Doubly nonlinear parabolic equations for a general class of Forchheimer gas flows in porous media
This paper is focused on the generalized Forchheimer flows of compressible fluids in porous media. The gravity effect and other general nonlinear forms of the source term and boundary flux are integrated into the model. We derive a doubly nonlinear parabolic equation for the so-called pseudo-pressur...
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| Published in | Nonlinearity Vol. 31; no. 8; pp. 3617 - 3650 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
IOP Publishing
01.08.2018
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0951-7715 1361-6544 |
| DOI | 10.1088/1361-6544/aabf05 |
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| Abstract | This paper is focused on the generalized Forchheimer flows of compressible fluids in porous media. The gravity effect and other general nonlinear forms of the source term and boundary flux are integrated into the model. We derive a doubly nonlinear parabolic equation for the so-called pseudo-pressure, and study its initial value problem subject to a general nonlinear Robin boundary condition. The growth rates in the source term and the boundary condition are arbitrarily large. The maximum of the solution, for positive time, is estimated in terms of certain Lebesgue norms of the initial and boundary data. The gradient estimates are obtained under a theoretical condition which, indeed, is relevant to the fluid flows in applications. In dealing with the complexity and generality of the equation and boundary condition, suitable trace theorems and Sobolev's inequalities are utilized, and a well-adapted Moser's iteration is implemented. |
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| AbstractList | This paper is focused on the generalized Forchheimer flows of compressible fluids in porous media. The gravity effect and other general nonlinear forms of the source term and boundary flux are integrated into the model. We derive a doubly nonlinear parabolic equation for the so-called pseudo-pressure, and study its initial value problem subject to a general nonlinear Robin boundary condition. The growth rates in the source term and the boundary condition are arbitrarily large. The maximum of the solution, for positive time, is estimated in terms of certain Lebesgue norms of the initial and boundary data. The gradient estimates are obtained under a theoretical condition which, indeed, is relevant to the fluid flows in applications. In dealing with the complexity and generality of the equation and boundary condition, suitable trace theorems and Sobolev's inequalities are utilized, and a well-adapted Moser's iteration is implemented. |
| Author | Celik, Emine Hoang, Luan Kieu, Thinh |
| Author_xml | – sequence: 1 givenname: Emine surname: Celik fullname: Celik, Emine email: ecelik@unr.edu organization: University of Nevada Department of Mathematics and Statistics, Reno, 1664 N. Virginia Street, Reno, NV 89557, United States of America – sequence: 2 givenname: Luan surname: Hoang fullname: Hoang, Luan email: luan.hoang@ttu.edu organization: Texas Tech University Department of Mathematics and Statistics, Box 41042, Lubbock, TX 79409-1042, United States of America – sequence: 3 givenname: Thinh surname: Kieu fullname: Kieu, Thinh email: thinh.kieu@ung.edu organization: University of North Georgia Department of Mathematics, Gainesville Campus, 3820 Mundy Mill Rd., Oakwood, GA 30566, United States of America |
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| Cites_doi | 10.1016/j.jde.2016.10.043 10.1007/s10958-015-2576-1 10.1007/BF02567072 10.57262/ade/1355703078 10.1016/j.jmaa.2012.12.055 10.1002/cpa.3160240507 10.1016/0022-247X(88)90053-4 10.1007/s10958-014-2045-2 10.1007/978-1-4614-5541-7 10.1002/num.21984 10.3934/dcdss.2014.7.737 10.1098/rspa.1999.0398 10.1063/1.4903002 10.1119/1.19408 10.1007/BF01176474 10.1063/1.3204977 10.1007/s00208-006-0053-3 10.1016/j.camwa.2016.06.029 10.1007/978-1-4612-0895-2 10.1002/num.20035 10.1007/s00021-016-0313-2 10.1137/S0036141091217731 10.1007/s00211-008-0157-7 10.1016/j.nonrwa.2012.07.003 10.1007/BF02673593 10.1088/0951-7715/29/3/1124 10.1088/0951-7715/24/1/001 10.1515/ans-2016-6027 10.1098/rspa.2003.1169 10.1007/s00028-009-0024-8 |
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| References | 22 44 23 24 25 26 Darcy H (7) 1856 Muskat M (35) 1937 Manfredi J J (33) 1994; 2 Ivanov A V (27) 1982; 160 Vázquez J L (43) 2007 Celik E (4) 2016; 29 30 31 10 32 34 Bear J (3) 1988 14 36 15 37 16 Hoang L (19) 2012; 17 38 17 39 Hoang L (18) 2011; 24 Surnachëv M D (41) 2012; 278 1 2 Ivanov A V (29) 1997; 28 Straughan B (40) 2008 5 Ivanov A V (28) 1997; 243 6 Douglas J J (11) 1993 8 Forchheimer P (12) 1901; 45 9 Ward J C (45) 1964; 90 Hoang L (21) 2015 Yaws C L (46) 2001 20 42 Forchheimer P (13) 1930 |
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| SubjectTerms | compressible fluids doubly nonlinear equation Forchheimer flows Moser iteration nonlinear Robin condition porous media |
| Title | Doubly nonlinear parabolic equations for a general class of Forchheimer gas flows in porous media |
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