TY - JOUR
T1 - Interior Estimates for Generalized Forchheimer Flows of Slightly Compressible Fluids
AU - Hoang, Luan T.
AU - Kieu, Thinh T.
N1 - Funding Information:
The first author acknowledges the support by NSF grant DMS-1412796.
Publisher Copyright:
© 2017 Walter de Gruyter GmbH, Berlin/Boston 2017.
PY - 2017/10/1
Y1 - 2017/10/1
N2 - The generalized Forchheimer flows are studied for slightly compressible fluids in porous media with time-dependent Dirichlet boundary data for the pressure. No restrictions are imposed on the degree of the Forchheimer polynomial. We derive, for all time, the interior L (L ∞} -estimates for the pressure, its gradient and time derivative, and the interior L 2 L 2 -estimates for its Hessian. The De Giorgi and Ladyzhenskaya-Uraltseva iteration techniques are used taking into account the special structures of the equations for both pressure and its gradient. These are combined with the uniform Gronwall-type bounds in establishing the asymptotic estimates when time tends to infinity.
AB - The generalized Forchheimer flows are studied for slightly compressible fluids in porous media with time-dependent Dirichlet boundary data for the pressure. No restrictions are imposed on the degree of the Forchheimer polynomial. We derive, for all time, the interior L (L ∞} -estimates for the pressure, its gradient and time derivative, and the interior L 2 L 2 -estimates for its Hessian. The De Giorgi and Ladyzhenskaya-Uraltseva iteration techniques are used taking into account the special structures of the equations for both pressure and its gradient. These are combined with the uniform Gronwall-type bounds in establishing the asymptotic estimates when time tends to infinity.
KW - Asymptotic
KW - Darcy-Forchheimer Equation
KW - Degenerate Parabolic Equation
KW - Nonlinear Differential Inequality
KW - Porous Media
KW - Stability
KW - Uniform Gronwall Inequality
UR - http://www.scopus.com/inward/record.url?scp=85019996971&partnerID=8YFLogxK
U2 - 10.1515/ans-2016-6027
DO - 10.1515/ans-2016-6027
M3 - Article
AN - SCOPUS:85019996971
VL - 17
SP - 739
EP - 767
JO - Advanced Nonlinear Studies
JF - Advanced Nonlinear Studies
SN - 1536-1365
IS - 4
ER -