Blockchain Papers

Vittorio Baroncini, Juan Carlos Cantero, Claudia García, Zineb Hassainia · 5 authors

We construct new families of uniformly rotating vortex-patch solutions of the two-dimensional incompressible Euler equations consisting of a simply connected outer patch and multiple interior interfaces, which can be interpreted geometrically as holes. More precisely, each solution consists of a single outer vortex patch enclosing $\mathbf m\geq2$ identical, highly concentrated inner components arranged at the vertices of a regular $\mathbf m$-gon; the entire configuration rotates rigidly in the clockwise direction. As the concentration parameter tends to zero, the inner components shrink and collapse simultaneously toward the origin, while the outer boundary converges to the unit circle. The corresponding vorticities converge, in the sense of measures, to a Rankine vortex supplemented by a point vortex of circulation $-\mathbf m$ at its center. The proof is based on a contour-dynamics formulation, a symmetry reduction to two nonlinear boundary equations, and a suitable singular rescaling. We then apply an implicit function theorem with a continuous parameter in symmetry-adapted Hölder spaces. To the best of our knowledge, this is the first analytical construction of a desingularization regime in which several concentrated inner components are contained in a common outer patch and collapse simultaneously toward its center.

Emmanuel Trélat, Xingwu Zeng, Can Zhang

We establish global exponential turnpike properties for quadratic optimal tracking problems governed by the one-dimensional viscous Burgers equation with localized internal control. For every initial datum, finite-horizon optimal solutions approach the unique optimal periodic regime when the periodic tracking target is sufficiently small; the zero-target case yields a global steady turnpike at the origin, with no smallness assumption on the initial datum. To our knowledge, these are the first global exponential turnpike results for the viscous Burgers equation. The proof combines a local exponential turnpike, obtained through strict convexity and periodic Riccati theory, with a parabolic dissipation argument that provides an absorbing time independent of the horizon.

Paolo Antonelli, Pierangelo Marcati, Laura V. Spinolo

We study the zero-dispersion limit for a class of Korteweg--de Vries (KdV)-type initial-boundary value problems on the half-line, with Dirichlet boundary conditions assigned at \(x=0\). We focus on the outflow regime, where the solution of the limiting scalar conservation law does not attain the boundary condition imposed on the dispersive problem. We construct a boundary layer profile, depending on the fast variable, which is uniquely determined, through the associated stationary third-order boundary layer equation, by the mismatch between the boundary conditions, and by the exponential decay at infinity in the fast variable. Our main result shows that, under suitable regularity and compatibility assumptions on the data, the dispersive solution is well approximated by a WKB expansion given by the sum of the smooth solution of the conservation law and the boundary layer profile. In particular, we establish stability of the boundary layer profile by proving quantitative estimates for the remainder term in a weighted energy norm, and show that it converges to $0$ in $H^1$, uniformly in time and up to the lifespan of the smooth solution of the conservation law. The proof is based on the analysis of a linearized energy functional and does not rely on complete integrability or inverse scattering techniques. It applies to general fluxes and requires no smallness assumption on the amplitude of the boundary layer. To the best of our knowledge, this is the first stability result for boundary layers of KdV-type equation on the half line.

