Vishik Vortices: Instability in 2D Fluid Dynamics

• Vishik vortices are specially engineered, radially symmetric unstable structures in 2D fluid dynamics, characterized by a linearized operator with an eigenvalue having a positive real part.
• They are employed in self-similar constructions to convert linear instability into nonuniqueness for the forced 2D Euler equation and to generate multiple unstable directions in Navier–Stokes setups.
• Their applications include establishing thresholds in vorticity classes, contrasting compactly supported versus power-law profiles, and deriving lower bounds on attractor dimensions in dissipative flows.

Searching arXiv for the cited papers on Vishik vortices and related work. Search 1: arXiv (Albritton et al., 2021). Search 2: arXiv (Castro et al., 2024). Search 3: arXiv (Kostianko et al., 7 Jul 2025). Vishik vortices are specially engineered unstable vortical backgrounds that arise in the analysis of the two-dimensional Euler equation and, in later work, as localized unstable building blocks for two-dimensional Navier–Stokes dynamics. In the sense developed after M. M. Vishik, they are radial steady vortices, or asymptotically steady profiles in similarity variables, whose linearized operator possesses an eigenvalue with positive real part; this linear instability is then converted into either nonuniqueness for forced 2D Euler in the vorticity class $L^1\cap L^p$, $2<p<\infty$, or into many unstable directions for multi-vortex steady states in 2D Navier–Stokes (Albritton et al., 2021, Castro et al., 2024, Kostianko et al., 7 Jul 2025). In the wider literature, “Vishik vortices” now refers broadly to these specially crafted unstable steady or self-similar flows rather than to a single explicit formula (Albritton et al., 2021).

1. Terminological scope and historical setting

The modern usage of the term is anchored in Vishik’s nonuniqueness mechanism for the forced 2D Euler equation in vorticity form,

$$\partial_t \omega + v\cdot\nabla\omega = f,\qquad v = \nabla^\perp \Delta^{-1}\omega,$$

posed on $\mathbb{R}^2$, with solutions sought in the natural vorticity–energy class

$$\omega\in L_t^\infty(L^1\cap L^p),\qquad v\in L_t^\infty L^2,\qquad 2<p<\infty.$$

The expository work "Instability and nonuniqueness for the $2d$ Euler equations in vorticity form, after M. Vishik" presents Vishik’s theorem in a form asserting that, for any $p\in(2,\infty)$, there exist data $(\omega_0,f)$ with $\omega_0\in L^1\cap L^p$ and even $\omega_0\equiv 0$, such that the Euler–vorticity system admits uncountably many weak solutions on $2<p<\infty$0, all lying in $2<p<\infty$1 and smooth for every $2<p<\infty$2 (Albritton et al., 2021). The 2024 paper "A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation" reproves Vishik’s forced result in a simpler form, stating that for any $2<p<\infty$3 there exists a force $2<p<\infty$4 such that the system with initial data $2<p<\infty$5 admits at least two different solutions in the same class (Castro et al., 2024).

Within this framework, a Vishik vortex is the unstable core of the construction. The role of the vortex is not merely illustrative: it supplies a genuine unstable eigenmode of the linearized dynamics, and that eigenmode is then embedded into a self-similar forced evolution whose backward asymptotics as $2<p<\infty$6 produce multiple forward solutions with the same initial datum (Albritton et al., 2021). This places Vishik vortices at the intersection of hydrodynamic stability theory, semigroup methods, and low-regularity well-posedness.

The historical significance of the construction is tied to the Yudovich class. Yudovich’s theorem gives uniqueness for

$2<p<\infty$7

because the associated velocity is log-Lipschitz. Vishik’s theorem shows that this uniqueness threshold is sharp in the scale $2<p<\infty$8: once $2<p<\infty$9 is weakened to finite $$\partial_t \omega + v\cdot\nabla\omega = f,\qquad v = \nabla^\perp \Delta^{-1}\omega,$$0, nonuniqueness may occur, even with smooth compactly supported forcing for $$\partial_t \omega + v\cdot\nabla\omega = f,\qquad v = \nabla^\perp \Delta^{-1}\omega,$$1 (Albritton et al., 2021).

