scholarly journals Blow up for the 2D Euler equation on some bounded domains

2015 ◽  
Vol 259 (7) ◽  
pp. 3490-3494 ◽  
Author(s):  
Alexander Kiselev ◽  
Andrej Zlatoš
Keyword(s):  
Euler Equation ◽  
Blow Up ◽  
Bounded Domains ◽  
2D Euler Equation

On the Existence for the Free Interface 2D Euler Equation with a Localized Vorticity Condition

2016 ◽  
Vol 73 (3) ◽  
pp. 523-544 ◽  
Author(s):  
Igor Kukavica ◽  
Amjad Tuffaha ◽  
Vlad Vicol ◽  
Fei Wang
Keyword(s):  
Euler Equation ◽  
Free Interface ◽  
2D Euler Equation

scholarly journals Homogeneous Euler equation: blow-ups, gradient catastrophes and singularity of mappings

2021 ◽  
Author(s):  
Boris G. Konopelchenko ◽  
Giovanni Ortenzi
Keyword(s):  
Euler Equation ◽  
Euler Equations ◽  
Blow Up ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Linear Combinations ◽  
First Occurrence ◽  
Different Dimensions ◽  
Characteristic Features

Abstract The paper is devoted to the analysis of the blow-ups of derivatives, gradient catastrophes and dynamics of mappings of ℝn → ℝn associated with the n-dimensional homogeneous Euler equation. Several characteristic features of the multi-dimensional case (n > 1) are described. Existence or nonexistence of blow-ups in different dimensions, boundness of certain linear combinations of blow-up derivatives and the first occurrence of the gradient catastrophe are among of them. It is shown that the potential solutions of the Euler equations exhibit blow-up derivatives in any dimenson n. Several concrete examples in two- and three-dimensional cases are analysed. Properties of ℝnu → ℝ nx mappings defined by the hodograph equations are studied, including appearance and disappearance of their singularities.

scholarly journals Stochastic stability of invariant measures: The 2D Euler equation

2019 ◽  
Vol 33 (17) ◽  
pp. 1950185
Author(s):  
F. Cipriano ◽  
H. Ouerdiane ◽  
R. Vilela Mendes
Keyword(s):  
Euler Equation ◽  
Coherent Structures ◽  
Stochastic Stability ◽  
Invariant Measures ◽  
Finite Dimensional ◽  
Stochastically Stable ◽  
2D Euler Equation ◽  
Stable Measures ◽  
Dissipative Dynamical Systems ◽  
Selection Of

In finite-dimensional dissipative dynamical systems, stochastic stability provides the selection of the physically relevant measures. That this might also apply to systems defined by partial differential equations, both dissipative and conservative, is the inspiration for this work. As an example, the 2D Euler equation is studied. Among other results this study suggests that the coherent structures observed in 2D hydrodynamics are associated with configurations that maximize stochastically stable measures uniquely determined by the boundary conditions in dynamical space.

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