Vortex Motion of the Euler and Lake Equations
We start by surveying the planar point vortex motion of the Euler equations in the whole plane, half-plane and quadrant. Then, we go on to prove the non-collision property of the 2-vortex system by using the explicit form of orbits of the 2-vortex system in the half-plane. We also prove that the N -...
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| Published in | Journal of nonlinear science Vol. 31; no. 3 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.06.2021
Springer Nature B.V |
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| Abstract | We start by surveying the planar point vortex motion of the Euler equations in the whole plane, half-plane and quadrant. Then, we go on to prove the non-collision property of the 2-vortex system by using the explicit form of orbits of the 2-vortex system in the half-plane. We also prove that the
N
-vortex system in the half-plane is nonintegrable for
N
>
2
, which was suggested previously by numerical experiments without rigorous proof. The skew-mean-curvature (or binormal) flow in
R
n
,
n
⩾
3
with certain symmetry can be regarded as point vortex motion of these 2D lake equations. We compare point vortex motions of the Euler and lake equations. Interesting similarities between the point vortex motion in the half-plane, quadrant and the binormal motion of coaxial vortex rings, sphere product membranes are addressed. We also raise some open questions in the paper. |
|---|---|
| AbstractList | We start by surveying the planar point vortex motion of the Euler equations in the whole plane, half-plane and quadrant. Then, we go on to prove the non-collision property of the 2-vortex system by using the explicit form of orbits of the 2-vortex system in the half-plane. We also prove that the
N
-vortex system in the half-plane is nonintegrable for
N
>
2
, which was suggested previously by numerical experiments without rigorous proof. The skew-mean-curvature (or binormal) flow in
R
n
,
n
⩾
3
with certain symmetry can be regarded as point vortex motion of these 2D lake equations. We compare point vortex motions of the Euler and lake equations. Interesting similarities between the point vortex motion in the half-plane, quadrant and the binormal motion of coaxial vortex rings, sphere product membranes are addressed. We also raise some open questions in the paper. We start by surveying the planar point vortex motion of the Euler equations in the whole plane, half-plane and quadrant. Then, we go on to prove the non-collision property of the 2-vortex system by using the explicit form of orbits of the 2-vortex system in the half-plane. We also prove that the N-vortex system in the half-plane is nonintegrable for N>2, which was suggested previously by numerical experiments without rigorous proof. The skew-mean-curvature (or binormal) flow in Rn,n⩾3 with certain symmetry can be regarded as point vortex motion of these 2D lake equations. We compare point vortex motions of the Euler and lake equations. Interesting similarities between the point vortex motion in the half-plane, quadrant and the binormal motion of coaxial vortex rings, sphere product membranes are addressed. We also raise some open questions in the paper. |
| ArticleNumber | 48 |
| Author | Yang, Cheng |
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| Cites_doi | 10.1098/rsta.1893.0020 10.1090/S0002-9947-1987-0891628-1 10.1063/1.3673800 10.1063/1.2425103 10.1088/0305-4470/28/8/013 10.1063/1.862605 10.4153/CJM-1949-022-2 10.1006/jcph.1995.1023 10.1007/978-1-4684-9290-3 10.1007/b97593 10.1063/1.869160 10.1063/1.166210 10.1007/BF03018608 10.1016/0167-2789(82)90067-7 10.1007/s00162-009-0148-z 10.1512/iumj.1996.45.1199 10.17323/1609-4514-2012-12-2-413-434 10.1088/1361-6544/abe1d1 10.1007/s40598-020-00162-8 10.1073/pnas.27.12.570 10.1098/rstl.1885.0015 |
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| Snippet | We start by surveying the planar point vortex motion of the Euler equations in the whole plane, half-plane and quadrant. Then, we go on to prove the... |
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| SubjectTerms | Analysis Classical Mechanics Computational fluid dynamics Economic Theory/Quantitative Economics/Mathematical Methods Euler-Lagrange equation Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Theoretical Vortex rings Vortices |
| Title | Vortex Motion of the Euler and Lake Equations |
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