arXiv subjects
Publications and source records attributed to Daniel Peralta-Salas.
We construct magnetohydrostatic equilibria in rotationally symmetric toroidal domains whose magnetic lines define an $(m,n)$ Seifert fibration for arbitrary coprime integers $m,n$.
We prove that the electric field generated in three-dimensional space by finitely many point charges of one sign has only finitely many equilibrium points, thereby answering a 1969 question of Morse and Cairns (restated by Eremenko in 2008 and, as a conjecture, by Shapiro in 2015). More generally, for nonzero charges $q_i$ of possibly mixed signs at distinct sites $\mathbf a_i\in\mathbf{R}^3$, we show that the Coulomb field has at most $2^{N-4}(N-1)(9N^2+9N+10)$ equilibria in the region where $S(\mathbf x):=\sum_iq_i|\mathbf x-\mathbf a_i|^{-3}$ does not vanish. Of course, for charges of one sign, $S$ is nonzero everywhere. The proof rules out curves of equilibria using algebraic geometry and complex analysis on an associated complex curve, and then applies a Bézout count to obtain a quantitative bound. Well-known examples show that, in the mixed-sign case, the Coulomb field can vanish on curves contained in the zero set of $S$.
We construct stationary solutions of the three-dimensional incompressible Euler equations with helical symmetry and support in a neighborhood of a helix. The solutions are piecewise smooth and arise from a nonlinear overdetermined elliptic boundary value problem associated with a stream-function formulation. A distinguishing feature is that the vortex cross-sections are intrinsically anisotropic: after rescaling, the leading-order shape is elliptic rather than radial, and the boundary exhibits a nontrivial third Fourier mode reflecting helical effects absent in previous axisymmetric constructions. A key step in the proof is the analysis of a genuinely anisotropic overdetermined elliptic problem with prescribed Dirichlet and nonconstant Neumann conditions.
A steady Euler flow is localizable if the pressure function is constant along its stream lines. This property was used by Gavrilov to construct the first smooth compactly supported steady states of 3D Euler. We prove that any analytic localizable 3D Euler flow in a bounded domain $Ω$ is axisymmetric and $Ω$ is a rotationally symmetric domain whose transverse section is a disk or an annulus with convex boundary curves. To the best of our knowledge, this is the first symmetry theorem for 3D steady Euler flows. In the context of MHD equilibria, this result shows that Grad's conjecture holds true for magnetic fields satisfying the isodynamic condition, a property introduced by Palumbo in the 1960's to minimize the effect of particle drifts in plasma confinement devices.
We explore the relationship between Turing completeness and topological entropy of dynamical systems. We first prove that a natural class of Turing machines that we call "branching Turing machines" (which includes most of the known examples of universal Turing machines) has positive topological entropy. Motivated by the recent construction of Turing complete Euler flows, we deduce that any Turing complete dynamics with a continuous encoding that simulates a universal branching machine is chaotic. On the other hand, we show that, unexpectedly, universal Turing machines with zero topological entropy (and even zero speed) can be constructed, unveiling the independence of chaos and universality at the symbolic level.
We prove the existence of weak solutions to the 3D ideal MHD equations, of class $C^α$ with $α=1/200$, for which the total energy and the cross helicity (i.e., the so-called Elsässer energies) are not conserved. The solutions do not possess any symmetry properties and the magnetic helicity, which is necessarily conserved for Hölder continuous solutions, is nonzero. The construction, which works both on the torus $\mathbb{T}^3$ and on $\mathbb{R}^3$ with compact spatial support, is based on a novel convex integration scheme in which the magnetic helicity is preserved at each step. This is the first construction of continuous weak solutions at a regularity level where one conservation law (here, the magnetic helicity) is necessarily preserved while another (here, the total energy or cross helicity) is not, and where the preservation of the former is nontrivial in the sense that it does not follow from symmetry considerations.
In this article, we establish the foundations of a computational field theory, which we term Topological Kleene Field Theory (TKFT), inspired by Stephen Kleene's seminal work on partial recursive functions and drawing parallels with Topological Field Theory. Our central result shows that any computable function can be simulated by the flow on a smooth bordism of a vector field with good local properties, setting an alternative model of computation to Turing machines. We thus establish that a computable function can be fully realized within a single go of a dynamical system, differing from previous works where computation is encoded as an iterative process. The output of the computable function emerges directly, laying the groundwork for potential applications that accelerate the physical realization of computation.
We prove that, on each low energy level, the natural Hamiltonian system defined by a generic smooth potential on $\mathbf{T}^2$ exhibits an arbitrarily high number of hyperbolic periodic orbits and a positive-measure set of invariant tori. Hence, quasi-periodic motion and hyperbolic behavior typically coexist in the low-energy dynamics of natural Hamiltonian systems with two degrees of freedom.
We construct solutions to the Navier-Stokes equations on $\mathbf{R}^2$ with an arbitrary number of stagnation points which merge and split along trajectories that can be prescribed freely, up to a small deformation.
We prove an extension theorem for local solutions of the 3d incompressible Euler equations. More precisely, we show that if a smooth vector field satisfies the Euler equations in a spacetime region $Ω\times(0,T)$, one can choose an admissible weak solution on $\mathbf R^3\times (0,T)$ of class $C^β$ for any $β<1/3$ such that both fields coincide on $Ω\times (0,T)$. Moreover, one controls the spatial support of the global solution. Our proof makes use of a new extension theorem for local subsolutions of the incompressible Euler equations and a $C^{1/3}$ convex integration scheme implemented in the context of weak solutions with compact support in space. We present two nontrivial applications of these ideas. First, we construct infinitely many admissible weak solutions of class $C^β_{\text{loc}}$ with the same vortex sheet initial data, which coincide with it at each time $t$ outside a turbulent region of width $O(t)$. Second, given any smooth solution $v$ of the Euler equation on $\mathbf T^3\times(0,T)$ and any open set $U \subset \mathbf T^3$, we construct admissible weak solutions which coincide with $v$ outside $U$ and are uniformly close to it everywhere at time 0, yet blow up dramatically on a subset of $U\times (0,T)$ of full Hausdorff dimension. These solutions are of class $C^β$ outside their singular set.
