It is well known that the incompressible Euler equations in two dimensions have globally regular solutions. The inviscid surface quasi-geostrophic (SQG) equation has a Biot-Savart law which is one derivative less regular than in the Euler case, and the question of global regularity for its solutions is still open. We study here the patch dynamics in the half-plane for a family of active scalars...

The global existence of weak solutions for the three-dimensional axisymmetric Euler-α (also known as Lagrangian-averaged Euler-α) equations, without swirl, is established, whenever the initial unfiltered velocity v0 satisfies ∇×v0 r is a finite Randon measure with compact support. Furthermore, the global existence and uniqueness, is also established in this case provided ∇×v0 r ∈ L c (R) with p...

We find a new class of data for which the Prandtl boundary layer equations and the hydrostatic Euler equations are locally in time well-posed. In the case of the Prandtl equations, we assume that the initial datum u0 is monotone on a number of intervals (on some strictly increasing on some strictly decreasing) and analytic on the complement and show that the local existence and uniqueness hold....

We introduce a variational principle for field theories, referred to as the HamiltonPontryagin principle, and we show that the resulting field equations are the Euler-Lagrange equations in implicit form. Secondly, we introduce multi-Dirac structures as a graded analog of standard Dirac structures, and we show that the graph of a multisymplectic form determines a multi-Dirac structure. We then d...

In this article we consider the evolution of vortex sheets in the plane both as a weak solution of the two dimensional incompressible Euler equations and as a (weak) solution of the Birkhoff-Rott equations. We begin by discussing the classical Birkhoff-Rott equations with respect to arbitrary parametrizations of the sheet. We introduce a notion of weak solution to the Birkhoff-Rott system and w...

In this article we consider the evolution of vortex sheets in the plane both as a weak solution of the two dimensional incompressible Euler equations and as a (weak) solution of the Birkhoff-Rott equations. We begin by discussing the classical Birkhoff-Rott equations with respect to arbitrary parametrizations of the sheet. We introduce a notion of weak solution to the Birkhoff-Rott system, and ...

The purpose of this article is to survey results concerning the unstable spectrum of the Euler equation linearized about a steady state. The Euler equations of the motion of an inviscid, incompressible fluid are the basic equations of fluid mechanics and they have been the object of much study by mathematicians over the centuries since Euler ”unveiled” them in 1755. However, many significant pr...

We present results concerning the local existence, regularity and possible blow up of solutions to incompressible Euler and NavierStokes equations.

and Applied Analysis 3 is called the auxiliary equation of the Euler differential equation 2.1 , and every solution of 2.1 is of the form yh x ⎧ ⎪⎨ ⎪⎩ c1x m1 c2x2 if m1 and m2 are distinct roots of 2.2 , c1 c2 lnx x 1−α /2 if 1 − α 2 is a double root of 2.2 , 2.3 where c1 and c2 are complex constants see 21, Section 2.7 . Theorem 2.1. Let α and β be complex constants such that no root of the au...

The purpose of this paper is to derive the anisotropic averaged Euler equations and to study their geometric and analytic properties. These new equations involve the evolution of a mean velocity field and an advected symmetric tensor that captures the fluctuation effects. Besides the derivation of these equations, the new results in the paper are smoothness properties of the equations in materi...