On the Null Asymptotic Stabilization of the Two-Dimensional Incompressible Euler Equations in a Simply Connected Domain

1999 ◽  
Vol 37 (6) ◽  
pp. 1874-1896 ◽  
Author(s):  
Jean-Michel Coron
Keyword(s):  
Euler Equations ◽  
Connected Domain ◽  
Two Dimensional ◽  
Asymptotic Stabilization ◽  
Incompressible Euler Equations ◽  
Simply Connected Domain ◽  
Simply Connected ◽  
Incompressible Euler

scholarly journals TWO-DIMENSIONAL INCOMPRESSIBLE IDEAL FLOWS IN A NONCYLINDRICAL MATERIAL DOMAIN

2007 ◽  
Vol 17 (12) ◽  
pp. 2035-2053 ◽  
Author(s):  
F. Z. FERNANDES ◽  
M. C. LOPES FILHO
Keyword(s):  
Weak Solution ◽  
Euler Equations ◽  
Connected Domain ◽  
Moving Boundary ◽  
Two Dimensional ◽  
Noncylindrical Domain ◽  
Simply Connected Domain ◽  
Initial Vorticity ◽  
Simply Connected ◽  
Incompressible Euler

The purpose of this work is to prove the existence of a weak solution of the two-dimensional incompressible Euler equations on a noncylindrical domain consisting of a smooth, bounded, connected and simply connected domain undergoing a prescribed motion. We prove the existence of a weak solution for initial vorticity in Lp, for p > 1. This work complements a similar result by C. He and L. Hsiao, who proved existence assuming that the flow velocity is tangent to the moving boundary, see Ref. 6.

scholarly journals Analytic materials

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160613 ◽  
Author(s):  
Graeme W. Milton
Keyword(s):  
Connected Domain ◽  
Free Field ◽  
Exact Results ◽  
Two Dimensional ◽  
Simply Connected Domain ◽  
Maximum Value ◽  
Two Dimensional Materials ◽  
The Subject ◽  
Simply Connected ◽  
Made In

The theory of inhomogeneous analytic materials is developed. These are materials where the coefficients entering the equations involve analytic functions. Three types of analytic materials are identified. The first two types involve an integer p . If p takes its maximum value, then we have a complete analytic material. Otherwise, it is incomplete analytic material of rank p . For two-dimensional materials, further progress can be made in the identification of analytic materials by using the well-known fact that a 90 ° rotation applied to a divergence-free field in a simply connected domain yields a curl-free field, and this can then be expressed as the gradient of a potential. Other exact results for the fields in inhomogeneous media are reviewed. Also reviewed is the subject of metamaterials, as these materials provide a way of realizing desirable coefficients in the equations.

scholarly journals On the approximation of vorticity fronts by the Burgers–Hilbert equation

2021 ◽  
pp. 1-37
Author(s):  
John K. Hunter ◽  
Ryan C. Moreno-Vasquez ◽  
Jingyang Shu ◽  
Qingtian Zhang
Keyword(s):  
Euler Equations ◽  
Asymptotic Expansions ◽  
Energy Method ◽  
Two Dimensional ◽  
Asymptotic Equation ◽  
Contour Dynamics ◽  
Incompressible Euler Equations ◽  
Incompressible Euler

This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.

scholarly journals ON BOUNDED-TYPE THIN LOCAL SETS OF THE TWO-DIMENSIONAL GAUSSIAN FREE FIELD

2017 ◽  
Vol 18 (3) ◽  
pp. 591-618 ◽  
Author(s):  
Juhan Aru ◽  
Avelio Sepúlveda ◽  
Wendelin Werner
Keyword(s):  
Harmonic Function ◽  
Connected Domain ◽  
Free Field ◽  
Mean Value ◽  
Two Dimensional ◽  
Gaussian Free Field ◽  
Simply Connected Domain ◽  
The Mean ◽  
Conformal Loop Ensemble ◽  
Simply Connected

We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply connected domain, and their relation to the conformal loop ensemble$\text{CLE}_{4}$and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the$\text{CLE}_{4}$carpet, and prove that all BTLS are necessarily connected to the boundary of the domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of$\text{CLE}_{4}$) are in fact measurable functions of the GFF.

scholarly journals On a decomposition of regular domains into John domains with uniform constants

2018 ◽  
Vol 24 (4) ◽  
pp. 1541-1583
Author(s):  
Manuel Friedrich
Keyword(s):  
Sobolev Spaces ◽  
Connected Domain ◽  
Two Dimensional ◽  
John Domain ◽  
Korn’S Inequality ◽  
Simply Connected Domain ◽  
John Domains ◽  
Simply Connected

We derive a decomposition result for regular, two-dimensional domains into John domains with uniform constants. We prove that for every simply connected domain Ω ⊂ ℝ2 with C1-boundary there is a corresponding partition Ω = Ω1 ⋃ … ⋃ ΩN with Σj=1NH1(∂Ωj\∂Ω)≤θ such that each component is a John domain with a John constant only depending on θ. The result implies that many inequalities in Sobolev spaces such as Poincaré’s or Korn’s inequality hold on the partition of Ω for uniform constants, which are independent of Ω.

scholarly journals A Characteristic Mapping method for the two-dimensional incompressible Euler equations

2021 ◽  
Vol 424 ◽  
pp. 109781
Author(s):  
Xi-Yuan Yin ◽  
Olivier Mercier ◽  
Badal Yadav ◽  
Kai Schneider ◽  
Jean-Christophe Nave
Keyword(s):  
Euler Equations ◽  
Mapping Method ◽  
Two Dimensional ◽  
Incompressible Euler Equations ◽  
Characteristic Mapping ◽  
Incompressible Euler
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