arXiv subjects

Daniel Peralta-Salas

Publications and source records attributed to Daniel Peralta-Salas.

At least 55 records · Page 3Linked to original sources

arXiv2022

Steady Euler flows on $\mathbb{R}^3$ with wild and universal dynamics

Understanding complexity in fluid mechanics is a major problem that has attracted the attention of physicists and mathematicians during the last decades. Using the concept of renormalization in dynamics, we show the existence of a locally dense set $\mathscr G$ of stationary solutions to the Euler equations in $\mathbb R^3$ such that each vector field $X\in \mathscr G$ is universal in the sense that any area preserving diffeomorphism of the disk can be approximated (with arbitrary precision) by the Poincaré map of $X$ at some transverse section. We remark that this universality is approximate but occurs at all scales. In particular, our results establish that a steady Euler flow may exhibit any conservative finite codimensional dynamical phenomenon; this includes the existence of horseshoes accumulated by elliptic islands, increasing union of horseshoes of Hausdorff dimension $3$ or homoclinic tangencies of arbitrarily high multiplicity. The steady solutions we construct are Beltrami fields with sharp decay at infinity. To prove these results we introduce new perturbation methods in the context of Beltrami fields that allow us to import deep techniques from bifurcation theory: the Gonchenko-Shilnikov-Turaev universality theory and the Newhouse and Duarte theorems on the geometry of wild hyperbolic sets. These perturbation methods rely on two tools from linear PDEs: global approximation and Cauchy-Kovalevskaya theorems. These results imply a strong version of V.I. Arnold's vision on the complexity of Beltrami fields in Euclidean space.

math.DS↗

arXiv2022

Inverse localization and global approximation for some Schrödinger operators on hyperbolic spaces

We consider the question of whether the high-energy eigenfunctions of certain Schrödinger operators on the $d$-dimensional hyperbolic space of constant curvature $-κ^2$ are flexible enough to approximate an arbitrary solution of the Helmholtz equation $Δh+h=0$ on $\mathbf{R}^d$, over the natural length scale $O(λ^{-1/2})$ determined by the eigenvalue $λ\gg 1$. This problem is motivated by the fact that, by the asymptotics of the local Weyl law, approximate Laplace eigenfunctions do have this approximation property on any compact Riemannian manifold. In this paper we are specifically interested in the Coulomb and harmonic oscillator operators on the hyperbolic spaces $\mathbf{H}^d(κ)$. As the dimension of the space of bound states of these operators tends to infinity as $κ$ tends to 0, one can hope to approximate solutions to the Helmholtz equation by eigenfunctions for some $κ> 0$ that is not fixed a priori. Our main result shows that this is indeed the case, under suitable hypotheses. We also prove a global approximation theorem with decay for the Helmholtz equation on manifolds that are isometric to the hyperbolic space outside a compact set, and consider an application to the study of the heat equation on $\mathbf{H}^d(κ)$. Although global approximation and inverse approximation results are heuristically related in that both theorems explore flexibility properties of solutions to elliptic equations on hyperbolic spaces, we will see that the underlying ideas behind these theorems are very different.

math.SP↗

arXiv2021

Localization properties of high energy eigenfunctions on flat tori

We consider the question of when the Laplace eigenfunctions on an arbitrary flat torus $\mathbf{T}_Γ:=\mathbf{R}^d/Γ$ are flexible enough to approximate, over the natural length scale of order $1/\sqrtλ$, where $λ\gg1$ is the eigenvalue, an arbitary solution of the Helmholtz equation $Δh + h=0$ on $\mathbf{R}^d$. This problem is motivated by the fact that, by the asymptotics for the local Weyl law, "approximate Laplace eigenfunctions" do have this approximation property on any compact Riemannian manifold. What we find is that the answer depends solely on the arithmetic properties of the spectrum. Specifically, recall that the eigenvalues of $\mathbf{T}_Γ$ are of the form $λ_k=Q_Γ(k)$, where $Q_Γ$ is a quadratic form and $k \in \mathbf{Z}^d$. Our main result is that the eigenfunctions of $\mathbf{T}_Γ$ have the desired approximation property if and only $Q_Γ$ is a multiple of a quadratic form with integer coefficients. In particular, the set of lattices $Γ$ for which this approximation property holds has measure zero but includes all rational lattices. A consequence of this fact is that when $Q_Γ$ is a multiple of a quadratic form with integer coefficients, Laplace eigenfunctions exhibit an extremely flexible behavior over scales of order $1/\sqrtλ$. In particular, there are eigenfunctions of arbitrarily high energy that exhibit nodal components diffeomorphic to any compact hypersurface of diameter $O(1/\sqrtλ)$.

