Gradient growth and relaxation to jump profiles for 3-fold symmetric scale-invariant Euler flows

Abstract: We consider the zero-homogeneous reduction of the two-dimensional Euler equation in the 3-fold symmetric case $m=3$, which is left unsolved in the $m\geq4$ relaxation theory of Said, Elgindi, and Murray \cite{EMS}. We prove that every nonconstant $W<sup>{1,p}$ solution satisfies $|g_θ(t)|<em>{L<sup>p}\to\infty$ as $t\to\pm\infty$ for $1&lt;p\leq\infty$. For $p=1$, the total variation is conserved, but the $L\log L$ modular tends to infinity whenever it is initially finite. If $D</em>θg_0$ is a summable sum of non-atomic one-sign components and atoms, every profile in the two omega-limit sets is a jump profile, and each half-orbit approaches its omega-limit set in $W<sup>{α,r}$ for $αr&lt;1$. The structural assumption on $D_θg_0$ is automatic for $C<sup>1$ data. Moreover, every weak $L<sup>2$ infinite-time limit generates a complete $L<sup>2$-precompact orbit.

• The paper proves that every nonconstant complete solution with initial data in W^{1,p}, 1<p≤∞, develops unbounded gradient norms as t→±∞, while an L log L refinement applies at p=1.
• It replaces the sign-definite kernel method with four-point cross-ratio monotonicity, showing that omega-limit profiles have purely atomic derivatives and attracting orbits are precompact in W^{α,r} for α<1/r.
• The classification of few-jump limits shows that two-jump profiles are traveling waves, while genuine three-jump trajectories are heteroclinic and have no periodic orbits, completing the m=3 relaxation theory.

The setting and the problem

The paper studies the zero-homogeneous reduction of the two-dimensional incompressible Euler equation. Under the ansatz $\omega(t,r,\theta)=g(t,\theta)$ with stream function $\psi=r^2G(t,\theta)$, the vorticity formulation reduces on the fundamental circle $\mathbb T_L=\mathbb R/L\mathbb Z$, $L=2\pi/3$, to

$$g_t+2Gg_\theta=0, \qquad (\partial_{\theta\theta}+4)G=g.$$

Because $A=\partial_{\theta\theta}+4$ is invertible on $2\pi/3$-periodic functions (frequencies are multiples of 3), the reduced dynamics are uniquely defined for $m=3$ without normalization. This case was left open by Said, Elgindi, and Murray (2608.16755), whose relaxation theory for $m\geq4$ relies on positivity of the symmetrized kernel of $(\partial_{\theta\theta}+4)^{-1}$. For $\psi=r^2G(t,\theta)$0 that kernel,

$\psi=r^2G(t,\theta)$1

changes sign, so their argument breaks down precisely at the lowest nonresonant symmetry. The paper resolves this remaining case and thereby answers Problem 5 in the survey of Drivas and Elgindi affirmatively for all nonconstant smooth data in the well-posed scale-invariant range $\psi=r^2G(t,\theta)$2.

Two-sided gradient growth

The first main result states that every nonconstant complete solution with $\psi=r^2G(t,\theta)$3, $\psi=r^2G(t,\theta)$4, satisfies

$\psi=r^2G(t,\theta)$5

This is a full two-sided limit, not a limsup statement, which distinguishes it from the generic Sobolev-growth results of Alazard–Said that hold only along dense $\psi=r^2G(t,\theta)$6 sets of data. At the endpoint $\psi=r^2G(t,\theta)$7, the total variation $\psi=r^2G(t,\theta)$8 is conserved — so the restriction $\psi=r^2G(t,\theta)$9 is sharp — but if the $\mathbb T_L=\mathbb R/L\mathbb Z$0 modular of $\mathbb T_L=\mathbb R/L\mathbb Z$1 is finite, then

$\mathbb T_L=\mathbb R/L\mathbb Z$2

The proof rests on an entropy identity: for $\mathbb T_L=\mathbb R/L\mathbb Z$3 with $\mathbb T_L=\mathbb R/L\mathbb Z$4, one computes

$\mathbb T_L=\mathbb R/L\mathbb Z$5

where the constant 10 comes from the spectral gap $\mathbb T_L=\mathbb R/L\mathbb Z$6 on $\mathbb T_L=\mathbb R/L\mathbb Z$7. If a derivative norm stayed bounded along a sequence tending to infinity, compactness of time windows around that sequence would force $\mathbb T_L=\mathbb R/L\mathbb Z$8 bounded below on infinitely many disjoint intervals, contradicting integrability of $\mathbb T_L=\mathbb R/L\mathbb Z$9. Backward time follows from the involution $L=2\pi/3$0. Via the annular norm identities for the lifted zero-homogeneous vorticity, this yields infinite-time gradient growth of the full planar Euler solution in every annulus.

Relaxation to jump profiles via cross-ratio monotonicity

The relaxation theory replaces kernel positivity with a four-point cross-ratio monotonicity. In the projective coordinate $L=2\pi/3$1, the third derivative of the rescaled velocity satisfies

$L=2\pi/3$2

so its sign matches the sign of $L=2\pi/3$3 on any arc where the latter has one sign. For four ordered material particles, the logarithmic derivative of the cross-ratio

$L=2\pi/3$4

admits a Peano-kernel representation against $L=2\pi/3$5 with a strictly positive kernel; the four-point functional annihilates all polynomials of degree at most two, which removes the ambiguity that sign-indefiniteness of the kernel creates. Consequently $L=2\pi/3$6 is nondecreasing on positive-sign arcs and $L=2\pi/3$7 is nondecreasing on negative-sign arcs.

