arXiv subjects
Publications and source records attributed to Svitlana Mayboroda.
The present paper establishes the Richardson dispersion law, predicting the particle separation with time according to $\E R_t^2 \sim t^3$, as well as anomalous dissipation for the associated tracer, for a random Hölder continuous velocity vectorfield with controlled sweeping.
We construct a divergence-free velocity field $\mathbf{u}(\mathbf{x},t)$ on $\mathbb{T}^{n}$ that is a genuine function of position and carries both defining properties of turbulent transport: Richardson pair dispersion and a dissipative anomaly. No synthetic field has carried both, because the frame that gives the first destroys the field the second requires. The key is a dichotomy: deforming a spectrally localised pattern smears it across scales, whereas pushing its generating centres forward and re-convolving leaves the spectral envelope exactly invariant --- so transporting the eddy centres preserves both cascade and frame. In the field the finite-size Lyapunov exponent obeys $ζ(R)=h(R)-1$ threshold by threshold within the scaling range, with a sweeping deficit $Δ_{\rm sw}=+0.038\pm0.019$, consistent with zero, against $+0.091\pm0.019$ for the frozen field on the same data, a gap that doubles at Navier--Stokes intermittency. Companion papers prove that this family shows Richardson dispersion and a dissipative anomaly (Fefferman and Mayboroda 2026), and that a scalar advected by this flow obeys Yaglom's relation (De Lellis 2026), at small Kubo number. Trajectories forget their starting point as the smallest eddy is removed: the median separation of pairs released from one point with independent Brownian kicks varies as $κ_{p}^{0.0333\pm0.0038}$ over 22 decades of particle diffusivity, against $κ_{p}^{1/2}$ for a smooth control. A scalar in the same field closes the exact scale-by-scale variance budget, four independently measured terms balancing the dissipation to 15\% over 1.2 decades with no adjustable constant, and reaches its inertial--convective asymptote $S_{L}=-χr$ to $0.9995\pm0.0018$ once the measured diffusive transfer is restored, while under an imposed mean gradient $χ$ changes by 14\% over a factor 5.8 in $κ$.
We develop a boundary value and potential theory for elliptic operators \[ L=-\Div(A\nabla)+\B\cdot\nabla \] with form-bounded drift in one-sided chord-arc domains. Assuming that the positive part of $\Div\B$ has sufficiently small form bound, we establish local regularity estimates and solvability of the Dirichlet problem, and construct the associated harmonic measure. Under the additional assumption $\Div\B\le0$, we construct a Green's function and prove pointwise and gradient estimates. For the characteristic distributional identity with a Dirac mass at the pole to hold, we will additionally assume either $\Div\B=0$ or $\B\in L^{n,1}_{\textup{loc}}(Ω)$. Moreover, in the divergence-free case, we obtain a local lower bound for the Green's function, CFMS estimates, doubling of harmonic measure, a comparison principle for nonnegative solutions with zero trace, and a change of pole formula. Finally, using AI, we construct, in the upper half-space, an operator of the form $-Δ+\B\cdot\nabla$ with $\B$ form-bounded and divergence-free, whose harmonic measure exhibits dimension drop.
We establish well posedness of the Dirichlet problem in weak local John domains $Ω\subset\R^\dmn$ with Ahlfors regular boundaries of possibly non integer dimension $0<\bdmn<\dmn$, for linear second order elliptic equations with real coefficients, and with data in Besov spaces with a very broad range of acceptable parameters.
We extend the Obukhov-Corrsin theory of scalar turbulence to compressible flows with spatially variable, anisotropic diffusivity. Combining density and diffusivity into a single positive matrix field, a transport landscape, permits an exact scale-by-scale balance in which the sign-indefinite commutator between filtering and diffusion is eliminated rather than estimated. If the density-weighted third-order velocity and scalar increments scale as $\ell^α$ and $\ell^β$, respectively, we prove that anomalous scalar dissipation is impossible when $α+2β>1$. Remarkably, this threshold is independent of the anisotropy and spatial regularity of the diffusivity, provided it remains uniformly elliptic. Codimension-one shocks in both velocity and scalar have $α=β=1/3$ and therefore lie exactly at the critical threshold. For statistically stationary turbulence we further obtain an exact density-weighted constant-flux relation, providing a compressible analogue of the relation underlying Yaglom's law. The results apply directly to passive-scalar transport in both gases and magnetized plasmas and provide testable diagnostics for simulations of compressible magnetohydrodynamic turbulence. Because the same operator governs anisotropic heat conduction, the results carry over to the electron temperature of a magnetized plasma, and they imply that gradient statistics in such flows must be contracted with the transport landscape rather than with the density.
In this paper we study the behavior of the solutions to the Robin problem in bounded $1$-sided NTA domains with Ahlfors-David regular boundary, generalizing the results of \cite{DavDEMM} to the case of a non constant Robin parameter. In particular, we will prove the mutual absolute continuity of the Robin harmonic measure with respect to the surface measure in the setting of variable permeability.