math.SP↗

arXiv2021

Looking at Euler flows through a contact mirror: Universality and undecidability

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. In recent papers [5, 6, 7, 8] several unknown facets of the Euler flows have been discovered, including universality properties of the stationary solutions to the Euler equations. The study of these universality features was suggested by Tao as a novel way to address the problem of global existence for Euler and Navier-Stokes [28]. Universality of the Euler equations was proved in [7] for stationary solutions using a contact mirror which reflects a Beltrami flow as a Reeb vector field. This contact mirror permits the use of advanced geometric techniques in fluid dynamics. On the other hand, motivated by Tao's approach relating Turing machines to Navier-Stokes equations, a Turing complete stationary Euler solution on a Riemannian $3$-dimensional sphere was constructed in [8]. Since the Turing completeness of a vector field can be characterized in terms of the halting problem, which is known to be undecidable [30], a striking consequence of this fact is that a Turing complete Euler flow exhibits undecidable particle paths [8]. In this article, we give a panoramic overview of this fascinating subject, and go one step further in investigating the undecidability of different dynamical properties of Turing complete flows. In particular, we show that variations of [8] allow us to construct a stationary Euler flow of Beltrami type (and, via the contact mirror, a Reeb vector field) for which it is undecidable to determine whether its orbits through an explicit set of points are periodic.

math.AP↗

arXiv2021

Critical point asymptotics for Gaussian random waves with densities of any Sobolev regularity

We consider Gaussian random monochromatic waves $u$ on the plane depending on a real parameter $s$ that is directly related to the regularity of its Fourier transform. Specifically, the Fourier transform of $u$ is $f\,dσ$, where $dσ$ is the Hausdorff measure on the unit circle and the density $f$ is a function on the circle that, roughly speaking, has exactly $s-\frac12$ derivatives in $L^2$ almost surely. When $s=0$, one recovers the classical setting for random waves with a translation-invariant covariance-kernel. The main thrust of this paper is to explore the connection between the regularity parameter $s$ and the asymptotic behavior of the number $N(\nabla u,R)$ of critical points that are contained in the disk of radius $R\gg1$. More precisely, we show that the expectation $\mathbb{E}N(\nabla u,R)$ grows like the area of the disk when the regularity is low enough ($s<\frac32$) and like the diameter when the regularity is high enough ($s>\frac52$), and that the corresponding exponent changes according to a linear interpolation law in the intermediate regime. The transitions occurring at the endpoint cases involve the square root of the logarithm of the radius. Interestingly, the highest asymptotic growth rate occurs only in the classical translation-invariant setting, $s=0$. A key step of the proof of this result is the obtention of precise asymptotic expansions for certain Neumann series of Bessel functions. When the regularity parameter is $s>5$, we show that in fact $N(\nabla u,R)$ grows like the diameter with probability 1, albeit the ratio is not a universal constant but a random variable.

math.SP↗

arXiv2021

Turing universality of the incompressible Euler equations and a conjecture of Moore

In this article we construct a compact Riemannian manifold of high dimension on which the time dependent Euler equations are Turing complete. More precisely, the halting of any Turing machine with a given input is equivalent to a certain global solution of the Euler equations entering a certain open set in the space of divergence-free vector fields. In particular, this implies the undecidability of whether a solution to the Euler equations with an initial datum will reach a certain open set or not in the space of divergence-free fields. This result goes one step further in Tao's programme to study the blow-up problem for the Euler and Navier-Stokes equations using fluid computers. As a remarkable spin-off, our method of proof allows us to give a counterexample to a conjecture of Moore dating back to 1998 on the non-existence of analytic maps on compact manifolds that are Turing complete.

math.AP↗

arXiv2021

Asymptotics for the nodal components of non-identically distributed monochromatic random waves

We study monochromatic random waves on $\mathbb{R}^n$ defined by Gaussian variables whose variances tend to zero sufficiently fast. This has the effect that the Fourier transform of the monochromatic wave is an absolutely continuous measure on the sphere with a suitably smooth density, which connects the problem with the scattering regime of monochromatic waves. In this setting, we compute the asymptotic distribution of the nodal components of random monochromatic waves, showing that the number of nodal components contained in a large ball $B_R$ grows asymptotically like $R/π$ with probability $p_n>0$, and is bounded uniformly in $R$ with probability $1-p_n$ (which is positive if and only if $n \geq 3$). In the latter case, we show the existence of a unique noncompact nodal component. We also provide an explicit sufficient stability criterion to ascertain when a more general Gaussian probability distribution has the same asymptotic nodal distribution law.

