The Second Iterate of the Muskat Equation in Supercritical Spaces

Abstract

The ill-posedness for the Muskat problem in spaces that are supercritical with respect to the scaling is studied. The main result of the paper establishes that for a sequence of approximations of the Muskat equation obtained via Taylor expansion, their corresponding second Picard's iterate is discontinuous around the origin in a certain family of supercritical spaces approaching a critical space.

arXiv comment29 pages

References in corpus (17)

• On the global existence for the Muskat problem
• Breakdown of smoothness for the Muskat problem
• On the Muskat problem: global in time results in 2D and 3D
• The Muskat problem in 2D: equivalence of formulations, well-posedness, and regularity results
• On The Muskat Problem With Viscosity Jump: Global In Time Results
• Global well-posedness for the 2D Muskat problem with slope less than 1
• A paradifferential approach for well-posedness of the Muskat problem
• Paralinearization of the Muskat equation and application to the Cauchy problem
• Turning waves and breakdown for incompressible flows
• On the Cauchy problem for the Muskat equation with non-Lipschitz initial data
• Large Time Decay Estimates for the Muskat Equation
• Well-posedness of the Muskat problem in subcritical $L_p$-Sobolev spaces
• Global wellposedness for the 3D Muskat problem with medium size slope
• Ill-posedness for subcritical hyperdissipative Navier-Stokes equations in the largest critical spaces
• Norm inflation for the Boussinesq system
• Global existence and decay of the inhomogeneous Muskat problem with Lipschitz initial data
• On the second iterate for active scalar equations