The Zero Surface Tension Limit of Two-Dimensional Interfacial Darcy Flow

We perform energy estimates for a sharp-interface model of two-dimensional, two-phase Darcy flow with surface tension. A proof of well-posedness of the initial value problem follows from these estimates. In general, the time of existence of these solutions will go to zero as the surface tension parameter vanishes. We then make two additional estimates, in the case that a stability condition is satisfied by the initial data: we make an additional energy estimate which is uniform in the surface tension parameter, and we make an estimate for the difference of two solutions with different values of the surface tension parameter. These additional estimates allow the zero surface tension limit to be taken, showing that solutions of the initial value problem in the absence of surface tension are the limit of solutions of the initial value problem with surface tension as surface tension vanishes.

[1]  Nader Masmoudi,et al.  The zero surface tension limit of three-dimensional water waves , 2009 .

[2]  A. Córdoba,et al.  The Rayleigh-Taylor condition for the evolution of irrotational fluid interfaces , 2009, Proceedings of the National Academy of Sciences.

[3]  Wolf-Patrick Düll.  Validity of the Korteweg–de Vries approximation for the two‐dimensional water wave problem in the arc length formulation , 2012 .

[4]  J. Ye,et al.  Global Existence for a Translating Near-Circular Hele-Shaw Bubble with Surface Tension , 2009, SIAM J. Math. Anal..

[5]  E. C. Childs.  Dynamics of fluids in Porous Media , 1973 .

[6]  G. Taylor,et al.  The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid , 1958, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[7]  Thomas Y. Hou,et al.  The long-time motion of vortex sheets with surface tension , 1997 .

[8]  Robert M. Strain,et al.  On the global existence for the Muskat problem , 2010, 1007.3744.

[9]  A. Córdoba,et al.  Interface evolution: the Hele-Shaw and Muskat problems , 2008, 0806.2258.

[10]  H. Helson.  Harmonic Analysis , 1983 .

[11]  Daniel Coutand,et al.  Well-Posedness of the Free-Boundary Compressible 3-D Euler Equations with Surface Tension and the Zero Surface Tension Limit , 2012, SIAM J. Math. Anal..

[12]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[13]  Inwon C. Kim.  Uniqueness and Existence Results on the Hele-Shaw and the Stefan Problems , 2003 .

[14]  Steven A. Orszag,et al.  Generalized vortex methods for free-surface flow problems , 1982, Journal of Fluid Mechanics.

[15]  G. Pedrizzetti,et al.  Vortex Dynamics , 2011 .

[16]  Michael Siegel,et al.  A non-stiff boundary integral method for 3D porous media flow with surface tension , 2012, Math. Comput. Simul..

[17]  David M. Ambrose,et al.  Well-posedness of two-phase Darcy flow in 3D , 2007 .

[18]  Jacques-Herbert Bailly.  Local existence of classical solutions to first-order parabolic equations describing free boundaries , 1998 .

[19]  E. Zeidler.  Asymptotic Fixed-Point Theorems , 1986 .

[20]  S. Howison.  Cusp Development in Hele–Shaw Flow with a Free Surface , 1986 .

[21]  Antonio Córdoba Barba,et al.  Porous media: the Muskat problem in 3D , 2013 .

[22]  M. SIAMJ.,et al.  CLASSICAL SOLUTIONS OF MULTIDIMENSIONAL HELE – SHAW MODELS , 1997 .

[23]  J. Bona,et al.  WELL-POSEDNESS OF A MODEL FOR WATER WAVES WITH VISCOSITY , 2012 .

[24]  P. Hartman.  Ordinary Differential Equations , 1965 .

[25]  Thomas Y. Hou,et al.  Convergence of a Boundary Integral Method for Water Waves , 1996 .

[26]  Xuming Xie,et al.  Local smoothing effect and existence for the one-phase Hele–Shaw problem with zero surface tension , 2012 .

[27]  Strichartz Estimates for the Water-Wave Problem with Surface Tension , 2009, 0908.3255.

[28]  J. Duchon,et al.  Évolution d’une interface par capillarité et diffusion de volume I. Existence locale en temps , 1984 .

[29]  The singular perturbation of surface tension in Hele-Shaw flows , 2000 .

[30]  S. Yau,et al.  Numerical study of interfacial problems with small surface tension , 2001 .

[31]  Thomas Y. Hou,et al.  Growth rates for the linearized motion of fluid interfaces away from equilibrium , 1993 .

[32]  S. Tanveer,et al.  Global solutions for a two-phase Hele-Shaw bubble for a near-circular initial shape , 2008, 0810.2980.

[33]  S. Shkoller,et al.  Well-posedness for the Classical Stefan Problem and the Zero Surface Tension Limit , 2011, 1112.5817.

[34]  Qing Nie,et al.  The nonlinear evolution of vortex sheets with surface tension in axisymmetric flows , 2001 .

[35]  Sijue Wu,et al.  Well-posedness in Sobolev spaces of the full water wave problem in 3-D , 1999 .

[36]  Mary C. Pugh,et al.  Global solutions for small data to the Hele-Shaw problem , 1993 .

[37]  Yan Guo,et al.  Dynamics near Unstable, Interfacial Fluids , 2007 .

[38]  Michael Siegel,et al.  Singular effects of surface tension in evolving Hele-Shaw flows , 1996, Journal of Fluid Mechanics.

[39]  Sijue Wu,et al.  Well-posedness in Sobolev spaces of the full water wave problem in 2-D , 1997 .

[40]  Nader Masmoudi,et al.  The zero surface tension limit two‐dimensional water waves , 2005 .

[41]  David M. Ambrose,et al.  Well-Posedness of Vortex Sheets with Surface Tension , 2003, SIAM J. Math. Anal..

[42]  Siegel,et al.  Singular perturbation of smoothly evolving Hele-Shaw solutions. , 1996, Physical review letters.

[43]  A. Córdoba,et al.  Interface evolution: Water waves in 2-D , 2008, 0810.5340.

[44]  Joachim Escher,et al.  ON HELE-SHAW MODELS WITH SURFACE TENSION , 1996 .

[45]  Antonio Córdoba,et al.  Porous media: The Muskat problem in three dimensions , 2013 .

[46]  F. Yi.  Global classical solution of Muskat free boundary problem , 2003 .

[47]  David M. Ambrose,et al.  Dispersion vs. anti-diffusion: well-posedness in variable coefficient and quasilinear equations of KdV-type , 2012, 1205.2710.

[48]  T. Hou,et al.  Removing the stiffness from interfacial flows with surface tension , 1994 .

[49]  Svetlana Tlupova,et al.  A small-scale decomposition for 3D boundary integral computations with surface tension , 2013, J. Comput. Phys..

[50]  R. Caflisch,et al.  Global existence, singular solutions, and ill‐posedness for the Muskat problem , 2004 .

[51]  Akira Ogawa,et al.  Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics , 2002 .

[52]  D. Ambrose.  Well-posedness of two-phase Hele–Shaw flow without surface tension , 2004, European Journal of Applied Mathematics.

[53]  Joachim Escher,et al.  Classical solutions for Hele-Shaw models with surface tension , 1997, Advances in Differential Equations.

[54]  G. Batchelor,et al.  An Introduction to Fluid Dynamics , 1968 .

[55]  Nader Masmoudi,et al.  Well-posedness of 3D vortex sheets with surface tension , 2007 .

[56]  Pingwen Zhang,et al.  Convergence of a boundary integral method for 3-D water waves , 2001 .