math.SP↗

arXiv2020

Constructing Turing complete Euler flows in dimension $3$

Can every physical system simulate any Turing machine? This is a classical problem which is intimately connected with the undecidability of certain physical phenomena. Concerning fluid flows, Moore asked in [15] if hydrodynamics is capable of performing computations. More recently, Tao launched a programme based on the Turing completeness of the Euler equations to address the blow up problem in the Navier-Stokes equations. In this direction, the undecidability of some physical systems has been studied in recent years, from the quantum gap problem [7] to quantum field theories [11]. To the best of our knowledge, the existence of undecidable particle paths of 3D fluid flows has remained an elusive open problem since Moore's works in the early 1990's. In this article we construct a Turing complete stationary Euler flow on a Riemannian $S^3$ and speculate on its implications concerning Tao's approach to the blow up problem in the Navier-Stokes equations.

math.DS↗

arXiv2020

On the singular Weinstein conjecture and the existence of escape orbits for $b$-Beltrami fields

Motivated by Poincaré's orbits going to infinity in the (restricted) three-body (see [26] and [6]), we investigate the generic existence of heteroclinic-like orbits in a neighbourhood of the critical set of a $b$-contact form. This is done by using the singular counterpart [3] of Etnyre--Ghrist's contact/Beltrami correspondence [9], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [29]. Specifically, we analyze the $b$-Beltrami vector fields on $b$-manifolds of dimension $3$ and prove that for a generic asymptotically exact $b$-metric they exhibit escape orbits. We also show that a generic asymptotically symmetric $b$-Beltrami vector field on an asymptotically flat $b$-manifold has a generalized singular periodic orbit and at least $4$ escape orbits. Generalized singular periodic orbits are trajectories of the vector field whose $α$- and $ω$-limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture.

math.SG↗

arXiv2020

Overdetermined boundary problems with nonconstant Dirichlet and Neumann data

In this paper we consider the overdetermined boundary problem for a general second order semilinear elliptic equation on bounded domains of $\mathbf{R}^n$, where one prescribes both the Dirichlet and Neumann data of the solution. We are interested in the case where the data are not necessarily constant and where the coefficients of the equation can depend on the position, so that the overdetermined problem does not generally admit a radial solution. Our main result is that, nevertheless, under minor technical hypotheses nontrivial solutions to the overdetermined boundary problem always exist.

math.AP↗

arXiv2020

Non-existence of axisymmetric optimal domains with smooth boundary for the first curl eigenvalue

We say that a bounded domain $Ω$ is optimal for the first positive curl eigenvalue $μ_1(Ω)$ if $μ_1(Ω)\leq μ_1(Ω')$ for any domain $Ω'$ with the same volume. In spite of the fact that $μ_1(Ω)$ is uniformly lower bounded in terms of the volume, in this paper we prove that there are no axisymmetric optimal (and even locally minimizing) domains with $C^{2,α}$ boundary that satisfies a mild technical assumption. As a particular case, this rules out the existence of $C^{2,α}$ optimal axisymmetric domains with a convex section. An analogous result holds in the case of the first negative curl eigenvalue.

math.AP↗

arXiv2020

Beltrami fields exhibit knots and chaos almost surely

In this paper we show that, with probability 1, a random Beltrami field exhibits chaotic regions that coexist with invariant tori of complicated topologies. The motivation to consider this question, which arises in the study of stationary Euler flows in dimension 3, is V.I. Arnold's 1965 conjecture that a typical Beltrami field exhibits the same complexity as the restriction to an energy hypersurface of a generic Hamiltonian system with two degrees of freedom. The proof hinges on the obtention of asymptotic bounds for the number of horseshoes, zeros, and knotted invariant tori and periodic trajectories that a Gaussian random Beltrami field exhibits, which we obtain through a nontrivial extension of the Nazarov--Sodin theory for Gaussian random monochromatic waves and the application of different tools from the theory of dynamical systems, including KAM theory, Melnikov analysis and hyperbolicity. Our results hold both in the case of Beltrami fields on $\mathbf{R}^3$ and of high-frequency Beltrami fields on the 3-torus.

math.SP↗

arXiv2020

Piecewise smooth stationary Euler flows with compact support via overdetermined boundary problems

