headerpos: 12198
 
 
 

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
Publisher
Journal Information
» Editorial Board
» Editorial Policy
» Archival Policy
» Article Publication Charges
» Copyright and Licensing Policy
Guidelines for Authors
» For Authors
» Instructions to Authors
» LaTex style files
Guidelines for Reviewers
» For Reviewers
» Review Form
Open Access
List of Issues
» 2019
» 2018
» 2017
» 2016
» 2015
» 2014
» 2013
» 2012
» 2011
» 2010
Vol. 59, Issue 4
Vol. 59, Issue 3
Vol. 59, Issue 2
Vol. 59, Issue 1
» 2009
» 2008
» Back Issues Phys. Math.
» Back Issues Chemistry
» Back issues (full texts)
  in Google. Phys. Math.
» Back issues (full texts)
  in Google. Chemistry
» Back issues (full texts)
  in Google Engineering
» Back issues (full texts)
  in Google Ecology
» Back issues in ETERA Füüsika, Matemaatika jt
Subscription Information
» Prices
Internet Links
Support & Contact
Publisher
» Staff
» Other journals

Simplified a priori estimate for the time periodic Burgers’ equation; pp. 34–41

(Full article in PDF format) doi: 10.3176/proc.2010.1.06


Authors

Magnus Fontes, Olivier Verdier

Abstract

We present here a version of the existence and uniqueness result of time periodic solutions to the viscous Burgers’ equation with irregular forcing terms (with Sobolev regularity –1 in space). The key result here is an a priori estimate which is simpler than the previously treated case of forcing terms with regularity –½ in time.

Keywords

periodic solutions, Burgers’ equation, distributions, Sobolev spaces, time-periodic.

References

  1. Burgers , J. M. Correlation problems in a one-dimensional model of turbulence. I. Nederl. Akad. Wetensch. , Proc. , 1950 , 53 , 247–260.

  2. Cole , J. D. On a quasi-linear parabolic equation occurring in aerodynamics. Quart. Appl. Math. , 1951 , 9 , 225–236.

  3. E , W. Aubry-Mather theory and periodic solutions of the forced Burgers equation. Comm. Pure Appl. Math. , 1999 , 52(7) , 811–828.
doi:10.1002/(SICI)1097-0312(199907)52:7<811::AID-CPA2>3.0.CO;2-D

  4. Fokas , A. S. and Stuart , J. T. The time periodic solution of the Burgers equation on the half-line and an application to steady streaming. J. Nonlinear Math. Phys. , 2005 , 12(1) , 302–314.
doi:10.2991/jnmp.2005.12.s1.24

  5. Fontes , M. and Verdier , O. Time-periodic solutions of the Burgers equation. J. Math. Fluid Mech. , 2009 , 11(2) , 303–323.
doi:10.1007/s00021-007-0260-z

  6. Fontes , M. and Saksman , E. Optimal results for the two dimensional Navier–Stokes equations with lower regularity on the data. In Actes des Journées Mathématiques à la Mémoire de Jean Leray , Vol. 9 of Sémin. Congr. , pp. 143–154. Soc. Math. France , Paris , 2004.

  7. Hopf , E. The partial differential equation ut + uux = μuxx. Comm. Pure Appl. Math. , 1950 , 3 , 201–230.
doi:10.1002/cpa.3160030302

  8. Jauslin , H. R. , Kreiss , H. O. , and Moser , J. On the forced Burgers equation with periodic boundary conditions. In Differential Equations: La Pietra 1996 (Florence) , Vol. 65 of Proc. Symp. Pure Math. , pp. 133–153. Amer. Math. Soc. , Providence , RI , 1999.

  9. Kreiss , H.-O. and Lorenz , J. Initial-boundary Value Problems and the Navier–Stokes Equations , Vol. 136 of Pure and Applied Mathematics. Academic Press Inc. , Boston , MA , 1989.

10. Sinaĭ , Ya. G. Two results concerning asymptotic behavior of solutions of the Burgers equation with force. J. Stat. Phys. , 1991 , 64(1–2) , 1–12.
doi:10.1007/BF01057866
 
Back

Current Issue: Vol. 68, Issue 3, 2019




Publishing schedule:
No. 1: 20 March
No. 2: 20 June
No. 3: 20 September
No. 4: 20 December