Blockchain Papers

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.

Bradford White

This preprint presents Invariant Ontodynamics (IOD), a structural field theory derived from a single minimal geometric primitive with zero continuously adjustable dimensionless fit parameters. To our knowledge, no prior framework derives both the Schrödinger equation and the Einstein field equations from a single uniqueness-selected geometric primitive without continuously adjustable fit parameters. The theory derives quantum dynamics, relativistic field structure, fermion spin-½, general relativity, and gauge symmetry as theorems rather than assumptions. A universal structural law — that the effective complexity of any system is a linear function of its structural curvature k, with a universal slope and fixed point derived from the same primitive — is empirically confirmed at R² = 0.978 across 15 pre-selected independent domains spanning 19 orders of magnitude in physical scale, under a pre-registration protocol with SHA-256 cryptographic locks. New results in this version include: A zero-free-parameter prediction of the Higgs boson mass, m_H = 125.33 GeV (0.06% from the observed 125.25 GeV), via a one-loop renormalization group trajectory anchored at a structurally derived UV scale A complete CPL dark-energy equation-of-state parameter pair (w₀ = −0.858, w_a = −0.411), both pre-registered before DESI DR3 Exact zero-free-parameter black hole thermodynamics: Schwarzschild radius, Hawking temperature, and surface gravity all derived from the primitive alone, with a falsifiable 29% Hawking temperature shift relative to the GR prediction A structural information measure (Heun log-coefficient) connecting the near-horizon field structure to the Brownian fixed-point evaporation endpoint, with exact Page curve endpoint M_Page = M₀/√2 Previously confirmed predictions — solar mixing angle (0.05σ), reactor angle (0.39σ), tau lepton mass (0.91σ), baryon asymmetry (−1.0σ), dark matter ratio (0.2%), inflationary spectral index (1.0σ) — remain confirmed. Three explicit tensions are stated without omission: atmospheric mixing angle (2.2σ, DUNE 2030 decisive), leptonic CP violation (J_CP = 0, DUNE 2030 decisive), and dark energy w₀ (0.4σ from DESI DR2 best fit, DESI DR3 decisive). Priority and legal status: This document is a public technical summary and priority disclosure. Full derivations, exact primitive specification, all coefficient values, and complete proofs are in US Provisional Patent No. 63/963,472 (filed January 2026) and Addenda 1–15 (through April 2026). The non-provisional application will be filed by January 2027.

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.

Giao Ky Duong, Nejla Nouaili, Hatem Zaag, Université Sorbonne Paris Nord, LAGA, CNRS(UMR7539), F-93430, Villetaneuse, France

In this paper, we revisit the proof of the existence of a solution to the semilinear heat equation in one space dimension with a flat blow-up profile, already proved by Bricmont and Kupainen together with Herrero and Velázquez. Though our approach relies on the well-celebrated method, based on the reduction of the problem to a finite-dimensional one, then the use of a topological 'shooting method' to solve the latter, the novelty of our approach lays in the use of a modulation technique to control the projection of the zero eigenmode arising in the problem. Up to our knowledge, this is the first time where modulation is used with this kind of profiles. We do hope that this simplifies the argument.

Yusheng Wei, Zongli Lin

No abstract is available for this record.

Yusheng Wei, Zongli Lin

In this paper, we propose a delay independent control scheme that regulates to zero the state and the control input of a linear input delayed system whose open loop poles are at the origin or in the open left-half plane. Two main features of our control scheme are its nondistributed nature in the sense that only the current state is used in the feedback and its delay independence in the sense that no knowledge of the delay, neither its exact value nor its upper bound, is required. The main ingredients of our control scheme and the regulation proof include a design of the delay independent truncated predictor feedback law with a time-varying feedback parameter, Lyapunov function based adaptation of the time-varying parameter, a mechanism for switching between two update laws of the time-varying parameter, and the partial differential equation based analysis for delayed systems.

Yusheng Wei, Zongli Lin

In this paper, we propose a delay independent control scheme that regulates to zero the state and the control input of a linear input delayed system whose open loop poles are at the origin. Two main features of our control scheme are its non-distributed nature in the sense that only the current state is used in the feedback, and its delay independence in the sense that no knowledge of the delay is required. The main ingredients of our control scheme and the regulation proof include a design of the delay independent truncated predictor feedback law with a time-varying feedback parameter, a Lyapunov function based adaptation of the time-varying parameter, a mechanism for switching between two update laws of the time-varying parameter, and the partial differential equation based analysis for delayed systems.

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.

