On the well-posedness of the incompressible flow in porous media
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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
ISRP Style
Fuyi Xu, Lishan Liu, On the well-posedness of the incompressible flow in porous media, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 12, 6371--6381
AMA Style
Xu Fuyi, Liu Lishan, On the well-posedness of the incompressible flow in porous media. J. Nonlinear Sci. Appl. (2016); 9(12):6371--6381
Chicago/Turabian Style
Xu, Fuyi, Liu, Lishan. "On the well-posedness of the incompressible flow in porous media." Journal of Nonlinear Sciences and Applications, 9, no. 12 (2016): 6371--6381
Keywords
- Well-posedness
- blow-up criterion
- particle trajectory mapping
- Fourier localization method
- Besov spaces of weak type.
MSC
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