The relationship between computational models and dynamics has captivated mathematicians and computer scientists since the earliest conceptualizations of computation. Recently, this connection has gained renewed attention, fueled by T. Tao's programme aiming to discover blowing-up solutions of the Navier-Stokes equations using an embedded computational model. In this survey paper, we review some of the recent works that introduce novel and exciting perspectives on the representation of computability through dynamical systems. Starting from dynamical universality in a classical sense, we shall explore the modern notions of Turing universality in fluid dynamics and Topological Kleene Field Theories as a systematic way of representing computable functions by means of dynamical bordisms. Finally, we will discuss some important open problems in the area.
In this article, we construct stationary solutions to the Navier-Stokes equations on certain Riemannian $3$-manifolds that exhibit Turing completeness, in the sense that they are capable of performing universal computation. This universality arises on manifolds admitting nonvanishing harmonic 1-forms, thus showing that computational universality is not obstructed by viscosity, provided the underlying geometry satisfies a mild cohomological condition. The proof makes use of a correspondence between nonvanishing harmonic $1$-forms and cosymplectic geometry, which extends the classical correspondence between Beltrami fields and Reeb flows on contact manifolds.
We construct families of rotationally symmetric toroidal domains in $\mathbb R^3$ for which the eigenfields associated to the first (positive) Ampèrian curl eigenvalue are symmetric, and others for which no first eigenfield is symmetric. This implies, in particular, that minimizers of the celebrated Woltjer's variational principle do not need to inherit the rotational symmetry of the domain. This disproves the folk wisdom that the eigenfields corresponding to the lowest curl eigenvalue must be symmetric if the domain is.
In this paper, we study the Chern-Hamilton energy functional on compact cosymplectic manifolds, fully classifying in dimension 3 those manifolds admitting a critical compatible metric for this functional. This is the case if and only if either the manifold is co-Kähler or if it is a mapping torus of the 2-torus by a hyperbolic toral automorphism and equipped with a suspension cosymplectic structure. Moreover, any critical metric has minimal energy among all compatible metrics. We also exhibit examples of manifolds with first Betti number $b_1 \geq 2$ admitting cosymplectic structures, but such that no cosymplectic structure admits a critical compatible metric.
We disprove the generalized Chern-Hamilton conjecture on the existence of critical compatible metrics on contact $3$-manifolds. More precisely, we show that a contact $3$-manifold $(M,α)$ admits a critical compatible metric for the Chern-Hamilton energy functional if and only if it is Sasakian or its associated Reeb flow is $C^\infty$-conjugate to an algebraic Anosov flow modeled on $\widetilde{SL}(2, \mathbb R)$. In particular, this yields a complete topological classification of compact $3$-manifolds that admit critical compatible metrics. As a corollary we prove that no contact structure on $\mathbb{T}^3$ admits a critical compatible metric and that critical compatible metrics can only occur when the contact structure is tight.
Berry's random wave conjecture posits that high energy eigenfunctions of chaotic systems resemble random monochromatic waves at the Planck scale. One important consequence is that, at the Planck scale around "many" points in the manifold, any solution to the Helmholtz equation $Δφ+φ=0$ can be approximated by high energy eigenfunctions. This property, sometimes called inverse localization, has useful applications to the study of the nodal sets of eigenfunctions. Alas, the only manifold for which the local limits of a sequence of high energy eigenfunctions are rigorously known to be given by random waves is the flat torus $(\mathbf{R}/\mathbf{Z})^2$, which is certainly not chaotic. Our objective in this paper is to study the validity of this "inverse localization" property in the class of integrable billiards, exploiting the fact that integrable polygonal billiards are classified and that Birkhoff conjectured that ellipses are the only smooth integrable billiards. Our main results show that, while there are infinitely many integrable polygons exhibiting good inverse localization properties, for "most" integrable polygons and ellipses, this property fails dramatically. We thus conclude that, in a generic integrable billiard, the local limits of Dirichlet and Neumann eigenfunctions do not match random waves, as one might expect in view of Berry's conjecture. Extensions to higher dimensions and nearly integrable polygons are discussed too.
Given any smooth solenoidal vector field $v_0$ on $\mathbf T^3$, we show the existence of infinitely many Hölder-continuous steady Euler flows $v$ with the same topology as $v_0$, in certain weak sense. In particular, we show that $v$ possesses a unique flow of the highest Hölder regularity, which is conjugate to the flow of $v_0$ via a volume-preserving Hölder homeomorphism of $\mathbf T^3$. This result extends to the case of Euler equations on toroidal domains, which has applications to the study of plasmas. The proof relies on a novel convex integration scheme incorporating the key idea that the velocity field of the subsolutions must remain diffeomorphic to $v_0$ at each iteration step.
We present new explicit tight and overtwisted contact structures on the (round) 3-sphere and the (flat) 3-torus for which the ambient metric is weakly compatible. Our proofs are based on the construction of nonvanishing curl eigenfields using suitable families of Jacobi or trigonometric polynomials. As a consequence, we show that the contact sphere theorem of Etnyre, Komendarczyk and Massot (2012) does not hold for weakly compatible metric as it was conjectured. We also establish a geometric rigidity for tight contact structures by showing that any contact form on the 3-sphere admitting a compatible metric that is the round one is isometric, up to a constant factor, to the standard (tight) contact form.