Sharp energy regularity and typicality results for Hölder solutions of incompressible Euler equations

On Derivation of Euler-lagrange Equations for Incompressible Energy-minimizers

We prove that any distribution q satisfying the equation ∇q = div f for some tensor f = (f i j), f i j ∈ h (U) (1 ≤ r < ∞) -the local Hardy space, q is in h, and is locally represented by the sum of singular integrals of f i j with Calderón-Zygmund kernel. As a consequence, we prove the existence and the local representation of the hydrostatic pressure p (modulo constant) associated with incomp...

Potentially Singular Solutions of the 3d Incompressible Euler Equations

Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view, by presenting a class of potentially singular solutions to the Euler equations computed in ax...

Incompressible Euler Equations : the blow - up problem and related results

The question of spontaneous apparition of singularity in the 3D incompressible Euler equations is one of the most important and challenging open problems in mathematical fluid mechanics. In this survey article we review some of recent approaches to the problem. We first review Kato’s classical local well-posedness result in the Sobolev space and derive the celebrated Beale-Kato-Majda criterion ...

Weak Solutions to the Stationary Incompressible Euler Equations

Some sharp Hölder estimates for two-dimensional elliptic equations

We present some recent sharp estimates for the Hölder exponent of solutions of linear second order elliptic equations in divergence form with measurable coefficients. We apply such results to planar Beltrami equations, and we exhibit a mapping of the “angular stretching” type for which our estimates are attained.