<p>In this paper, we consider the finite-time blowup of hollow vortices. These are solutions of the two-dimensional Euler equations for which the fluid domain is the complement of finitely many Jordan curves <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Gamma _1, \ldots , \Gamma _M$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi mathvariant="normal">Γ</mi> <mi>M</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, and such that the flow is irrotational and incompressible, but with a nonzero circulation around each boundary component. The region bounded by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Gamma _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Γ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> is a “vortex core”, modeled as a bubble of ideal gas: the pressure is constant in space and inversely proportional to the area of the vortex. This can be thought of as the isobaric approximation assuming isothermal flow. Our results come in two parts. There exist explicit families of purely circular rotating and imploding hollow vortices. Implosion means more precisely that the vortex core shrinks to the origin in finite time, while the absolute value of the pressure simultaneously diverges to infinity. We prove that for any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$m \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, there exist near-circular <i>m</i>-fold symmetric rotating hollow vortices. By contrast, for all <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$m \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, the purely circular imploding vortices are locally unique among all collapsing vortices with uniform velocity at infinity. The second part concerns configurations of multiple hollow vortices. The existence of configurations of point vortices that collapse into a common point in finite time is classical. We prove that generically, these can be desingularized to yield families of hollow vortex configurations exhibiting self-similar finite-time implosion. Specific examples of an imploding trio and quartet of hollow vortices are given.</p>

Finite-time self-similar implosion of hollow vortices

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2026年7月23日

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Robin Ming Chen,
Samuel Walsh,
Miles H. Wheeler

摘要

In this paper, we consider the finite-time blowup of hollow vortices. These are solutions of the two-dimensional Euler equations for which the fluid domain is the complement of finitely many Jordan curves $\Gamma _1, \ldots , \Gamma _M$ $Γ_{1}, \dots, Γ_{M}$ , and such that the flow is irrotational and incompressible, but with a nonzero circulation around each boundary component. The region bounded by $\Gamma _k$ $Γ_{k}$ is a “vortex core”, modeled as a bubble of ideal gas: the pressure is constant in space and inversely proportional to the area of the vortex. This can be thought of as the isobaric approximation assuming isothermal flow. Our results come in two parts. There exist explicit families of purely circular rotating and imploding hollow vortices. Implosion means more precisely that the vortex core shrinks to the origin in finite time, while the absolute value of the pressure simultaneously diverges to infinity. We prove that for any $m \ge 2$ $m \geq 2$ , there exist near-circular m-fold symmetric rotating hollow vortices. By contrast, for all $m \ge 2$ $m \geq 2$ , the purely circular imploding vortices are locally unique among all collapsing vortices with uniform velocity at infinity. The second part concerns configurations of multiple hollow vortices. The existence of configurations of point vortices that collapse into a common point in finite time is classical. We prove that generically, these can be desingularized to yield families of hollow vortex configurations exhibiting self-similar finite-time implosion. Specific examples of an imploding trio and quartet of hollow vortices are given.