Now showing items 1-10 of 27
Incompressible flow in porous media with fractional diffusion [Article]
(IOP Publishing, 2009-08)
In this paper we study the heat transfer with a general fractional diffusion term of an incompressible fluid in a porous medium governed by Darcy’s law. We show formation of singularities with infinite energy and for finite ...
Interface evolution: the Hele-Shaw and Muskat problems [Article]
(Princeton University, 2011)
We study the dynamics of the interface between two incompressible 2-D flows where the evolution equation is obtained from Darcy’s law. The free boundary is given by the discontinuity among the densities and viscosities of ...
Splash singularity for water waves [Article]
(National Academy of Sciences, 2012-01-17)
We exhibit smooth initial data for the two-dimensional (2D) waterwave equation for which we prove that smoothness of the interface breaks down in finite time. Moreover, we show a stability result together with numerical ...
Turning waves and breakdown for incompressible flows [Article]
(National Academy of Sciences, 2011-03-22)
We consider the evolution of an interface generated between two immiscible, incompressible, and irrotational fluids. Specifically we study the Muskat and water wave problems. We show that starting with a family of initial ...
Lack of uniqueness for weak solutions of the incompressible porous media equation [Article]
In this work we consider weak solutions of the incompressible 2-D porous media equation. By using the approach of De Lellis-Sz´ekelyhidi we prove non-uniqueness for solutions in L∞ in space and time.
Breakdown of smoothness for the Muskat problem [Article]
In this paper we show that there exist analytic initial data in the stable regime for the Muskat problem such that the solution turns to the unstable regime and later breaks down, i.e., no longer belongs to C4.
Finite time singularities for the free boundary incompressible Euler equations [Article]
(Princeton University, 2013)
In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem), for which the smoothness of the ...
The Rayleigh-Taylor condition for the evolution of irrotational fluid interfaces [Article]
(National Academy of Sciences, 2009-07-07)
For the free boundary dynamics of the two-phase Hele-Shaw and Muskat problems, and also for the irrotational incompressible Euler equation, we prove existence locally in time when the Rayleigh-Taylor condition is initially ...
Interface evolution: water waves in 2-D [Article]
We study the free boundary evolution between two irrotational, incompressible and inviscid fluids in 2-D without surface tension. We prove local-existence in Sobolev spaces when, initially, the difference of the gradients ...