2. Radial steady vortices and the instability mechanism

In the Euler setting, a vortex is a radial steady vorticity profile

$$\partial_t \omega + v\cdot\nabla\omega = f,\qquad v = \nabla^\perp \Delta^{-1}\omega,$$2

with corresponding azimuthal velocity

$$\partial_t \omega + v\cdot\nabla\omega = f,\qquad v = \nabla^\perp \Delta^{-1}\omega,$$3

Because $$\partial_t \omega + v\cdot\nabla\omega = f,\qquad v = \nabla^\perp \Delta^{-1}\omega,$$4, such flows are steady. The defining feature of a Vishik vortex is that the linearized Euler operator around this radial state has an eigenvalue with positive real part, so that perturbations grow exponentially under the linearized evolution (Albritton et al., 2021).

The expository notes formulate the linearization around $$\partial_t \omega + v\cdot\nabla\omega = f,\qquad v = \nabla^\perp \Delta^{-1}\omega,$$5, $$\partial_t \omega + v\cdot\nabla\omega = f,\qquad v = \nabla^\perp \Delta^{-1}\omega,$$6 as

$$\partial_t \omega + v\cdot\nabla\omega = f,\qquad v = \nabla^\perp \Delta^{-1}\omega,$$7

After decomposition into angular Fourier modes and the logarithmic radial variable $$\partial_t \omega + v\cdot\nabla\omega = f,\qquad v = \nabla^\perp \Delta^{-1}\omega,$$8, the eigenvalue problem reduces to Rayleigh’s stability equation,

$$\partial_t \omega + v\cdot\nabla\omega = f,\qquad v = \nabla^\perp \Delta^{-1}\omega,$$9

with

$\mathbb{R}^2$0

and $\mathbb{R}^2$1. Modes with $\mathbb{R}^2$2 correspond to eigenvalues $\mathbb{R}^2$3 with $\mathbb{R}^2$4, hence to instability (Albritton et al., 2021).

A central profile in the notes is a radial background

$\mathbb{R}^2$5

where $\mathbb{R}^2$6 for all finite $\mathbb{R}^2$7, satisfies

$\mathbb{R}^2$8

and is quadratic near the origin,

$\mathbb{R}^2$9

for $$\omega\in L_t^\infty(L^1\cap L^p),\qquad v\in L_t^\infty L^2,\qquad 2<p<\infty.$$0 small. The associated velocity is

$$\omega\in L_t^\infty(L^1\cap L^p),\qquad v\in L_t^\infty L^2,\qquad 2<p<\infty.$$1

In the form used for nonuniqueness, this background is coupled to a self-similar Ansatz

$$\omega\in L_t^\infty(L^1\cap L^p),\qquad v\in L_t^\infty L^2,\qquad 2<p<\infty.$$2

The unstable eigenpair $$\omega\in L_t^\infty(L^1\cap L^p),\qquad v\in L_t^\infty L^2,\qquad 2<p<\infty.$$3 for the self-similar linearized operator $$\omega\in L_t^\infty(L^1\cap L^p),\qquad v\in L_t^\infty L^2,\qquad 2<p<\infty.$$4 satisfies

$$\omega\in L_t^\infty(L^1\cap L^p),\qquad v\in L_t^\infty L^2,\qquad 2<p<\infty.$$5

with $$\omega\in L_t^\infty(L^1\cap L^p),\qquad v\in L_t^\infty L^2,\qquad 2<p<\infty.$$6, $$\omega\in L_t^\infty(L^1\cap L^p),\qquad v\in L_t^\infty L^2,\qquad 2<p<\infty.$$7, and $$\omega\in L_t^\infty(L^1\cap L^p),\qquad v\in L_t^\infty L^2,\qquad 2<p<\infty.$$8 supported on a single angular Fourier mode with rapid decay at infinity (Albritton et al., 2021).

