The Muskat problem, in its general setting, concerns the interface evolution between two incompressible fluids of different densities and viscosities porous media. motion is driven by gravity capillarity forces, where latter due to surface tension. To leading order, both problems with without tension effect are scaling invariant Sobolev space $$H^{1+\frac{d}{2}}({\mathbb {R}}^d)$$ , d dimension...

We study the Muskat problem describing vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium an L p -setting with ∈ (1, ∞). The Sobolev space $W_p^s(\mathbb R)$ s = 1+1/ is critical for this problem. prove, each (1+1/ , 2) that Rayleigh–Taylor condition identifies open subset within which parabolic type. This enables us to establish local well-posedness all the...

We consider the one-phase Muskat problem modeling dynamics of free boundary a single fluid in porous media. In stable regime, we prove local well-posedness for interfaces that are general curves and can have singularities. particular, acute angle corners or cusps. Moreover, show isolated corners/cusps on interface must be rigid, meaning corner is preserved finite time, there no rotation at tip,...

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an L(R) maximum principle, in the form of a new “log” conservation law (3) which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence of Lipschitz continuous sol...

a relationship based on the modified couple stress theory is developed to investigate the flexural sensitivity of an atomic force microscope (afm) with assembled cantilever probe (acp). this acp comprises a horizontal cantilever, two vertical extensions and two tips located at the free ends of the extensions which form a caliper. an approximate solution to the flexural vibration problem is obta...

We study the two-dimensional multiphase Muskat problem describing motion of three immiscible fluids with equal viscosities in a vertical homogeneous porous medium identified $\mathbb{R}^2$ under effect gravity. first formulate governing equations as strongly coupled evolution for functions that parameterize sharp interfaces between fluids. Afterwards we prove is parabolic type and establish its...