Samantha Xu

Consider the radial nonlinear wave equation $-\partial_t^2 u + Δu = u^3$, $u :\mathbb{R}_t \times \mathbb{R}_x^3 \to \mathbb{R}$, $u(t,x) = u(t,|x|)$. In this paper, we construct a Gibbs measure for this system and prove its invariance under the flow of the NLW. In particular, we are in the infinite volume setting. For the finite volume analogue, specifically on the unit ball with zero boundary values, an invariant Gibbs measure was constructed by Burq, Tvetkov, and de Suzzoni as a Borel measure on super-critical Sobolev spaces. In this paper, we advocate that the finite volume Gibbs measure be considered on a space of weighted Hölder continuous functions. The measure is supported on this space and the NLW is locally well-posed there, a counter-point to the Sobolev super-criticality noted by Burq and Tzvetkov. Furthermore, the flow of the NLW leaves this measure invariant. We use a multi-time Feynman--Kac formula to construct the infinite volume limit measure by computing the asymptotics of the fundamental solution of an appropriate parabolic PDE. We use finite speed of propagation and results from descriptive set theory to establish invariance of the infinite volume measure. To the best of our knowledge, this paper provides the first construction and proof of invariance of a Gibbs measure in infinite volume outside of the 1D case.

Nathalie Verdière, Lilianne Denis-Vidal, Ghislaine Joly-Blanchard

This paper is devoted to the identification of a pollution source in a river. A simple mathematical model of such a problem is given by a one-dimensional linear advection-dispersion-reaction equation with a right hand side spatially supported in a point (the source) and a time variant intensity, both unknown. The identifiability of the distributed system was established for two points of observations one upstream, the other downstream from the source provided the pollutant flow rate is zero on an interval [T, T + deltaT] (T > 0). But the distributed system has to be discretized in order to do a numerical estimation of the unknown parameters. It is why this paper is devoted to the identifiability of the differential system obtained by using a semi-discretization scheme in space. The proof of the identifiability does not require the restrictive assumption about the pollutant flow rate but the knowledge of the initial condition and one observation located upstream from the source. Moreover, from this study, a numerical procedure is deduced for estimating the unknown parameters. It does not necessitate a priori knowledge about the parameters and the unknown function is not expanded on a basis of special functions. Both aspects play an important role in the real applications

Luís Barreira, Clàudìa Valls

No abstract is available for this record.

Charles R. Doering, John Gibbon

Introduction In this chapter we show how the dimension of the global attractor ℕ can be estimated for the Navier-Stokes equations. The approach is an extension of that developed in Chapter 4 for ordinary differential equations where it was shown that if N -dimensional volume elements in the system phase space contract to zero, then the attractor dimension d L (ℕ) must be bounded by N . For partial differential equations the technical chore remains the same; namely, to derive estimates on the spectrum of the linearized evolution operator, linearized around solutions on the attractor, and to perform this operation in some function space instead of an a priori finite dimensional phase space. As we saw in Chapter 4 in the context of the Lorenz equations, this requires some knowledge of the location of the attractor, i.e., a priori estimates on the solutions. This approach is pursued in section 9.2 which deals with the 2 d Navier-Stokes equations. It was shown in Chapter 7 that a global attractor si exists in this case, and we have good control of the solutions on the attractor. It turns out that the result for periodic boundary conditions is quite sharp, within logarithms of both the conventional heuristic estimate for the number of degrees of freedom in a 2 d turbulent flow and rigorous lower bounds. The 3 d Navier-Stokes equations on a periodic domain are the concern of section 9.3. The lack of a regularity proof for this case results in some uncertainty concerning the very existence of a compact attractor. To achieve any formal estimate of the attractor dimension it is necessary to assume that H 1 remains bounded for all t .

Kurt Kreith

1. Introduction.The classical Sturmian theorem of ordinary differential equations deals with functions u(x) and v(x) which are, respectively, solutions of differential equations(1) Um-^^j+tumO,(2) Mv=-l{J^+yv = 0.Under the assumption that "F is larger than M" (in the sense that a(x) ä a(x) > 0 and c(x) £ y(x)) one can infer information about all solutions of (2) from knowledge about a particular nontrivial solution of (1)-i.e. if u(xx) = u(x2)=0 then every solution of (2) has a zero in [xx, x2].These ideas have been generalized to second order elliptic equations by several authors ([l]-[4]) considering elliptic operators and also by Protter [5] and Swanson [6] considering the nonselfadjoint case.Given a proper relation among the coefficients of F and M and that Lu=0 has a nontrivial solution with nodal domain Ü, then it can be shown that every solution of Mv = 0 has a zero in Í2.While all the above proofs of this fact make essential use of some sort of ordering among elliptic operators, the nature of this ordering is never defined in operator-theoretic terms.The results of §2 below suggest that it is an order relationship between certain resolvents of the differential operators F and M which underlies the separation properties characteristic of Sturmian theorems.It will be shown that quite general operator equations in a Banach space 3S satisfy a type of Sturmian theorem if the operators' resolvents satisfy prescribed positivity requirements with respect to a cone SP.In order to apply this theory to differential operators, one must first establish the corresponding positivity properties for their resolvents.This is done in §3 for sufficiently regular nonselfadjoint second order elliptic operators, and the general theory of §2 is then applied in the proof of two Sturmian theorems and the establishment of criteria for certain Green's functions to be positive.