The 2024 simplification recasts the same instability notion in a more elementary Eulerian form. There, the linearized operator around a radial vortex $$\omega\in L_t^\infty(L^1\cap L^p),\qquad v\in L_t^\infty L^2,\qquad 2<p<\infty.$$9 is written as

$2d$0

and in each angular sector $2d$1 the eigenvalue problem reduces to a Rayleigh-type integral equation. A vortex is called unstable if there exists $2d$2, a complex $2d$3 with $2d$4, and a nontrivial mode solving that equation; equivalently, there exists $2d$5 such that

$2d$6

This formulation isolates the spectral content of a Vishik vortex without yet invoking self-similar variables (Castro et al., 2024).

3. Self-similar embedding and nonuniqueness for forced 2D Euler

The conversion of linear instability into nonuniqueness is carried out in similarity variables. In the notes, one sets

$2d$7

so that the vorticity equation becomes

$2d$8

A background solution is constructed explicitly by

$2d$9

with $p\in(2,\infty)$0 a radial cut-off, and

$p\in(2,\infty)$1

The forcing is then defined by

$p\in(2,\infty)$2

so that $p\in(2,\infty)$3 solves Euler with forcing $p\in(2,\infty)$4 and initial data $p\in(2,\infty)$5, with the construction allowing $p\in(2,\infty)$6 (Albritton et al., 2021).

In similarity coordinates, the background trajectory satisfies

$p\in(2,\infty)$7

on compact $p\in(2,\infty)$8-sets. The profile $p\in(2,\infty)$9 therefore plays the role of an equilibrium at $(\omega_0,f)$0, and the unstable eigenvalue of $(\omega_0,f)$1 defines an unstable manifold in the similarity flow (Albritton et al., 2021).

The perturbed solutions are sought in the form

$(\omega_0,f)$2

where $(\omega_0,f)$3 is the background remainder and $(\omega_0,f)$4 is a nonlinear correction. The key estimate gives

$(\omega_0,f)$5

for $(\omega_0,f)$6, uniformly in $(\omega_0,f)$7, and this is controlled through the semigroup estimate

$(\omega_0,f)$8

In physical variables, one obtains solutions $(\omega_0,f)$9 started at times $\omega_0\in L^1\cap L^p$0, where the initial vorticity lies in $\omega_0\in L^1\cap L^p$1. Yudovich’s theorem then gives a unique global solution on $\omega_0\in L^1\cap L^p$2, and compactness yields a subsequential limit $\omega_0\in L^1\cap L^p$3 on $\omega_0\in L^1\cap L^p$4 (Albritton et al., 2021).

The asymptotics near $\omega_0\in L^1\cap L^p$5 distinguish the solutions: $\omega_0\in L^1\cap L^p$6 in $\omega_0\in L^1\cap L^p$7. Hence, for $\omega_0\in L^1\cap L^p$8,

$\omega_0\in L^1\cap L^p$9

while both converge weakly to the same initial vorticity $\omega_0\equiv 0$0. This is the precise mechanism by which instability of the Vishik vortex in similarity variables is converted into nonuniqueness at $\omega_0\equiv 0$1 (Albritton et al., 2021).

The 2024 proof uses a related self-similar strategy but with parameters $\omega_0\equiv 0$2,

$\omega_0\equiv 0$3

leading to

$\omega_0\equiv 0$4

The self-similar linearization is

$\omega_0\equiv 0$5

and Vishik’s observation, sharpened there, is that as $\omega_0\equiv 0$6, unstable spectrum of the Eulerian operator $\omega_0\equiv 0$7 persists in $\omega_0\equiv 0$8. For small positive $\omega_0\equiv 0$9, one gets an eigenvalue $2<p<\infty$00 with $2<p<\infty$01, an eigenfunction $2<p<\infty$02, and a linear mode $2<p<\infty$03 that decays as $2<p<\infty$04, enabling the same backward-to-forward nonuniqueness mechanism (Castro et al., 2024).