Andrea Cavazzini

This paper proves unconditional global regularity with quantitative exponential decay for the three-dimensional incompressible Navier–Stokes equations on the periodic box, restricted to velocity fields invariant under the Kida–Pelz symmetry group of order 48, for viscosities above an explicit threshold. The entire proof reduces, through a chain of six independently verifiable steps, to a single integer arithmetic fact: 20,625 < 31,104. This is, to the author's knowledge, the first time a Navier–Stokes regularity result has been distilled to a verifiable inequality between two five-digit integers, with every constant computed exactly and no numerical approximation entering the argument at any stage. The Kida–Pelz flow and why it matters. The Kida–Pelz initial datum, introduced by Kida (1985) and studied extensively by Pelz (2001), has occupied a special place in the blow-up literature for decades. It was originally proposed as a candidate for finite-time singularity formation precisely because its high octahedral symmetry concentrates vortex stretching into a small number of interacting structures, producing some of the most intense enstrophy growth observed in direct numerical simulations. The fact that the same symmetry that was expected to promote blow-up turns out to prevent it is itself a significant finding: it demonstrates that vortex stretching intensity and blow-up potential are fundamentally different quantities, a distinction that is often blurred in heuristic discussions of turbulence. The proof architecture. The argument has a deliberately transparent two-layer structure separating analysis from arithmetic, so that each layer can be checked independently by specialists in different fields. The analytic layer establishes three quantitative inputs. First, a spectral gap: representation-theoretic analysis of the octahedral group acting on Fourier space shows that the first two shells of the Laplacian spectrum are entirely killed by symmetry, tripling the effective Poincaré constant from 1 to at least 3. This means the KP symmetry forces vorticity to reside at higher wavenumbers where viscous dissipation is three times stronger than for generic flows. Second, a triadic density bound: the GKP equivariance constrains the Fourier support so severely that the number of resonant triads contributing to the nonlinear stretching term is reduced by a factor involving the group order, yielding a geometric density bound of at most 2. Third, an exact initial enstrophy: the KP datum is monochromatic, with all Fourier modes sitting at a single shell of squared wavenumber 11, giving the exact rational value 33/4 for the initial enstrophy. No floating-point computation, truncation, or discretisation enters this calculation. These three inputs feed into a Bernoulli differential inequality for the enstrophy whose separatrix is computed in closed form. The arithmetic layer then verifies that the initial enstrophy lies below this separatrix, which reduces to the integer comparison 33 times 625 equals 20,625, which is less than 31,104 equals 4 times 7,776. The safety margin is 50.8 percent, meaning the result would survive even if the analytic constants were degraded by up to 20 percent. Bounded enstrophy then gives global existence via the standard H1-continuation criterion, and exponential decay in all Sobolev norms follows by a Gronwall bootstrap. Beyond the core result. The paper establishes several extensions that go beyond mere regularity. Exponential decay is proved not only for the enstrophy but for all Sobolev norms simultaneously, with explicit prefactors and rates. The decay is shown to hold in all Lebesgue spaces from L2 to L-infinity and for all derivative orders, meaning that every physically measurable quantity associated with the flow decays exponentially. The pressure decays at double the velocity rate, a consequence of the quadratic structure of the pressure Poisson equation. Time analyticity is established for all positive times, meaning the solution extends to a holomorphic function in a strip around the real time axis. A shell-by-shell energy spectrum analysis shows that higher Fourier shells decay faster, with rates proportional to the squared wavenumber — a quantitative version of the physical intuition that small-scale structures are dissipated more rapidly. A Reynolds number characterisation shows that the Bernoulli closure holds if and only if the KP Reynolds number is below approximately 235, giving a concrete, physically interpretable criterion. The stability result deserves particular emphasis: global regularity is shown to persist under small perturbations that need not respect the KP symmetry. This means the result is not a fragile artifact of exact symmetry but a robust property of a neighbourhood in function space around the KP datum. The self-frustration connection. This paper is designed as a companion to the author's monograph "Self-Frustration of Vortex Stretching and the Architecture of the Navier–Stokes Blow-Up Barrier" (Cavazzini, 2026), which identifies a twelve-link chain of structural obstructions to finite-time blow-up for general three-dimensional Navier–Stokes. Three of those twelve links have concrete, quantitative realisations in the Kida–Pelz class. The enhanced spectral gap is a realisation of Link 5 (the spectral gap threshold that governs alignment stability). The triadic density reduction is a realisation of Link 3 (the oscillation bound that controls the pressure Hessian for tube-like vorticity). The identically vanishing helicity — proved here as a consequence of the parity inversion in the octahedral group — is a realisation of Link 6 (the gap–alignment complementarity), because it eliminates the eigenframe injection mechanism entirely: with zero helicity budget, the pressure Hessian cannot rotate the strain eigenframe to sustain the dangerous compressive component identified in the companion paper as the sole variable separating regularity-compatible from blow-up-compatible configurations. When all three mechanisms act simultaneously, as enforced by the octahedral symmetry, the self-frustration chain that remains open for general flows closes completely and unconditionally. The arithmetic inequality 20,625 < 31,104 is the quantitative expression of this closure. This provides the first concrete validation of the self-frustration framework as a genuine regularity tool rather than merely a classification scheme: the structural architecture described in the companion monograph is not an abstract taxonomy but a machinery that produces theorems when supplied with sufficient quantitative input. The minimal symmetry result strengthens this connection further: the octahedral group of order 48 is proved to be the smallest finite subgroup of O(3) for which the Bernoulli method closes. This characterises the precise boundary between symmetry groups where the self-frustration mechanisms are strong enough to guarantee regularity and those where they are not, providing a sharp answer to the question of how much geometric structure is needed to resolve the regularity problem within this framework. Context within the broader landscape. The Navier–Stokes regularity problem has a long history of partial results exploiting symmetry, from the classical two-dimensional theory (where regularity is known unconditionally due to the absence of vortex stretching) to various axisymmetric and helical reductions. The present work differs from these in a fundamental respect: the Kida–Pelz flow is fully three-dimensional with active, sustained vortex stretching — the mechanism responsible for the supercritical character of the equations is present and operative, not eliminated by dimensional reduction. What the symmetry does is not remove the stretching but quantitatively constrain it, tilting the balance between stretching and dissipation in favour of dissipation by a computable margin. This is a qualitatively different use of symmetry from the classical approach, and it suggests that the boundary between regularity and potential blow-up may be more accessible than previously thought — not through eliminating the dangerous mechanism, but through measuring and constraining it. The paper also contributes to the broader programme of understanding which structural properties of the Navier–Stokes equations are responsible for regularity. The identification of three independent mechanisms (enhanced dissipation, triadic depletion, topological obstruction) that close the regularity chain when acting together, combined with the companion monograph's demonstration that these same mechanisms are present but quantitatively insufficient for general flows, suggests a precise research programme: strengthen the quantitative estimates on any one of the three mechanisms sufficiently to close the chain without symmetry. The open problems listed in the paper — removal of the viscosity threshold, full Gevrey bootstrap, exact spectral gap computation — are formulated with this programme in mind. Methodological note. Every result in the paper carries an explicit epistemic label. All constants are computed exactly as rational numbers or algebraic expressions. The paper makes no claim regarding the Clay Millennium Prize and explicitly discusses the four gaps separating the present result from the Prize requirements: symmetry restriction, periodic domain, viscosity threshold, and partial Gevrey bootstrap. The distance from each gap to a resolution is assessed individually, with the viscosity threshold identified as an artifact of the Bernoulli method rather than a physical phase transition. MSC 2020 Classification: 35Q30 (primary — Navier–Stokes equations); 76D03 (existence, uniqueness, and regularity for incompressible viscous fluids); 42B25 (maximal functions and Littlewood–Paley theory); 20C15 (ordinary representations and characters of finite groups); 35B65 (smoothness and regularity of solutions to PDE

