Non uniform Rotating Vortices and Periodic Orbits for the Two-Dimensional Euler Equations
@article{Garcia2020NonUR, title={Non uniform Rotating Vortices and Periodic Orbits for the Two-Dimensional Euler Equations}, author={Claudia Garc'ia and Taoufik Hmidi and Juan Soler}, journal={Archive for Rational Mechanics and Analysis}, year={2020} }
This paper concerns the study of some special ordered structures in turbulent flows. In particular, a systematic and relevant methodology is proposed to construct non trivial and non radial rotating vortices with non necessarily uniform densities and with different $m$--fold symmetries, $m\ge 1$. In particular, a complete study is provided for the truncated quadratic density $(A|x|^2+B){\bf{1}}_{\mathbb{D}}(x)$, with $\mathbb{D}$ the unit disc. We exhibit different behaviors with respect to the…
17 Citations
Quantitative estimates for uniformly-rotating vortex patches
- Mathematics
- 2020
In this paper, we derive some quantitative estimates for uniformly-rotating vortex patches. We prove that if a non-radial simply-connected patch $D$ is uniformly-rotating with small angular velocity…
Vortex Patches Choreography for Active Scalar Equations
- MathematicsJ. Nonlinear Sci.
- 2021
The desingularization of the Thomsom polygon for the point vortex system, that is, point vortices located at the vertex of a regular polygon with N sides, is proposed and the study of the contour dynamics equation combined with the application of the infinite dimensional Implicit Function theorem and the well--chosen of the function spaces.
Symmetry in stationary and uniformly rotating solutions of active scalar equations
- MathematicsDuke Mathematical Journal
- 2021
In this paper, we study the radial symmetry properties of stationary and uniformly-rotating solutions of the 2D Euler and gSQG equations, both in the smooth setting and the patch setting. For the 2D…
Global solutions for the generalized SQG equation and rearrangements
- Mathematics
- 2021
In this paper, we study the existence of rotating and traveling-wave solutions for the generalized surface quasi-geostrophic (gSQG) equation. The solutions are obtained by maximization of the energy…
Traveling waves near Couette flow for the 2D Euler equation
- Mathematics
- 2021
In this paper we reveal the existence of a large family of new, nontrivial and smooth traveling waves for the 2D Euler equation at an arbitrarily small distance from the Couette flow in H, with s <…
Time periodic doubly connected solutions for the 3D quasi-geostrophic model
- Mathematics
- 2022
In this paper, we construct time periodic doubly connected solutions for the 3D quasi-geostrophic model in the patch setting. More specifically, we prove the existence of nontrivial m-fold doubly…
Multipole vortex patch equilibria for active scalar equations
- Mathematics
- 2021
. We study how a general steady configuration of finitely-many point vortices, with Newtonian interaction or generalized surface quasi-geostrophic interactions, can be desingularized into a steady…
Global bifurcation for corotating and counter-rotating vortex pairs
- Mathematics
- 2022
Abstract. The existence of a local curve of corotating and counter-rotating vortex pairs was proven by Hmidi and Mateu in [HM17] via a desingularization of a pair of point vortices. In this paper, we…
Time Periodic Solutions for 3D Quasi-Geostrophic Model
- MathematicsCommunications in Mathematical Physics
- 2022
This paper aims to study time periodic solutions for 3D inviscid quasi-geostrophic model. We show the existence of non trivial rotating patches by suitable perturbation of stationary solutions given…
Steady asymmetric vortex pairs for Euler equations
- MathematicsDiscrete & Continuous Dynamical Systems - A
- 2021
In this paper, we study the existence of co-rotating and counter-rotating unequal-sized pairs of simply connected patches for Euler equations. In particular, we prove the existence of curves of…
References
SHOWING 1-10 OF 72 REFERENCES
Global Bifurcation of Rotating Vortex Patches
- MathematicsCommunications on Pure and Applied Mathematics
- 2019
We rigorously construct continuous curves of rotating vortex patch solutions to the two‐dimensional Euler equations. The curves are large in that, as the parameter tends to infinity, the minimum…
Symmetry in stationary and uniformly rotating solutions of active scalar equations
- MathematicsDuke Mathematical Journal
- 2021
In this paper, we study the radial symmetry properties of stationary and uniformly-rotating solutions of the 2D Euler and gSQG equations, both in the smooth setting and the patch setting. For the 2D…
Steady symmetric vortex pairs and rearrangements
- Mathematics, Physics
- 1988
We prove an existence theorem for a steady planar flow of an ideal fluid, containing a bounded symmetric pair of vortices, and approaching a uniform flow at infinity. The data prescribed are the…
Imperfect Bifurcation for the Quasi-Geostrophic Shallow-Water Equations
- Mathematics, Environmental ScienceArchive for Rational Mechanics and Analysis
- 2018
We study analytical and numerical aspects of the bifurcation diagram of simply connected rotating vortex patch equilibria for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations…
Vortex Axisymmetrization, Inviscid Damping, and Vorticity Depletion in the Linearized 2D Euler Equations
- Physics, MathematicsAnnals of PDE
- 2019
Coherent vortices are often observed to persist for long times in turbulent 2D flows even at very high Reynolds numbers and are observed in experiments and computer simulations to potentially be…
Stability Results, Almost Global Generalized Beltrami Fields and Applications to Vortex Structures in the Euler Equations
- Mathematics
- 2016
Strong Beltrami fields, that is, vector fields in three dimensions whose curl is the product of the field itself by a constant factor, have long played a key role in fluid mechanics and…
On the V-states for the Generalized Quasi-Geostrophic Equations
- Mathematics
- 2015
We prove the existence of the V-states for the generalized inviscid SQG equations with $${\alpha \in ]0, 1[.}$$α∈]0,1[. These structures are special rotating simply connected patches with m-fold…
2D Homogeneous Solutions to the Euler Equation
- Mathematics
- 2014
In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form u = ∇⊥Ψ, Ψ(r, θ) = r λψ(θ), for λ > 0, we show that…
Existence of Corotating and Counter-Rotating Vortex Pairs for Active Scalar Equations
- Mathematics, Physics
- 2016
In this paper, we study the existence of corotating and counter-rotating pairs of simply connected patches for Euler equations and the $${{\rm (SQG)}_{\alpha}}$$(SQG)α equations with $${\alpha \in…
Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations
- Mathematics, Environmental Science
- 2014
Motivated by the recent work of Hassainia and Hmidi [Z. Hassainia, T. Hmidi - On the {V}-states for the generalized quasi-geostrophic equations,arXiv preprint arXiv:1405.0858], we close the question…