Related papers: An Eulerian-Lagrangian Form for the Euler Equation…
We study regularity properties of the data-to-solution maps of the family of generalized surface quasi-geostrophic equations which includes both the 2D incompressible Euler and the standard surface quasi-geostrophic equations. We prove that…
One of the most remarkable features of known nonstationary solutions to the incompressible Euler equations is the phenomenon known as the Taylor hypothesis, which predicts that coarse scale averages of the velocity carry the fine scale…
It has been known since work of Lichtenstein [42] and Gunther [29] in the 1920's that the $3D$ incompressible Euler equation is locally well-posed in the class of velocity fields with H\"older continuous gradient and suitable decay at…
We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has…
We study explicit solutions to the 2 dimensional Euler equations in the Lagrangian framework. All known solutions have been of the separation of variables type, where time and space dependence are treated separately. The first such…
In this paper we consider the problem of finding periodic solutions of certain Euler-Lagrange equations, which include, among others, equations involving the $p$-Laplace and, more generality, the $(p,q)$-Laplace operator. We employ the…
The 3D incompressible Euler equation is an important research topic in the mathematical study of fluid dynamics. Not only is the global regularity for smooth initial data an open issue, but the behaviour may also depend on the presence or…
In this article, we will study unbounded solutions of the 2D incompressible Euler equations. One of the motivating factors for this is that the usual functional framework for the Euler equations (e.g. based on finite energy conditions, such…
We show that, for two-dimensional space-periodic incompressible flow, the solution can be evaluated numerically in Lagrangian coordinates with the same accuracy achieved in standard Eulerian spectral methods. This allows the determination…
In this paper we consider the Euler-Poisson system (describing a plasma made of ions with a negligible ion temperature) on the Sobolev spaces $H^s(\R^3)$, $s > 5/2$. Using a geometric approach we show that for any time $T > 0$ the…
We construct solutions of the 2D incompressible Euler equations in $\mathds{R}^2\times [0,\infty)$ such that initially the velocity is in the super-critical Sobolev space $H^\beta$ for $1<\beta<2$, but are not in $H^{\beta'}$ for…
We introduce many families of explicit solutions to the three dimensional incompressible Euler equations for nonviscous fluid flows using the Lagrangian framework. Almost no exact Lagrangian solutions exist in the literature prior to this…
In this note, we establish Yudovich's existence and uniqueness result for bounded (as well as mildly unbounded) vorticity weak solution of the two-dimensional incompressible Euler equations. As a biproduct of our proof, we establish some…
In this paper we study the uniqueness property of solutions to the steady incompressible Euler equations with perturbations in $\Bbb R^N$. Our perturbations include as special cases the Euler equations with a `single signed' nonlinear term,…
We are concerned with how regular initial data have to be to ensure local existence for Einstein's equations in wave coordinates. Klainerman-Rodnianski and Smith-Tataru showed that there in general is local existence for data in Sobolev…
Building on the recent work of C. De Lellis and L. Sz\'{e}kelyhidi, we construct global weak solutions to the three-dimensional incompressible Euler equations which are zero outside of a finite time interval and have velocity in the…
A new perspective on the classical mechanical formulation of particle trajectories in lorentz-violating theories is presented. Using the extended hamiltonian formalism, a Legendre Transformation between the associated covariant Lagrangian…
The Cauchy problem for the two dimensional compressible Euler equations with data in the Sobolev space $H^s(\mathbb R^2)$ is known to have a unique solution of the same Sobolev class for a short time, and the data-to-solution map is…
The first two sections of this work review the framework of [6] for approximate solutions of the incompressible Euler or Navier-Stokes (NS) equations on a torus T^d, in a Sobolev setting. This approach starts from an approximate solution…
We consider the incompressible Euler or Navier-Stokes (NS) equations on a torus T^d in the functional setting of the Sobolev spaces H^n(T^d) of divergence free, zero mean vector fields on T^d, for n > d/2+1. We present a general theory of…