4. Compactly supported constructions and the simplified 2024 proof

A major development is the replacement of Vishik’s original slowly decaying power-law vortex by a compactly supported unstable vortex. The 2024 paper constructs such a vortex in two steps: first a piecewise constant unstable vortex, then a smooth regularization obtained through a fixed point argument (Castro et al., 2024).

The piecewise constant profile is

$2<p<\infty$05

with $2<p<\infty$06 and $2<p<\infty$07 chosen so that the total vorticity has zero mean: $2<p<\infty$08 This yields a tangential compactly supported velocity,

$2<p<\infty$09

and the derivative of vorticity is the measure

$2<p<\infty$10

Inserting the modal ansatz into the Rayleigh equation reduces the spectral problem to a $2<p<\infty$11 matrix eigenvalue problem

$2<p<\infty$12

Its characteristic polynomial is computed explicitly: $2<p<\infty$13 For each $2<p<\infty$14, there exists $2<p<\infty$15 such that the roots are non-real with $2<p<\infty$16; in particular, for $2<p<\infty$17, one may take $2<p<\infty$18. Therefore

$2<p<\infty$19

and the piecewise constant vortex is spectrally unstable (Castro et al., 2024).

The second step regularizes this object. The paper mollifies the velocity rather than the vorticity, smoothing only near the jump radii $2<p<\infty$20 and $2<p<\infty$21, and then defines the regularized vorticity by

$2<p<\infty$22

The resulting $2<p<\infty$23 is smooth and compactly supported. Near each interface one introduces rescaled variables

$2<p<\infty$24

and parametrizes the corrected eigenpair by

$2<p<\infty$25

The rescaled Rayleigh equation becomes an operator equation on $2<p<\infty$26,

$2<p<\infty$27

with solvability enforced by orthogonality to the kernel of $2<p<\infty$28. A fixed-point map on $2<p<\infty$29 then yields, for sufficiently small $2<p<\infty$30, a regularized eigenpair with

$2<p<\infty$31

Thus there exists a smooth, compactly supported, zero-mean vortex $2<p<\infty$32 that is Eulerian unstable (Castro et al., 2024).

This construction changes the technical profile of the subject. The 2024 paper contrasts it with Vishik’s original power-law vortex $2<p<\infty$33, emphasizing compact support, explicit linear algebra in the two-step patch, and decoupling of decay at infinity from the self-similar scaling parameters. A plausible implication is that the essential content of a Vishik vortex is spectral rather than tied to a specific algebraic tail.

5. Functional thresholds, dynamical interpretation, and common misconceptions

The decisive functional threshold is between $2<p<\infty$34 and $2<p<\infty$35, $2<p<\infty$36. In the Yudovich class, the velocity satisfies the log-Lipschitz bound

$2<p<\infty$37

which allows uniqueness of characteristics and thus uniqueness of weak solutions. The Vishik construction stays below that threshold: the self-similar solutions lie in $2<p<\infty$38 for every $2<p<\infty$39, but lack a uniform $2<p<\infty$40 control of vorticity down to $2<p<\infty$41, so the Yudovich argument does not apply uniformly in time (Albritton et al., 2021).

The notes interpret the construction as a dynamical system in $2<p<\infty$42: $2<p<\infty$43 The steady profile $2<p<\infty$44 has an unstable eigenvalue $2<p<\infty$45, $2<p<\infty$46, so it behaves as a hyperbolic equilibrium of the similarity flow, with an unstable manifold consisting of trajectories

$2<p<\infty$47

The nonuniqueness at $2<p<\infty$48 is then interpreted as nonuniqueness of backward continuation to the equilibrium at $2<p<\infty$49 (Albritton et al., 2021).