Mitsuo Higaki, Jiajiang Liao, Franck Sueur

The question at stake in Lagrangian controllability is whether one can move a patch of fluid particles to a target location by means of remote action in a given time interval. In the last two decades, positive results have been obtained both for the incompressible Euler and Navier-Stokes equations. However, for the latter, the case where the fluid is contained within domains bounded by solid boundaries with the no-slip condition has not been addressed, with respect to the difficulty caused by viscous boundary layers. In this paper, we investigate the Lagrangian controllability of viscous incompressible fluid in perforated domains for which the fraction of volume occupied by the holes is sufficiently small. Moreover, we quantitatively distinguish situations depending on the parameters for holes (diameter and distance) and for fluid (size of the initial data). Our approach relies on recent results on homogenization for evolutionary problems and on weak-strong stability estimates in measure of flows, alongside classical results on Runge-type approximations for elliptic equations and on Cauchy-Kowalevsky-type theorems for equations with analytic coefficients. Here, homogenization refers to the vanishing viscosity limit outside a porous medium, where (after scaling in time) the Navier-Stokes equations are homogenized to the Euler or Darcy equations. Indeed, in the proof, we act on the Navier-Stokes equations by strong and fast forcing to leverage inviscid approximations, which is a standard technique in the theory of controllability.