These monotonicities force atomicity of limiting components. Along any sequence $L=2\pi/3$8, each transported positive component $L=2\pi/3$9 converges to a single atom of mass $$g_t+2Gg_\theta=0, \qquad (\partial_{\theta\theta}+4)G=g.$$0, while each negative component converges to a measure supported on at most two points; the roles exchange under time reversal. Combined with summability of component masses, this excludes both absolutely continuous and Cantor parts from every omega-limit profile: every $$g_t+2Gg_\theta=0, \qquad (\partial_{\theta\theta}+4)G=g.$$1 has purely atomic distributional derivative. Each half-orbit is attracted to its omega-limit set in $$g_t+2Gg_\theta=0, \qquad (\partial_{\theta\theta}+4)G=g.$$2 for all $$g_t+2Gg_\theta=0, \qquad (\partial_{\theta\theta}+4)G=g.$$3, $$g_t+2Gg_\theta=0, \qquad (\partial_{\theta\theta}+4)G=g.$$4 — a range shown to be sharp when limits carry jumps. Moreover, every weak $$g_t+2Gg_\theta=0, \qquad (\partial_{\theta\theta}+4)G=g.$$5 infinite-time limit generates a complete solution whose orbit is precompact in these spaces, which is exactly the compact-orbit conclusion requested in Problem 5 of (2608.16755).

For $$g_t+2Gg_\theta=0, \qquad (\partial_{\theta\theta}+4)G=g.$$6 data the structural decomposition of $$g_t+2Gg_\theta=0, \qquad (\partial_{\theta\theta}+4)G=g.$$7 into signed non-atomic components is automatic, so Corollary 1.5 applies to all nonconstant smooth profiles. For Morse data with $$g_t+2Gg_\theta=0, \qquad (\partial_{\theta\theta}+4)G=g.$$8 maxima and minima, the jump count of limiting profiles is explicitly $$g_t+2Gg_\theta=0, \qquad (\partial_{\theta\theta}+4)G=g.$$9; since nonconstant Morse functions form an open dense subset of $A=\partial_{\theta\theta}+4$0, relaxation to finitely many jumps holds generically among smooth profiles. The results also lift to genuine whole-plane Euler solutions: for compactly supported, 3-fold symmetric vorticity smooth off the origin with nonconstant radial trace, the rescaled gradient norm satisfies $A=\partial_{\theta\theta}+4$1 as $A=\partial_{\theta\theta}+4$2.

Classification of few-jump orbits

For step-function data the velocity is explicit through the Green kernel primitive $A=\partial_{\theta\theta}+4$3, and the jump positions obey a finite ODE. Two-jump profiles have constant separation and hence are travelling waves. Three-jump profiles satisfy, on the simplex of interval lengths $A=\partial_{\theta\theta}+4$4,

$A=\partial_{\theta\theta}+4$5

so after the time change $A=\partial_{\theta\theta}+4$6 the lengths move linearly along an open segment. Since $A=\partial_{\theta\theta}+4$7 vanishes linearly at edges and quadratically at vertices of the simplex, physical time diverges at both endpoints: every genuine three-jump orbit is heteroclinic modulo rotations between travelling waves or constants, and no three-jump orbit is periodic. Every limit of time translates of a solution starting from data with one positive and one negative component lies in one of these three classes.

Limitations and open questions

Several qualifications bear directly on the strength of the results. The growth theorem requires $A=\partial_{\theta\theta}+4$8; at $A=\partial_{\theta\theta}+4$9 the total variation is conserved and only the conditional $2\pi/3$0 divergence is available, so no unconditional growth statement exists at the Lebesgue endpoint. The relaxation conclusions depend on the summability condition on component masses; it is automatic for $2\pi/3$1 and finite-decomposition BV data, but the paper does not claim attraction for arbitrary BV data outside this class. The omega-limit sets are characterized as sets of jump profiles with prescribed component masses, not as single profiles — whether individual orbits converge to a specific jump profile, rather than to the full invariant set, remains open. Likewise, the paper does not determine the jump locations $2\pi/3$2 in the limiting profiles, nor the rate at which the $2\pi/3$3 distance decays. Finally, the analysis is confined to the $2\pi/3$4 scale-invariant reduction; extending the cross-ratio mechanism beyond symmetric reductions, or obtaining quantitative growth rates comparable to the double-exponential bounds known in other geometries, is not addressed.

Conclusion

The paper completes the long-time theory of scale-invariant 2D Euler flows at the last unresolved symmetry. It proves unconditional two-sided infinite-time gradient growth for every nonconstant $2\pi/3$5 profile ($2\pi/3$6), with an $2\pi/3$7 refinement at the endpoint, and establishes relaxation of half-orbits to compact invariant sets of jump profiles with purely atomic derivatives, together with precompactness of complete orbits through weak infinite-time limits. The key technical innovation — replacing the sign-definite kernel used for $2\pi/3$8 by four-point cross-ratio monotonicity in projective coordinates — resolves the sign-indefiniteness specific to $2\pi/3$9 and, combined with the earlier theory, settles the growth and compact-orbit questions for all nonconstant smooth data in the well-posed range $m=3$0.