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Daniel Faraco

Publications and source records attributed to Daniel Faraco.

At least 19 recordsLinked to original sources
arXiv2026

Turbulent Dynamos on Bounded Domains and Their Generalization to the Geometric Transport Equation

For any smooth bounded domain $\Omega \subset \mathbb{R}^3$, we construct a divergence-free velocity field $u \in L_t^1 W^{1,p}(\Omega)$ for all $p < \infty$, and magnetic fields $B^\epsilon \in L_t^p C^{m}(\Omega)$ for all $p < \infty$ and $m\in \mathbb{N}$, that solve the kinematic dynamo equation and exhibit arbitrarily fast growth of any magnetic energy mode, uniformly in the vanishing-diffusivity limit $\epsilon \to 0$. The construction is based on the convex integration scheme of Modena-Sz\'ekelyhidi and Cheskidov-Luo. The main novelty lies in the introduction of explicit potentials, which allow the solutions to be localized and avoid the need to work with the anti-curl operator. In addition, we present a unified scheme for the geometric transport equation (GTE), which encompasses both the transport and Maxwell equations.

math.AP↗
arXiv2025

Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation

We prove non-uniqueness of weak solutions to the forced $\alpha$-SQG equation with Sobolev regularity $W^{s,p}$ in the supercritical regime $s < \alpha + \frac{2}{p}$, covering the 2D Euler equation ($\alpha = 0$), the Surface Quasi-Geostrophic equation ($\alpha = 1$), and the intermediate cases. A key step is the construction of smooth, compactly supported vortices that exhibit non-linear instability. As a by-product, we show existence of global smooth solutions to the (unforced) $\alpha$-SQG equation that are neither rotating nor traveling.

math.AP↗
arXiv2025

Recovering nonsmooth coefficients for higher-order perturbations of a polyharmonic operator

We consider an inverse problem for a higher order elliptic operator where the principal part is the polyharmonic operator $(-\Delta)^m$ with $ m \geq 2$. We show that the map from the coefficients to a certain bilinear form is injective. We have a particular focus on obtaining these results under lower regularity on the coefficients. It is known that knowledge of this bilinear form is equivalent to a knowledge of a Dirichlet to Neumann map or the Cauchy data for solutions.

math.AP↗
arXiv2024

A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation

We give a simpler proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation in the vorticity class $L^1\cap L^p$ with $2<p<\infty$. The main simplification is an alternative construction of a smooth and compactly supported unstable vortex, which is split into two steps: Firstly, we construct a piecewise constant unstable vortex, and secondly, we find a regularization through a fixed point argument. This simpler structure of the unstable vortex yields a simplification of the other parts of Vishik's proof.

math.AP↗
arXiv2024

Homogenization on parallelizable Riemannian manifolds

We consider the problem of finding the homogenization limit of oscillating linear elliptic equations on an arbitrary parallelizable manifold $(M,g,\Gamma)$. We replicate the concept of two-scale convergence by pulling back tensors $T$ defined on the torus bundle $\mathbb{T}M$ to $M$. The process consist of two steps: localization in the slow variable through Voronoi domains, and inducing local periodicity in the fast variable from the local exponential map in combination with the geometry of the torus bundle. The procedure yields explicit cell formulae for the homogenization limit and as a byproduct a theory of two-scale convergence of tensors of arbitrary order.

math.AP↗
arXiv2024

Lower semicontinuity, Stoilow factorization and principal maps

We consider a strengthening of the usual quasiconvexity condition of Morrey in two dimensions, which allows us to prove lower semicontinuity for functionals which are unbounded as the determinant vanishes. This notion, that we call principal quasiconvexity, arose from the planar theory of quasiconformal mappings and mappings of finite distortion. We compare it with other quasiconvexity conditions that have appeared in the literature and provide a number of concrete examples of principally quasiconvex functionals that are not polyconvex. The Stoilow factorization, that in the context of maps of integrable distortion was developed by Iwaniec and \v{S}ver\'ak, plays a prominent role in our approach.

math.AP↗
arXiv2023

The local Burkholder functional, quasiconvexity and Geometric Function Theory

We show that the local Burkholder functional $\mathcal B_K$ is quasiconvex. In the limit of $p$ going to 2 we find a class of non-polyconvex functionals which are quasiconvex on the set of matrices with positive determinant. In order to prove the validity of lower semicontinuity arguments in this setting, we show that the Burkholder functionals satisfy a sharp extension of the classical function theoretic area formula. As a corollary, in addition to functionals in geometric function theory, one finds new classes of non-polyconvex functionals, degenerating as the determinant vanishes, for which there is existence of minimizers.

math.AP↗
arXiv2023

Entropy solutions to macroscopic IPM

We investigate maximal potential energy dissipation as a selection criterion for subsolutions (coarse grained solutions) in the setting of the unstable Muskat problem. We show that both, imposing this criterion on the level of convex integration subsolutions, and the strategy of Otto based on a relaxation via minimizing movements, lead to the same nonlocal conservation law. Our main result shows that this equation admits an entropy solution for unstable initial data with an analytic interface.

