## Differential and Integral Equations

### Global solvability for two-dimensional filtered Euler equations with measure valued initial vorticity

Takeshi Gotoda

#### Abstract

We study the filtered Euler equations that are the regularized Euler equations derived by filtering the velocity field. The filtered Euler equations are a generalization of two well-known regularizations of incompressible inviscid flows, the Euler-$\alpha$ equations and the vortex blob method. We show the global existence of a unique weak solution for the two-dimensional (2D) filtered Euler equations with initial vorticity in the space of Radon measure that includes point vortices and vortex sheets. Moreover, a sufficient condition for the global well-posedness is described in terms of the filter and thus our result is applicable to various filtered models. We also show that weak solutions of the 2D filtered Euler equations converge to those of the 2D Euler equations in the limit of the regularization parameter provided that initial vorticity belongs to the space of bounded functions.

#### Article information

Source
Differential Integral Equations, Volume 31, Number 11/12 (2018), 851-870.

Dates
First available in Project Euclid: 25 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.die/1537840872

Mathematical Reviews number (MathSciNet)
MR3857867

Zentralblatt MATH identifier
06986981

#### Citation

Gotoda, Takeshi. Global solvability for two-dimensional filtered Euler equations with measure valued initial vorticity. Differential Integral Equations 31 (2018), no. 11/12, 851--870. https://projecteuclid.org/euclid.die/1537840872