On the well-posedness of the incompressible flow in porous media

Volume 9, Issue 12, pp 6371--6381 http://dx.doi.org/10.22436/jnsa.009.12.37

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1520 Downloads
• 2354 Views

Authors

Fuyi Xu - School of Mathematics Science, Qufu normal University, Qufu, 263516, China. - School of Science, Shandong University of Technology, Zibo, 255049, China. Lishan Liu - School of Mathematics Science, Qufu normal University, Qufu, 263516, China.

Abstract

In this paper, we are concerned with the two-dimensional (2D) incompressible flow in porous media (IPM) in the whole space. We prove the local well-posedness of the solutions for the system in Besov spaces of weak type and obtain blow-up criterion of solutions by particle trajectory method and Fourier localization method.

Share and Cite

Keywords

• Well-posedness
• blow-up criterion
• particle trajectory mapping
• Fourier localization method
• Besov spaces of weak type.

MSC

•  35Q35
•  76B03

References

• [1] J. Beale, T. Kato, A. Majda, Remark on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 94 (1984), 61--66


• [2] J. Bear, Dynamics of Fluids in Porous Media, Elsevier, New York (1972)


• [3] J. Bergh, J. Löfstrom, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New York (1976)


• [4] D. Chae, On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces, Comm. Pure Appl. Math., 55 (2002), 654--678


• [5] D. Chae, On the Euler equations in the critical Triebel-Lizorkin spaces, Arch. Rational Mech. Anal., 170 (2003), 185--210


• [6] D. Chae, Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptot. Anal., 38 (2004), 339--358


• [7] J.-Y. Chemin, Perfect incompressible fluids, Oxford Lectures Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York (1998)


• [8] D. Córdoba, F. Gancedo, R. Orive, Analytical behavior of the two-dimensional incompressible flow in porous media, J. Math. Phys., 48 (2007), 19 pages


• [9] D. Nield, A. Bejan, Convection in Porous Media, Springer-Verlag, New York (1999)


• [10] M. Price, Introducing Groundwater, 2nd ed., Chapman and Hall, London (1996)


• [11] E. M. Stein, Harmonic Analysis, Princeton University Press, Princeton (1993)


• [12] R. Takada, Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type, J. Evol. Equ., 8 (2008), 693--725


• [13] W. Yu, Y. He, On the well-posedness of the incompressible porous media equation in Triebel-Lizorkin spaces, Boundary Value Problems, 2014 (2014), 11 pages