Contour Dynamics of Incompressible 3-D Fluids in a Porous Medium with Different Densities

By D. C'ordoba, Francisco Gancedo. 2007

169 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/3b702a1c72381b25d2c3dd0d986cb01293653613

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In Applied Mathematics

In Computational Mechanics

In Computational Theory and Mathematics

Citations in the corpus

Relaxation of the incompressible porous media equation (2011, background): “Evolution of Microstructure In this section we show how to construct solutions to (IPM1)-(IPM4) in the spatial domain Ω := (−1, 1) × (−1, 1) which exhibit the type of mixing behaviour that is expected ([2, 14, 15]), when one starts with a horizontal interface with the heavier fluid on top.”
Endpoint Sobolev Theory for the Muskat Equation (2020, methodology): “In the Córdoba-Gancedo formulation [33], the evolution equation for f simplifies to ∂t f = 1 π pv ˆ”
Global regularity for 2D Muskat equations with finite slope (2015, background): “It can incorporate capillary forces that deal with surface tension effects [CP93, C93, KM15], density jumps [CG07], viscosity [SCH04], and permeability discontinuities [BCGB14].”
Interface evolution: the Hele-Shaw and Muskat problems (2008, background): “Finally, in the case where the viscosities are the same, the character of the interphase as the graph of a function is preserved and in [9] [10] this fact has been used to prove local existence and a maximum principle, in the stable case, together with ill-posedness in the unstable situation.”
Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves (2011, background): “In particular f is Lipschitz continuous.”
On the Euler equations of incompressible fluids (2007, methodology): “The modern well-posedness theory for free surfaces uses harmonic analysis and geometric PDEs ([3], [4], [35], [63], [65], [66], [102], [126], [127]) and is a subject in full swing.”
Lack of Uniqueness for Weak Solutions of the Incompressible Porous Media Equation (2009, background): “It is known the existence of weak solutions, where the motion takes place in the interface between fluids with different constant densities, modeling the contour dynamics Muskat problem [3].”
Global well-posedness for the two-dimensional Muskat problem with slope less than 1 (2017, unknown): “The ﬂuids evolve according to Darcy’s law, giving an evolution of the interface (see [Córdoba and Gancedo 2007] for derivation of equations), and the problem in two dimensions is analogous to the two-phase Hele-Shaw cell (see [Saffman and Taylor 1958]).”
Paralinearization of the Muskat Equation and Application to the Cauchy Problem (2019, background): “The well-posedness of the Cauchy problem was first proved in [21] by Córdoba and Gancedo for initial data in H3(R) in the stable regime ρ2 > ρ1 (they also proved that the problem is ill-posed in Sobolev spaces when ρ2 < ρ1).”
On the global existence for the Muskat problem (2010, methodology): “The key point for our result, in comparison to previous work [7, 19, 9, 14], is that the constant c0 can be easily explicitly computed (see (8)).”
Well-posedness of the Muskat problem with H2 initial data (2014, unknown): “In particular, for the infinitely-deep case, Córdoba & Gancedo proved the local existence of solutions for H(3)(R) initial data in the stable Rayleigh-Taylor regime and the ill-posed character of the Muskat problem in the unstable RayleighTaylor regime in [16], a maximum principle for ‖h(t)‖L∞ in [1”
On the Two‐Dimensional Muskat Problem with Monotone Large Initial Data (2016, methodology): “Moreover, it is proved in [10] that even for arbitrary small initial data in Hs some solutions leave the space Hs right away in the unstable situation.”