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A note on stability shifting for the Muskat problem.

Diego Córdoba1, Javier Gómez-Serrano2, Andrej Zlatoš3

  • 1Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, C/ Nicolas Cabrera, 13-15, 28049 Madrid, Spain Department of Mathematics, Princeton University, 804 Fine Hall, Washington Road, Princeton, NJ 08544, USA dcg@icmat.es.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|August 12, 2015
PubMed
Summary
This summary is machine-generated.

Solutions to the Muskat problem can exhibit dynamic stability shifts, transitioning from unstable to stable and back to unstable. Numerical evidence also reveals turning singularities in solutions with specific initial conditions.

Keywords:
Muskat problemRayleigh–Taylorincompressible fluidinterfaceporous media

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Area of Science:

  • Fluid dynamics
  • Mathematical analysis
  • Partial differential equations

Background:

  • The Muskat problem describes the interface between two immiscible fluids of different densities in porous media.
  • Understanding the stability of these interfaces is crucial for various geophysical and engineering applications.

Purpose of the Study:

  • To demonstrate the existence of Muskat problem solutions that exhibit a shift in stability regimes.
  • To provide numerical evidence for the development of turning singularities in certain solutions.

Main Methods:

  • Analytical investigation of the Muskat problem.
  • Numerical simulations to observe solution behavior and singularity formation.

Main Results:

  • Existence of solutions that transition from unstable to stable, then back to unstable.
  • Numerical observation of turning singularities for solutions with medium-sized L(∞) norm of the initial condition derivative.

Conclusions:

  • The stability of Muskat problem solutions is not monotonic and can evolve over time.
  • Turning singularities represent a complex phenomenon in fluid dynamics that warrants further investigation.