We construct new stationary weak solutions of the 3D Euler equation with compact support. The solutions, which are piecewise smooth and discontinuous across a surface, are axisymmetric with swirl. The range of solutions we find is different from, and larger than, the family of smooth stationary solutions recently obtained by Gavrilov and Constantin-La-Vicol; in particular, these solutions are not localizable. A key step in the proof is the construction of solutions to an overdetermined elliptic boundary value problem where one prescribes both Dirichlet and (nonconstant) Neumann data.

math.AP↗

arXiv2020

The helicity uniqueness conjecture in 3D hydrodynamics

We prove that the helicity is the only regular Casimir function for the coadjoint action of the volume-preserving diffeomorphism group $\text{SDiff}(M)$ on smooth exact divergence-free vector fields on a closed three-dimensional manifold $M$. More precisely, any regular $C^1$ functional defined on the space of $C^\infty$ (more generally, $C^k$, $k\ge 4$) exact divergence-free vector fields and invariant under arbitrary volume-preserving diffeomorphisms can be expressed as a $C^1$ function of the helicity. This gives a complete description of Casimirs for adjoint and coadjoint actions of $\text{SDiff}(M)$ in 3D and completes the proof of Arnold-Khesin's 1998 conjecture for a manifold $M$ with trivial first homology group. Our proofs make use of different tools from the theory of dynamical systems, including normal forms for divergence-free vector fields, the Poincaré-Birkhoff theorem, and a division lemma for vector fields with hyperbolic zeros.

math.DG↗

arXiv2020

A characterization of 3D steady Euler flows using commuting zero-flux homologies

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a $3$-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan's homological characterization of geodesible flows in the volume-preserving case. As an application, we show that the steady Euler flows cannot be constructed using plugs (as in Wilson's or Kuperberg's constructions). Analogous results in higher dimensions are also proved.

math.DG↗

arXiv2019

Global, local and dense non-mixing of the 3D Euler equation

We prove a non-mixing property of the flow of the 3D Euler equation which has a local nature: in any neighbourhood of a "typical" steady solution there is a generic set of initial conditions, such that the corresponding Euler flows will never enter a vicinity of the original steady one. More precisely, we establish that there exist stationary solutions $u_0$ of the Euler equation on $\mathbb S^3$ and divergence-free vector fields $v_0$ arbitrarily close to $u_0$, whose (non-steady) evolution by the Euler flow cannot converge in the $C^k$ Hölder norm ($k>10$ non-integer) to any stationary state in a small (but fixed a priori) $C^k$-neighbourhood of $u_0$. The set of such initial conditions $v_0$ is open and dense in the vicinity of $u_0$. A similar (but weaker) statement also holds for the Euler flow on $\mathbb T^3$. Two essential ingredients in the proof of this result are a geometric description of all steady states near certain nondegenerate stationary solutions, and a KAM-type argument to generate knotted invariant tori from elliptic orbits.

math.DS↗

arXiv2019

Beltrami fields with hyperbolic periodic orbits enclosed by knotted invariant tori

We prove that there exist Beltrami fields in Euclidean space, with sharp decay at infinity, which have a prescribed set of invariant tori (possibly knotted or linked) that enclose an arbitrarily large number of hyperbolic periodic orbits. These hyperbolic orbits are cablings over the core curve of each torus. Moreover, the domain bounded by each invariant torus is covered by an almost full measure set of invariant tori. We show that an analogous result holds for high-frequency Beltrami fields on the flat 3-torus.

math.DS↗

arXiv2019

Approximation theorems for the Schrödinger equation and quantum vortex reconnection

We prove the existence of smooth solutions to the Gross-Pitaevskii equation on $\mathbf{R}^3$ that feature arbitrarily complex quantum vortex reconnections. We can track the evolution of the vortices during the whole process. This permits to describe the reconnection events in detail and verify that this scenario exhibits the properties observed in experiments and numerics, such as the $t^{1/2}$ and change of parity laws. We are mostly interested in solutions tending to1 at infinity, which have finite Ginzburg-Landau energy and physically correspond to the presence of a background chemical potential, but we also consider the cases of Schwartz initial data and of the Gross-Pitaevskii equation on the torus. An essential ingredient in the proofs is the development of novel global approximation theorems for the Schrödinger equation on $\mathbf{R}^n$. Specifically, we prove a qualitative approximation result that applies for solutions defined on very general spacetime sets and also a quantitative result for solutions on product sets in spacetime $D \times \mathbf{R}$. This hinges on frequency-dependent estimates for the Helmholtz-Yukawa equation that are of independent interest.

math.AP↗