Differential and Integral Equations

Regularity of stagnation-point form solutions of the two-dimensional Euler equations

Alejandro Sarria

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Abstract

A class of semi-bounded solutions of the two-dimensional incompressible Euler equations, satisfying either periodic or Dirichlet boundary conditions, is examined. For smooth initial data, new blowup criteria in terms of the initial concavity profile is presented and the effects that the boundary conditions have on the global regularity of solutions is discussed. In particular, by deriving a formula for a general solution along Lagrangian trajectories, we describe how periodicity can prevent blow-up. This is in opposition to Dirichlet boundary conditions which, as we will show, allow for the formation of singularities in finite time. Lastly, regularity of solutions arising from non-smooth initial data is briefly discussed.

Article information

Source
Differential Integral Equations, Volume 28, Number 3/4 (2015), 239-254.

Dates
First available in Project Euclid: 4 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.die/1423055226

Mathematical Reviews number (MathSciNet)
MR3306561

Zentralblatt MATH identifier
1363.35059

Subjects
Primary: 35B44: Blow-up 35B65: Smoothness and regularity of solutions 35B10: Periodic solutions 35Q35: PDEs in connection with fluid mechanics

Citation

Sarria, Alejandro. Regularity of stagnation-point form solutions of the two-dimensional Euler equations. Differential Integral Equations 28 (2015), no. 3/4, 239--254. https://projecteuclid.org/euclid.die/1423055226


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