REGULARIZATION OF PLANAR VORTICES FOR THE INCOMPRESSIBLE FLOW

摘要/Abstract

摘要： In this paper, we continue to construct stationary classical solutions for the incompressible planar flows approximating singular stationary solutions of this problem. This procedure is carried out by constructing solutions for the following elliptic equations

where 0 < p < 1, Ω ? R² is a bounded simply-connected smooth domain, κ_i (i=1, …, k) is prescribed positive constant. The result we prove is that for any given non-degenerate critical point x₀=(x_0,1, …, x_0,k) of the Kirchhoff-Routh function defined on Ω^k corresponding to (κ₁, …, κ_k), there exists a stationary classical solution approximating stationary k points vortex solution. Moreover, as λ → +∞, the vorticity set {y:u_λ > κ_j} ∩ B_δ(x_0,j) shrinks to {x_0,j}, and the local vorticity strength near each x_0,j approaches κ_j, j=1, …, k. This result makes the study of the above problem with p ≥ 0 complete since the cases p > 1, p=1, p=0 have already been studied in [11, 12] and [13] respectively.

关键词: regularization, planar vortices, vorticity sets, reduction

Abstract: In this paper, we continue to construct stationary classical solutions for the incompressible planar flows approximating singular stationary solutions of this problem. This procedure is carried out by constructing solutions for the following elliptic equations

where 0 < p < 1, Ω ? R² is a bounded simply-connected smooth domain, κ_i (i=1, …, k) is prescribed positive constant. The result we prove is that for any given non-degenerate critical point x₀=(x_0,1, …, x_0,k) of the Kirchhoff-Routh function defined on Ω^k corresponding to (κ₁, …, κ_k), there exists a stationary classical solution approximating stationary k points vortex solution. Moreover, as λ → +∞, the vorticity set {y:u_λ > κ_j} ∩ B_δ(x_0,j) shrinks to {x_0,j}, and the local vorticity strength near each x_0,j approaches κ_j, j=1, …, k. This result makes the study of the above problem with p ≥ 0 complete since the cases p > 1, p=1, p=0 have already been studied in [11, 12] and [13] respectively.

Key words: regularization, planar vortices, vorticity sets, reduction

曹道珉, 彭双阶, 严树森. REGULARIZATION OF PLANAR VORTICES FOR THE INCOMPRESSIBLE FLOW[J]. 数学物理学报(英文版), 2018, 38(5): 1443-1467.

Daomin CAO, Shuangjie PENG, Shuangjie PENG. REGULARIZATION OF PLANAR VORTICES FOR THE INCOMPRESSIBLE FLOW[J]. Acta Mathematica Scientia, 2018, 38(5): 1443-1467.

参考文献

[1] Ambrosetti A, Struwe M. Existence of steady vortex rings in an ideal fluid. Arch Rational Mech Anal, 1989, 108:97-109
[2] Amick C J, Fraenkel L E. The uniqueness of a family of steady vortex rings. Arch Rational Mech Anal, 1988, 100:207-241
[3] Amick C J, Turner R E L. A global branch of steady vortex rings. J Reine Angew Math, 1988, 384:1-23
[4] Arnold V I, Khesin B A. Topological Methods in Hydrodynamics. Applied Mathematical Sciences, Vol 125. New York:Springer, 1998
[5] Badiani T V. Existence of steady symmetric vortex pairs on a planar domain with an obstacle. Math Proc Cambridge Philos Soc, 1998, 123:365-384
[6] Bartsch T, Pistoia A, Weth T. N-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-Poisson and the Lane-Emden-Fowler equations. Comm Math Phys, 2010, 297:653-686
[7] Berger M S, Fraenkel L E. Nonlinear desingularization in certain free-boundary problems. Comm Math Phys, 1980, 77:149-172
[8] Burton G R. Vortex rings in a cylinder and rearrangements. J Diff Equ, 1987, 70:333-348
[9] Burton G R. Rearrangements of functions, saddle points and uncountable families of steady configurations for a vortex. Acta Math, 1989, 163:291-309
[10] Caffarelli L, Friedman A. Asymptotic estimates for the plasma problem. Duke Math J, 1980, 47:705-742
[11] Cao D, Liu Z, Wei J. Regularization of point vortices for the Euler equation in dimension two. Arch Ration Mech Anal, 2014, 212:179-217
[12] Cao D, Peng S, Yan S. Multiplicity of solutions for the plasma problem in two dimensions. Adv Math, 2010, 225:2741-2785
[13] Cao D, Peng S, Yan S. Planar vortex patch problem in incompressible steady flow. Adv Math, 2015, 270:263-301
[14] Chang K C. The obstacle problem and partial differential equations with discontinuous nonlinearities. Comm Pure Appl Math, 1980, 33:117-146
[15] Chang K C. Variational methods for nondifferentiable functionals and their applications to partial differential equations. J Math Anal Appl, 1981, 80:102-129
[16] Dancer E N, Yan S. The Lazer-McKenna conjecture and a free boundary problem in two dimensions. J London Math Soc, 2008, 78:639-662
[17] Fraenkel L E, Berger M S. A global theory of steady vortex rings in an ideal fluid. Acta Math, 1974, 132:13-51
[18] Fraenkel L E, Berger M S. A global theory of steady vortex rings in an ideal fluid. Bull Amer Math Soc, 1973, 79:806-810
[19] Flucher M, Wei J. Asymptotic shape and location of small cores in elliptic free-boundary problems. Math Z, 1998, 228:683-703
[20] Friedman A, Turkington B. Vortex rings:existence and asymptotic estimates. Trans Amer Math Soc, 1981, 268:1-37
[21] Li G, Yan S, Yang J. An elliptic problem related to planar vortex pairs. SIAM J Math Anal, 2005, 36:1444-1460
[22] Li Y, Peng S. Multiple solutions for an elliptic problem related to vortex pairs. J Diff Equ, 2011, 250:3448-3472
[23] Lin C. C. On the motion of vortices in two dimension-I. Existence of the Kirchhoff-Routh function. Proc Nat Acad Sci, 1941, 27:570-575
[24] Marchioro C, Pulvirenti M. Euler evolution for singular initial data and vortex theory. Comm Math Phys, 1983, 91:563-572
[25] Ni W-M. On the existence of global vortex rings. J Anal Math, 1980, 37:208-247
[26] Norbury J. Steady planar vortex pairs in an ideal fluid. Comm Pure Appl Math, 1975, 28:679-700
[27] Norbury J. A steady vortex ring close to Hill's spherical vortex. Proc Cambridge Philos Soc, 1972, 72:253-284
[28] Smets D, Van Schaftingen J. Desingulariation of vortices for the Euler equation. Arch Rational Mech Anal, 2010, 198:869-925
[29] Turkington B. On steady vortex flow in two dimensions, I. Comm Partial Diff Eqt, 1983, 8:999-1030
[30] Turkington B. On steady vortex flow in two dimensions, Ⅱ. Comm Partial Diff Equ, 1983, 8:1031-1071
[31] Yang J. Existence and asymptotic behavior in planar vortex theory. Math Models Methods Appl Sci, 1991, 1:461-475