How does rough axisymmetric Euler flow behave across regularity breakdown?

A kinetic limit that outlives the Euler reference flow

Classical derivations of the incompressible Euler limit from kinetic theory assume a regular macroscopic reference flow: spatial Lipschitz control drives the weak–strong stability argument, and Hilbert-expansion methods demand still higher regularity. The paper asks whether that kinetic-to-Euler passage can survive a finite-time loss of regularity of the limiting three-dimensional Euler flow.

The question is not artificial. Within the same axisymmetric no-swirl symmetry class, sufficiently regular flows are globally regular, yet finite-time loss of regularity is known: Elgindi established singularity formation in the C1,α class, Elgindi–Ghoul–Masmoudi obtained finite-energy singular solutions, and Córdoba–Martínez-Zoroa–Zheng constructed a distinct finite-energy, non-self-similar blow-up mechanism. The regular Euler reference required by classical arguments can therefore genuinely break down in finite time.

A second obstruction appears at rough regularity. Conservation of kinetic energy is itself nontrivial for rough Euler solutions, and the relevant failure here is not small-scale anomalous dissipation but large-scale loss of energy: even under strong local compactness, an order-one amount of energy may escape to spatial infinity. A singularity-crossing kinetic limit must control both the loss of the regular reference flow and the possible escape of energy at infinity.

Local compactness leaves the energy tail uncontrolled

For an axisymmetric velocity without swirl, write u = ur er + uz ez, ω = ωϑ eϑ, and ϖ = ωϑ/r. The relative vorticity ϖ is then transported, ∂tϖ + u·∇ϖ = 0, with u = Bϖ = ∇×(−∆)^(−1)(rϖ eϑ). Axisymmetry restores a two-dimensional transport structure in the meridional variables (r, z), but not a two-dimensional energy theory: the Biot–Savart reconstruction and the L2 energy ½∫|u(t,x)|²dx retain three-dimensional geometry.

That mismatch is largely invisible to local compactness but decisive for global L2 compactness. For relative vorticity in L1∩Lp with p>1, natural approximations already yield strong local L2 compactness of the velocity, and global weak Euler solutions exist in the same range. But an order-one amount of energy may escape to arbitrarily large cylindrical radius while remaining invisible on every fixed region, so un → u strongly in L2loc does not by itself imply ‖un(t)‖² → ‖u(t)‖². The missing issue is compactness at spatial infinity.

A classical remedy uses a sign condition together with finite impulse: the signed impulse (1/2π)∫r²ϖ(t,x)dx is conserved by the axisymmetric Euler evolution, and when ϖ has one sign this invariant becomes a coercive radial moment that prevents a fixed amount of vorticity from moving to arbitrarily large cylindrical radius. The paper instead asks whether the transported L1∩Lp structure alone can enforce global L2 compactness, without a sign condition or an a priori radial moment.

Above 4/3, energy cannot escape to infinity

The answer is moment-free for every p≥4/3. Theorem A fixes T<∞ and 1<p<∞, takes regular axisymmetric no-swirl approximate Euler flows (un, ϖn, fn, πn) with supn‖un‖L∞(0,T;L2)<∞ and fn→0 in L1(0,T;L2), and assumes ϖn→ϖ in C([0,T];L1∩Lp) with un(0)→u0 in L2. Then un→u in C([0,T];L2(R3)) and ‖u(t)‖²=‖u0‖² for 0≤t≤T, provided either p≥4/3 or supn sup0≤t≤T M2(ϖn(t))<∞, where M2(ϖ) := (1/2π)∫r²|ϖ(x)|dx is the absolute impulse.

The two alternatives correspond to different mechanisms. For p≥4/3, energy escape is ruled out directly, without first confining the vorticity. A radial localization of the energy identity shows that energy reaching large cylindrical radius must cross a shell r∼R; for 4/3≤s≤3/2 the nonlinear flux obeys the sharp shell estimate (1/R)∫CR|u|³dx ≲ R^(−δs)‖u‖2^(αs) ‖r^(−1)∇u‖Ls(CR′)^(γs) + R^(−5/2)‖u‖2³ with δs=(3s−4)/(4(s−1)), where CR⊂CR′ are comparable cylindrical shells r∼R and αs, γs>0 are interpolation exponents. The radial gain is positive precisely for s>4/3.

At the endpoint s=4/3 the power gain vanishes, but strong CtLx^(4/3) convergence supplies uniform far-field tightness of r^(−1)∇u; together with the corresponding pressure localization this rules out energy escape to r=∞ and upgrades interior compactness to global CtLx² compactness. The paper stresses that 4/3 is a large-scale no-escape threshold at spatial infinity, not a small-scale Onsager regularity threshold. Below 4/3, Proposition 3 constructs, for every 1<p<4/3, smooth incompressible passive transports with vanishing L1∩Lp scalar norm and vanishing initial Biot–Savart energy but order-one terminal Biot–Savart energy — a matching obstruction, though not a self-consistent Euler counterexample.

