### Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation.

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In these notes we give some examples of the interaction of mathematics with experiments and numerical simulations on the search for singularities.

We consider the dynamics of an interface given by two incompressible fluids with different characteristics evolving by Darcy’s law. This scenario is known as the Muskat problem, being in 2D mathematically analogous to the two-phase Hele-Shaw cell. The purpose of this paper is to outline recent results on local existence, weak solutions, maximum principles and global existence.

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 ${L}^{2}\left(\mathbb{R}\right)$ maximum principle, in the form of a new “log” conservation law which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance ${\u2225f\u2225}_{1}\le 1/5$. Previous results of this...

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