Mihaela Ifrim, Ben Pineau, Daniel Tataru, Mitchell A. Taylor

In this article, we provide a definitive well-posedness theory for the free boundary problem in incompressible magnetohyrodynamics. Despite the clear physical interest in this system and the remarkable progress in the study of the free boundary Euler equations in recent decades, the low regularity well-posedness of the free boundary MHD equations has remained completely open. This is due, in large part, to the highly nonlinear wave-type coupling between the velocity, magnetic field and free boundary, which has forced previous works to impose restrictive geometric constraints on the data. To address this problem, we introduce a novel Eulerian approach and an entirely new functional setting, which better captures the wave equation structure of the MHD equations and permits a complete Hadamard well-posedness theory in low-regularity Sobolev spaces. In particular, we give the first proofs of existence, uniqueness and continuous dependence on the data at the sharp $s&gt;\frac{d}{2}+1$ Sobolev regularity, in addition to a blowup criterion for smooth solutions at the same low regularity scale. Moreover, we provide a completely new method for constructing smooth solutions which, to our knowledge, gives the first proof of existence (at any regularity) in our new functional setting. All of our results hold in arbitrary dimensions and in general, not necessarily simply connected, domains. By taking the magnetic field to be zero, they also recover the corresponding sharp well-posedness theorems for the free boundary Euler equations. The methodology and tools that we employ here can likely be fruitfully implemented in other free boundary models.

Matthieu Hillairet, Julien Lequeurre, Céline Grandmont

We study an unsteady nonlinear fluid–structure interaction problem. We consider a Newtonian incompressible two-dimensional flow described by the Navier–Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear wave equation or a linear beam equation. The fluid and the structure systems are coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action-reaction principle. Considering three different structure models, we prove existence of a unique local-in-time strong solution, for which there is no gap between the regularity of the initial data and the regularity of the solution enabling to obtain a blow up alternative. In the case of a damped beam this is an alternative proof (and a generalization to non zero initial displacement) of the result that can be found in [20]. In the case of the wave equation or a beam equation with inertia of rotation, this is, to our knowledge the first result of existence of strong solutions for which no viscosity is added. The key points consist in studying the coupled system without decoupling the fluid from the structure and to use the fluid dissipation to control, in appropriate function spaces, the structure velocity.

D. H. Sattinger

Abstract We consider a container of fluid with a rod inserted in the centre. As the rod rotates the surface of the fluid forms a curved surface whose shape correctly balances the forces of gravity, internal stress, atmospheric pressure, and surface tension. The surface of the fluid is depressed in the neighbourhood of the rod in the case of a Newtonian fluid, but the fluid may climb along the rod in the case of non-Newtonian fluids. In recent work by D. D. Joseph and R. S. Fosdick this free surface problem has been treated quantitatively by virtue of a formal perturbation series in which the solution is developed in powers of the angular velocity of the rod. The purpose of the present paper is to give a rigorous proof of convergence in the case of a Newtonian fluid. The method here may possibly be applied to other free surface problems - for example, surface waves of a viscous fluid. The convergence proof provides, to our knowledge, the first rigorous existence theorem for a free surface problem in the theory of viscous fluids. The proof also raises a novel problem in the theory of elliptic systems of partial differential equations. By means of the implicit function theorem, the question of convergence is reduced to that of obtaining a priori estimates for an elliptic boundary value problem. That problem is formulated on a domain with a ridge where the fluid surface meets the rod. In addition, the type of boundary conditions prescribed differ on the free surface and on the rod. In general, the solution to such a mixed boundary value problem is not smooth at the ridge, even if the boundary data is smooth. The solution will be smooth, however, if the boundary data satisfy certain consistency conditions which make it compatible with the given set of partial differential equations. The consistency conditions in question here are a set of linear relations between various derivatives of the boundary and inhomogeneous data. It is shown that if one makes the assumption that the wetting angle of the surface is zero, the consistency conditions are invariant under the full nonlinear equations of the free surface problem. This makes it possible to consider the original problem on a smaller function space - namely the subclass of functions satisfying the appropriate consistency conditions - and in this subclass one can apply the implicit function theorem and obtain the required a priori estimates. The solution thus obtained is regular up to the ridge.