Sign-changing vorticity shifts the threshold to 5/3

Below 4/3, global energy compactness is recovered from control of the absolute impulse M2(ϖ). Theorem B fixes T<∞ and 1<p<∞ and takes a regular finite-energy axisymmetric no-swirl Euler flow on [0,T] with ϖ∈L∞(0,T;L1∩Lp) and M2(ϖ(0))<∞. If p≥5/3, then (1+M2(ϖ(t)))^(1/4) ≤ (1+M2(ϖ(0)))^(1/4) + Ct for 0≤t≤T, without a sign condition. If 1<p<5/3, the same estimate holds provided min{‖ϖ+(0)‖Lp♯, ‖ϖ−(0)‖Lp♯}<∞, where p♯ = 4p/(3p−1).

The mechanism is a different geometry. With X(ϖ,η) := ∫(Bη)r rϖ dx, the meridional kernel is skew, X(ϖ,η) = −X(η,ϖ), so same-sign interactions cancel exactly and, for a smooth Euler flow, d/dt M2(ϖ(t)) = (2/π)X(ϖ−,ϖ+). The key mixed estimate is |X(ϖ,η)| ≲p ‖ϖ‖1‖η‖1 + ‖ϖ‖p^(1/4) M2(ϖ)^(3/4) ‖η‖p♯. The far interaction is controlled at the L1 level; near the diagonal, redistributing the radial weight reduces the singular interaction to a two-dimensional Hardy–Littlewood–Sobolev pairing on the meridional half-plane, and the balance forces p♯.

With only the ambient L1∩Lp control, the estimate closes exactly when p♯≤p, whose fixed point is p=5/3. Below 5/3, one-sided Lp♯ control supplies the missing integrability. The paper reports that both thresholds are sharp: the exponent p♯ is optimal for the mixed criterion, since for p≤q<p♯ uniform L1∩Lp, absolute-impulse, and one-sided Lq bounds do not yield a class-uniform bound on the instantaneous growth of M2; and the sign-free threshold 5/3 is dynamically sharp, since below it uniformly controlled smooth sign-changing families can develop arbitrarily large absolute impulse at a fixed positive time. Sign-free propagation of the first radial moment holds for p≥3/2, with 3/2 sharp for class-uniform instantaneous control.

A consequence the paper highlights: for one-sign finite-impulse vorticity the additional condition is automatic, so the resulting global energy compactness holds throughout the full range p>1. This extends the energy-conservation range of Nobili–Seis from p>3/2 to every p>1 for one-sign vorticity.

Kinetic solutions realize prescribed rough Euler data

The paper considers ε∂tFε + ξ·∇xFε = (1/κε)Q(Fε,Fε) with κ=κ(ε)→0, where Q is either the Coulomb Landau operator or a non-cutoff Boltzmann operator with soft potentials in the parameter range considered. Writing Fε = Mε + Gε with Mε the exact local Maxwellian, and splitting the relative vorticity as ϖε = ϖεA + ϖεB, the transported component satisfies ∂tϖεA + vε·∇xϖεA = 0, so its Lq norms and sign structure are transported exactly. The acoustic, baroclinic, and microscopic-stress effects are carried by ϖεB, whose velocity uεB tends to zero on every fixed time interval.

Theorem C takes 1<p<∞ and axisymmetric ϖ0∈L1(R3)∩Lp(R3), s0∈L2(R3) with u0 := Bϖ0 ∈ L2(R3); if 1<p<4/3 it additionally assumes M2(ϖ0)<∞ and min{‖(ϖ0)+‖Lp♯, ‖(ϖ0)−‖Lp♯}<∞. For either collision operator there exists a family of smooth nonnegative axisymmetric no-swirl kinetic initial data Fε0 whose strong solutions Fε exist uniquely on [0,Tε] with Tε→∞, and whose macroscopic initial data satisfy ϖε0→ϖ0 in L1∩Lp, vε0→u0 in L2, sε0→s0 in L2. No vanishing acoustic preparation is required.

Along the whole kinetic lifespan the microscopic relaxation Gε→0 holds. For every finite T>0 and every sequence εj↓0, one may extract a subsequence and a renormalized finite-energy axisymmetric no-swirl Euler pair (u,ϖ) on [0,T] with initial data (u0,ϖ0) such that (ϖεjA, vεj, sεj) → (ϖ,u,s) strongly in C([0,T]; (L1∩Lp)×L2×L2). The limiting thermodynamic mode solves ∂ts + u·∇xs = 0 with s(0)=s0, and the limiting Euler energy is conserved: ‖u(t)‖²=‖u0‖² for 0≤t≤T.