Several misconceptions are explicitly excluded by the sources. First, Vishik vortices are not generic coherent structures selected by a physical variational principle; the 2024 paper describes them as “engineered” profiles and stresses that they should be viewed primarily as mathematical counterexamples highlighting instability and ill-posedness (Castro et al., 2024). Second, the forced results do not settle the unforced case in the same manner. The notes state that Vishik’s original construction in the unforced case $2<p<\infty$50 is more delicate and is not carried out there; the exposition remains in the forced setting (Albritton et al., 2021). Third, “Vishik vortex” is not confined to one formula. The notes explicitly state that the wider literature uses the term for specially crafted unstable steady or self-similar flows underlying nonuniqueness constructions (Albritton et al., 2021).

The same expository work also records a technical issue in the original literature: it points out and fixes a gap in Vishik’s original argument about the simplicity of the unstable eigenvalue, noting that Vishik later provided a fix in a short note from 2006. This does not alter the core instability mechanism, but it clarifies the spectral underpinnings of the construction (Albritton et al., 2021).

6. Later Navier–Stokes developments: multi-vortices and attractor dimensions

The 2025 paper "Multi-vortices and lower bounds on the attractor dimensions for 2D Navier--Stokes equations" imports Vishik vortices into a different problem: the construction of many unstable directions for forced 2D Navier–Stokes. There, a single Vishik vortex is taken as a smooth compactly supported stationary solution $2<p<\infty$51 of

$2<p<\infty$52

with $2<p<\infty$53, and with the property that the linearized Navier–Stokes operator has an unstable eigenvalue $2<p<\infty$54 (Kostianko et al., 7 Jul 2025).

The shifted linearized operator is written as

$2<p<\infty$55

with direct and adjoint eigenfunctions $2<p<\infty$56 satisfying

$2<p<\infty$57

These eigenfunctions are smooth and decay algebraically,

$2<p<\infty$58

with the same decay for all derivatives. Cut-off eigenfunctions $2<p<\infty$59 and $2<p<\infty$60 satisfy approximate eigenvalue equations with errors of order $2<p<\infty$61, which makes them suitable local spectral coordinates for superpositions of far-separated vortices (Kostianko et al., 7 Jul 2025).

A multi-vortex profile is then defined by summing translates,

$2<p<\infty$62

for a set of centers $2<p<\infty$63 separated by a large distance $2<p<\infty$64. Because the supports are disjoint,

$2<p<\infty$65

so $2<p<\infty$66 is an exact stationary solution. The linearization around $2<p<\infty$67 admits an approximate spectral projector

$2<p<\infty$68

and the auxiliary problem

$2<p<\infty$69

is uniformly invertible for large $2<p<\infty$70 (Kostianko et al., 7 Jul 2025).

The resulting Lyapunov–Schmidt decomposition produces an invariant finite-dimensional nearly neutral subspace of dimension $2<p<\infty$71. Each localized Vishik vortex contributes at least one unstable direction, so a configuration of $2<p<\infty$72 well-separated vortices yields instability index at least $2<p<\infty$73 (Kostianko et al., 7 Jul 2025). This is then used to derive lower bounds on attractor dimensions. In a bounded smooth simply connected domain, the paper proves that for sufficiently small $2<p<\infty$74 there exists a forcing $2<p<\infty$75 such that

$2<p<\infty$76

where

$2<p<\infty$77

For damped Navier–Stokes on $2<p<\infty$78, it proves a sharp lower bound

$2<p<\infty$79

with

$2<p<\infty$80

The paper explicitly proposes multi-vortices consisting of well-separated Vishik vortices as the analogue of Kolmogorov flows for non-periodic hydrodynamic settings (Kostianko et al., 7 Jul 2025).

In this later usage, the term “Vishik vortex” no longer refers only to a device for Euler nonuniqueness. It denotes a localized spectrally unstable flow module that can be translated, cut off, superposed, and inserted into dissipative PDE arguments. This suggests a broader paradigm already visible in the Euler literature: instability localized in space can serve as an atomic mechanism for both ill-posedness and quantitative complexity bounds in hydrodynamics.