060-8
LA  - en
ID  - CRMATH_2003__336_4_309_0
ER  -

%0 Journal Article
%A Hassan Allouba
%T A linearized Kuramoto–Sivashinsky PDE via an imaginary-Brownian-time-Brownian-angle process
%J Comptes Rendus. Mathématique
%D 2003
%P 309-314
%V 336
%N 4
%I Elsevier
%R 10.1016/S1631-073X(03)00060-8
%G en
%F CRMATH_2003__336_4_309_0

Hassan Allouba. A linearized Kuramoto–Sivashinsky PDE via an imaginary-Brownian-time-Brownian-angle process. Comptes Rendus. Mathématique, Volume 336 (2003) no. 4, pp. 309-314. doi : 10.1016/S1631-073X(03)00060-8. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00060-8/

Bibliographie
Cité par

[1] H. Allouba Brownian-time processes: the PDE connection II and the corresponding Feynman–Kac formula, Trans. Amer. Math. Soc., Volume 354 (2002) no. 11, pp. 4627-4637

[2] H. Allouba, On the connection between the Kuramoto–Sivashinsky PDE and imaginary-Brownian-time-Brownian-angle processes, 2003, in preparation

[3] H. Allouba; W. Zheng Brownian-time processes: the PDE connection and the half-derivative generator, Ann. Probab., Volume 29 (2001) no. 4, pp. 1780-1795

[4] S. Benachour; B. Roynette; P. Vallois Explicit solutions of some fourth order partial differential equations via iterated Brownian motion, Seminar on Stochastic Analysis, Random Fields and Applications, Ascona, 1996, Progr. Probab., 45, Birkhäuser, Basel, 1999, pp. 39-61

[5] K. Burdzy Some path properties of iterated Brownian motion, Seminar on Stochastic Processes, 1992, Birkhäuser, 1993, pp. 67-87

[6] K. Burdzy Variation of iterated Brownian motion, Workshop and Conf. on Measure-Valued Processes, Stochastic PDEs and Interacting Particle Systems, CRM Proc. Lecture Notes, 5, 1994, pp. 35-53

[7] H. Changbing; R. Temam Robust control of the Kuramoto–Sivashinsky equation, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, Volume 8 (2001) no. 3, pp. 315-338

[8] A. Cheskidov; C. Foias On the non-homogeneous stationary Kuramoto–Sivashinsky equation, Phys. D, Volume 154 (2001) no. 1–2, pp. 1-14

[9] C. Foias; I. Kukavica Determining nodes for the Kuramoto–Sivashinsky equation, J. Dynamics Differential Equations, Volume 7 (1995) no. 2, pp. 365-373

[10] C. Foias; B. Nicolaenko; G.R. Sell; R. Temam Inertial manifolds for the Kuramoto–Sivashinsky equation and an estimate of their lowest dimension, J. Math. Pures Appl. (9), Volume 67 (1988) no. 3, pp. 197-226

[11] T. Funaki Probabilistic construction of the solution of some higher order parabolic differential equation, Proc. Japan Acad. Ser. A Math. Sci., Volume 55 (1979) no. 5, pp. 176-179

[12] K. Hochberg; E. Orsingher Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations, J. Theoret. Probab., Volume 9 (1996) no. 2, pp. 511-532

[13] M.S. Jolly; R. Rosa; R. Temam Evaluating the dimension of an inertial manifold for the Kuramoto–Sivashinsky equation, Adv. Differential Equations, Volume 5 (1–3) (2000), pp. 31-66

[14] D. Khoshnevisan; T. Lewis Iterated Brownian motion and its intrinsic skeletal structure, Seminar on Stochastic Analysis, Random Fields and Applications, Ascona, 1996, Progr. Probab., 45, Birkhäuser, Basel, 1999, pp. 201-210

[15] D. Khoshnevisan; T. Lewis Stochastic calculus for Brownian motion on a Brownian fracture, Ann. Appl. Probab., Volume 9 (1999) no. 3, pp. 629-667

[16] Z. Shi Lower limits of iterated Wiener processes, Statist. Probab. Lett., Volume 23 (1995) no. 3, pp. 259-270

[17] Z. Shi; M. Yor Integrability and lower limits of the local time of iterated Brownian motion, Studia Sci. Math. Hungar., Volume 33 (1997) no. 1–3, pp. 279-298

[18] R. Temam; X. Wang Estimates on the lowest dimension of inertial manifolds for the Kuramoto–Sivashinsky equation in the general case, Differential Integral Equations, Volume 7 (1994) no. 3–4, pp. 1095-1108

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Sommaire tome 336, janvier–juin 2003

C. R. Math (2003)

Semimartingale attractors for generalized Allen–Cahn SPDEs driven by space–time white noise

Hassan Allouba; José A. Langa

C. R. Math (2003)

Dynamics of crystal steps

Olivier Pierre-Louis

C. R. Phys (2005)