The kinetic estimates therefore produce the approximate Euler sequence required by Theorem A rather than taking one as an external input. For p≥4/3 the no-escape result applies directly; below 4/3 the same splitting lets the absolute-impulse bounds be propagated while the uεB errors are controlled, and finite-impulse Biot–Savart stability supplies the global L2 upgrade.

Convergence is rigid up to the singular time

Theorem D assumes the setting of Theorem C and supposes that the prescribed Euler initial datum generates a regular finite-energy axisymmetric no-swirl solution on [0,T*) that loses regularity at some 0<T*<∞. Then, for each fixed admissible kinetic realization, the projected velocity and transported relative vorticity converge, without subsequence extraction, to the regular Euler solution on [0,T*) and reach the same strong traces at T*. From every sequence εj↓0 one may extract a subsequence converging on every finite time interval to a global energy-conserving Euler solution that agrees with the regular solution for t<T* and attains the same trace at T*. No uniqueness beyond T* is asserted.

The paper also reports quantitative features of this passage. In shrinking time windows around the breakdown, the kinetic family approaches the same Euler trace from both sides of T*, a control obtained from explicit fixed-scale estimates including spatial Lipschitz bounds for the solenoidal velocity with a quantitatively controlled blow-up rate. In the p=2 branch, Proposition 39 shows that the limiting velocity satisfies a local-in-space almost-Lipschitz temporal modulus. Corollary 40 applies the results to three finite-time singularity constructions.

Before T*, weak–strong uniqueness identifies every cluster path with the regular Euler solution; compactness then fixes the common trace at T*, without invoking any uniqueness theory beyond the singular time. The paper's summary of the architecture is that the kinetic approximation is rigid up to the singular time, while compactness survives across and beyond it.

What the thresholds do and do not cover

The results are stated for three-dimensional axisymmetric Euler flow without swirl, with relative vorticity in L1∩Lp and 1<p<∞. The moment-free p≥4/3 statement of Theorem A requires no sign condition and no spatial-moment assumption; the below-4/3 statement requires uniform control of the absolute impulse M2(ϖn(t)) over the approximating family. The sharp absolute-impulse dynamics of Theorem B requires ϖ∈L∞(0,T;L1∩Lp) and M2(ϖ(0))<∞, and for 1<p<5/3 additionally a one-sided Lp♯ bound on the initial positive or negative part.

The kinetic side is restricted to the Coulomb Landau equation and a class of non-cutoff soft-potential Boltzmann equations in the parameter range considered. The kinetic solutions are constructed on lifespans Tε→∞, and the Euler limits are obtained along subsequences on every fixed time interval; the energy-conserving continuation past T* is asserted for subsequences, and no uniqueness beyond T* is claimed. The below-4/3 obstruction in Proposition 3 is a passive-transport construction, not a self-consistent Euler counterexample, so it delimits the direct shell mechanism rather than the Euler equations themselves.

The paper also distinguishes the nature of its thresholds: 4/3 is a large-scale no-escape threshold at spatial infinity rather than a small-scale Onsager regularity threshold, while 5/3 and p♯ are sharp thresholds for propagation of the absolute impulse under sign-changing vorticity. These are different questions, and the paper does not treat them as interchangeable.

The threshold that matters most is the sign condition

The paper's most consequential split is not between kinetic and fluid language but between sign-changing and one-sign vorticity. For p≥4/3, global energy compactness is moment-free and sign-free. For 1<p<4/3, it depends on the absolute impulse, and the sharp absolute-impulse dynamics then bifurcates: sign-free propagation holds for p≥5/3, while 1<p<5/3 requires one-sided Lp♯ integrability with p♯=4p/(3p−1). For one-sign finite-impulse vorticity the extra condition is automatic, and global energy compactness holds throughout p>1.

That is the comparison a reader should carry away: the same transported L1∩Lp structure yields global L2 compactness for free above 4/3, but below 4/3 the decisive variable is whether the vorticity changes sign and, if it does, whether one sign component has the sharp Lp♯ integrability. The kinetic construction inherits exactly this split, since below 4/3 Theorem C imposes the absolute-impulse and one-sided Lp♯ assumptions on ϖ0 while the p≥4/3 branch needs neither.

The thresholds 4/3, 3/2, and 5/3 are therefore not interchangeable cutoffs. 4/3 governs direct no-escape of energy at spatial infinity; 3/2 governs sign-free propagation of the first radial moment; 5/3 governs sign-free propagation of the absolute impulse. The paper reports each as sharp in its own sense, and the practical reading is that a rough axisymmetric Euler or kinetic-to-Euler problem should first be classified by which of these quantities it actually controls.

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