Axi‐symmetrization near Point Vortex Solutions for the <scp>2D</scp> Euler Equation

Ionescu, Alexandru D.; Jia, Hao

doi:10.1002/cpa.21974

Cited by 43 publications

(30 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For sufficiently regular data some interesting phenomena appear, such as vortex axysimmetrization (the vorticity weakly converges back to radial symmetry) and vorticity depletion (faster inviscid damping rates than those possible with passive scalar evolution). Finally, for perturbations around coherent vortex structures we also refer to [20,31,46,47,61] and references therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Traveling waves near Couette flow for the 2D Euler equation

Castro¹,

Lear²

2021

Preprint

View full text Add to dashboard Cite

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 s , with s < 3/2, at the level of the vorticity. The speed of these waves is of order 1 with respect to this distance. This result strongly contrasts with the setting of very high regularity in Gevrey spaces (see [7]), where the problem exhibits an inviscid damping mechanism that leads to relaxation of perturbations back to nearby shear flows. It also complements the fact that there not exist nontrivial traveling waves in the H 3 2 + neighborhoods of Couette flow (see [51]). Contents 1. Introduction and main result 1.1. Sketch of the proof 1.2. Organization 2. Formulation of the problem 2.1. The 2D Euler as an equation for the level curves of the vorticity 2.2. The equation for the traveling wave 2.3. The profile function ε,κ . 3. Bifurcation theory and Crandall-Rabinowitz 4. Functional setting and regularity 4.1. The functional setting 4.2. Hypothesis 1 and 2 5. Analysis of the linear part 5.1. Decomposition of the linear operator 5.2. Rescaling of the decomposition 5.3. One dimensionality of the kernel of the linear operator 5.4. Codimension of the image of the linear operator 5.5. The transversality property 6. Main theorem 6.1. Distance of the traveling wave to the Couette flow 6.2. Full regularity of the solution 7. Appendix References

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

Traveling waves near Couette flow for the 2D Euler equation

Castro¹,

Lear²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…All these stability results will likely be modified if point vortices are present in the flow, and the stability of hybrid solutions with multiple point vortices is an important question that needs to be addressed. Some recent progress for a single point vortex in a perturbed background vorticity field is reported in Ionescu & Jia (2021).…”

Section: Summary and Future Directionsmentioning

confidence: 99%

Liouville chains: new hybrid vortex equilibria of the two-dimensional Euler equation

et al. 2021

View full text Add to dashboard Cite

show abstract

“…For strictly monotone radial profiles in R 2 , Bedrossian-Coti Zelati-Vicol [5] obtained the linear inviscid damping. We also mention asymptotic results for a point vortex by Coti Zelati-Zillinger [20] and by Ionescu-Jia [27]. For the infinite channel T × R, Beichman-Denisov [7] obtained stability for long rectangular patches.…”

Section: Resultsmentioning

confidence: 91%

Stability of radially symmetric, monotone vorticities of 2D Euler equations

Choi¹,

Lim²

2021

Preprint

View full text Add to dashboard Cite

We consider the incompressible Euler equations in R 2 when the initial vorticity is bounded, radially symmetric and non-increasing in the radial direction. Such a radial distribution is stationary, and we show that the monotonicity produces stability in some weighted norm related to the angular impulse. For instance, it covers the cases of circular vortex patches and Gaussian distributions. Our stability does not depend on L ∞ -bound or support size of perturbations. The proof is based on the fact that such a radial monotone distribution minimizes the impulse of functions having the same level set measure.

show abstract

Axi‐symmetrization near Point Vortex Solutions for the 2D Euler Equation

Cited by 43 publications

References 40 publications

Traveling waves near Couette flow for the 2D Euler equation

Traveling waves near Couette flow for the 2D Euler equation

Liouville chains: new hybrid vortex equilibria of the two-dimensional Euler equation

Stability of radially symmetric, monotone vorticities of 2D Euler equations

Contact Info

Product

Resources

About