math.AP↗
arXiv2021

Magnetic helicity, weak solutions and relaxation of ideal MHD

We revisit the issue of conservation of magnetic helicity and the Woltjer-Taylor relaxation theory in magnetohydrodynamics in the context of weak solutions. We introduce a relaxed system for the ideal MHD system, which decouples the effects of hydrodynamic turbulence such as the appearance of a Reynolds stress term from the magnetic helicity conservation in a manner consistent with observations in plasma turbulence. As by-products we answer two open questions in the field: We show the sharpness of the $L^3$ integrability condition for magnetic helicity conservation and provide turbulent bounded solutions for MHD dissipating energy and cross helicity but with (arbitrary) constant magnetic helicity.

math.AP↗
arXiv2021

On the proof of Taylor's conjecture in multiply connected domains

In this Letter we extend the proof, by Faraco and Lindberg, of Taylor's conjecture in multiply connected domains to cover arbitrary vector potentials and remove the need to impose conditions on the magnetic field due to gauge invariance. This extension allows us to treat general magnetic fields in closed domains that are important in laboratory plasmas and brings closure to a problem that has been open for almost 50 years.

math-ph↗
arXiv2021

Semiclassical estimates for pseudodifferential operators and the Muskat problem in the unstable regime

We obtain new semiclassical estimates for pseudodifferential operators with low regular symbols. Such symbols appear naturally in a Cauchy Problem related to recent weak solutions to the unstable Muskat problem constructed via convex integration in [CCF16]. In particular, our new estimates reveal the tight relation between the speed of opening of the mixing zone and the regularity of the interphase.

math.AP↗
arXiv2021

Mixing solutions for the Muskat problem

We prove the existence of mixing solutions of the incompressible porous media equation for all Muskat type $H^5$ initial data in the fully unstable regime. The proof combines convex integration, contour dynamics and a basic calculus for non smooth semiclassical type pseudodifferential operators which is developed.

math.AP↗
arXiv2021

Localized mixing zone for Muskat bubbles and turned interfaces

We construct mixing solutions to the incompressible porous media equation starting from Muskat type data in the partially unstable regime. In particular, we consider bubble and turned type interfaces with Sobolev regularity. As a by-product, we prove the continuation of the evolution of IPM after the Rayleigh-Taylor and smoothness breakdown exhibited in [18,17]. At each time slice the space is split into three evolving domains: two non-mixing zones and a mixing zone which is localized in a neighborhood of the unstable region. In this way, we show the compatibility between the classical Muskat problem and the convex integration method.

math.AP↗
arXiv2019

Bounded solutions of ideal MHD with compact support in space-time

We show that in 3-dimensional ideal magnetohydrodynamics there exist infinitely many bounded solutions that are compactly supported in space-time and have non-trivial velocity and magnetic fields. The solutions violate conservation of total energy and cross helicity, but preserve magnetic helicity. For the 2-dimensional case we show that, in contrast, no nontrivial compactly supported solutions exist in the energy space.

math.AP↗
arXiv2019

Improved Hölder regularity for strongly elliptic PDEs

We establish surprising improved Schauder regularity properties for solutions to the Leray-Lions divergence type equation in the plane. The results are achieved by studying the nonlinear Beltrami equation and making use of special new relations between these two equations. In particular, we show that solutions to an autonomous Beltrami equation enjoy a quantitative improved degree of Hölder regularity, higher than what is given by the classical exponent $1/K$.

math.CV↗
arXiv2018

Limiting Carleman weights and conformally transversally anisotropic manifolds

We analyze the structure of the set of limiting Carleman weights in all conformally flat manifolds, 3-manifolds, and 4-manifolds. In particular we give a new proof of the classification of Euclidean limiting Carleman weights, and show that there are only three basic such weights up to the action of the conformal group. In dimension three we show that if the manifold is not conformally flat, there could be one or two limiting Carleman weights. We also characterize the metrics that have more than one limiting Carleman weight. In dimension four we obtain a complete spectrum of examples according to the structure of the Weyl tensor. In particular, we construct unimodular Lie groups whose Weyl or Cotton-York tensors have the symmetries of conformally transversally anisotropic manifolds, but which do not admit limiting Carleman weights.

math.AP↗
arXiv2018

Proof of Taylor's conjecture on magnetic helicity conservation

We prove Taylor's conjecture which says that in 3D MHD, magnetic helicity is conserved in the ideal limit in bounded, simply connected, perfectly conducting domains. When the domain is multiply connected, magnetic helicity depends on the vector potential of the magnetic field. In that setting we show that magnetic helicity is conserved for a large and natural class of vector potentials but not in general for all vector potentials. As an analogue of Taylor's conjecture in 2D, we show that mean square magnetic potential is conserved in the ideal limit, even in multiply connected domains.

math.AP↗