Archiv der Mathematik

, Volume 112, Issue 6, pp 661–672 | Cite as

Well-posedness to 3D Burgers’ equation in critical Gevrey Sobolev spaces

  • Ridha SelmiEmail author
  • Abdelkerim Châabani


We prove that the three-dimensional periodic Burgers’ equation has a unique global in time solution in a critical Gevrey–Sobolev space. Comparatively to Navier–Stokes equations, the main difficulty is the lack of an incompressibility condition. In our proof of existence, we overcome the bootstrapping argument, which was a technical step in a precedent proof in Sololev spaces. This makes our proof shorter and gives sense of considering the Gevrey class for a mathematical study to Burgers’ equation. To prove that the unique solution is global in time, we use the maximum principle. Energy methods, Sobolev product laws, compactness methods, and Fourier analysis are the main tools.


Burgers’ equation Critical Gevrey–Sobolev space Existence Uniqueness Global solution 

Mathematics Subject Classification

Primary 35A01 35A02 Secondary 35B10 35B50 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Benameur, J.: On the exponential type explosion of Navier–Stokes equations. Nonlinear Anal. 103, 87–97 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benameur, J., Jlaili, L.: On the blow up criterion of 3D-NSE in Sobolev–Gevrey spaces. J. Math. Fluid Mech. 18, 805–822 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chemin, J.Y., Desjardins, B., Gallagher, I., Grenier, E.: Mathematical Geophysics: An Introduction to Rotating Fluids and the Navier–Stokes Equations. Oxford University Press, Oxford (2006)zbMATHGoogle Scholar
  4. 4.
    Fabes, E.B., Jones, B.F., Rivière, N.M.: The initial value problem for the Navier–Stokes equations with data in \(L^p\). Arch. Rational. Mech. Anal. 45, 222–240 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Levermore, C.D., Oliver, M.: Analyticity of solutions for a generalized Euler equation. J. Differ. Equ. 133, 321–339 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Oliver, M.: A mathematical investigation of models of shallow water with a varying bottom. Ph.D. Thesis, University of Arizona, Tucson, Arizona (1996)Google Scholar
  7. 7.
    Pooley, B.C., Robinson, J.C.: Well-posedness for the diffusive 3D Burgers equations with initial data in \(H^{1/2}\). Lond. Math. Soc. Lect. Note Ser. 430, 137–153 (2016)zbMATHGoogle Scholar
  8. 8.
    Robinson, J., Rodrigo, J.L., Sadowski, W.: Local Existence and Uniqueness in \(H^{1/2}\). The Three-Dimensional Navier–Stokes Equations: Classical Theory, pp. 192–205. Cambridge University Press, Cambridge (2016)CrossRefGoogle Scholar
  9. 9.
    Temam, R.: Navier–Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and Its Applications, vol. 2, 3rd edn. North-Holland Publishing, Amsterdam (1984)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics College of ScienceNorthern Border UniversityArarKingdom of Saudi Arabia
  2. 2.Department of Mathematics Faculty of Mathematical, Physical and Natural Sciences of TunisUniversity of Tunis El ManarTunisTunisia

Personalised recommendations