We prove the absence of anomalous dissipation for passive scalars driven by some random autonomous divergence-free vector fields in $\mathbb T^d$. In dimension $d=2$ we just need continuity almost surely and a mild nondegeneracy condition on the randomness. In dimension $d\geq 3$ we assume a special geometric structure and almost sure Hölder regularity with a Hölder exponent bigger than $\frac{1}{8}$. No regularity is assumed on the passive scalar except for boundedness in the initial data. The proof relies on dimension-theoretic arguments, as opposed to commutator estimates. A consequence of these results is that the same assumptions prevent (almost surely) many other expected properties of turbulent flows, such as anomalous regularization, the Yaglom-Obukhov-Corrsin law, and Richardson diffusion.
We introduce a physics-informed framework for learning finite-dimensional embeddings of solution families of partial differential equations. The method uses a multihead Physics-Informed Neural Network in which a shared body learns a latent manifold representing the solution space, while linear heads reconstruct individual solutions associated with different initial conditions. A head-orthogonalization penalty removes degeneracies in the latent representation and stabilizes the principal-component spectrum across training realizations. Because the initial condition is built into the network output by construction, these principal components measure the additional variability the network learns on top of the initial profile, not the full solution itself. We apply the method to the one-dimensional viscous Burgers equation, with the heat and wave equations as robustness checks. For a latent dimension $n_b=20$, the learned manifolds exhibit pronounced effective dimensional reduction: for Burgers dynamics, only $2$-$4$ principal components capture about $95\%$ of the latent-space variance, while $4$-$7$ capture about $99\%$, depending on the initial-condition family; the same qualitative compression holds for the heat and wave equations. We also split the wavenumber axis into bands (``Fourier shells'') and measure how much each band contributes to every principal component. The resulting frequency profile is invariant under the change-of-basis freedom that the orthogonalization penalty leaves in the latent space, and is therefore reproducible across independent training runs. More broadly, this establishes the learned spectral profiles and principal components as robust observables of solution-manifold geometry.
We prove a coarea-type result for scalar functions $f$ in fractional Sobolev spaces $W^{s, p} (Ω)$ with $Ω\subset \mathbb R^n$, $0<s<1$, and $1\leq p < \infty$. Our theorem shows that a.e. level set has zero Hausdorff $\mathcal{H}^{n-s}$ measure, where the level set $f^{-1} (y)$ is defined as the set all points at which $y$ is between the $\liminf$ and the $\limsup$ (as $r\downarrow 0$) of the averages of $f$ over the balls $B_r (y)$. A quite general construction of random series of wavelets shows also that with probability $1$ (many) level sets have indeed dimension $n-s$.
We present here the Robin boundary condition and its significance in mathematics and physics.
We establish well posedness of the Poisson problem in weak local John domains, for linear second order elliptic equations with real coefficients, and with data in weighted Lebesgue spaces with a very broad range of acceptable parameters.
Mesoscale structures can often be described as fractional dimensional across a wide range of scales. We consider a $γ$ dimensional measure embedded in an $N$ dimensional space and discuss how to determine its dimension, both in $N$ dimensions and projected into $D$ dimensions. It is a highly non-trivial problem to decode the original geometry from lower dimensional projection of a high-dimensional measure. The projections are space-feeling, the popular box-counting techniques do not apply, and the Fourier methods are contaminated by aliasing effects. In the present paper we demonstrate that under the "Copernican hypothesis'' that we are not observing objects from a special direction, projection in a wavelet basis is remarkably simple: the wavelet power spectrum of a projected $γ$ dimensional measure is $P_j \propto 2^{-jγ}$. This holds regardless of the embedded dimension, $N$, and the projected dimension, $D$. This approach could have potentially broad applications in data sciences where a typically sparse matrix encodes lower dimensional information embedded in an extremely high dimensional field and often measured in projection to a low dimensional space. Here, we apply this method to JWST and Chandra observations of the nearby supernova Cas A. We find that the emissions can be represented by projections of mesoscale substructures with fractal dimensions varying from $γ= 1.7$ for the warm CO layer observed by JWST, up to $γ= 2.5$ for the hot X-ray emitting gas layer in the supernova remnant. The resulting power law indicates that the emission is coming from a fractal dimensional mesoscale structure likely produced by magneto-hydrodynamical instabilities in the expanding supernova shell.
The present paper establishes delicate properties of the Green function with Robin boundary conditions, in particular, elucidating the nature of the passage between the Dirichlet-like and Neumann-like behavior. This yields sharp quantifiable bounds on the corresponding harmonic measure and proves the phase transition in the behavior of the total flow earlier conjectured in physics literature in concert with the efficacy of mammalian lungs.
We present an announcement of some recent results concerning well-posedness of the Poisson-Dirichlet problem with boundary data in Besov spaces with fractional smoothness. This is a far-reaching generalization as previously known theorems concerning well-posedness of the Poisson problem in such intermediate smoothness classes were mostly restricted to the context of Lipschitz domains and coefficients satisfying strong regularity assumptions.
Since the seminal work of Kenig and Pipher, the Dahlberg-Kenig-Pipher (DKP) condition on oscillations of the coefficient matrix became a standard threshold in the study of absolute continuity of the harmonic measure with respect to the Hausdorff measure on the boundary. It has been proved sufficient for absolute continuity in the domains with increasingly complex geometry, and known counterexamples show that in a certain sense it is necessary as well. In the present note, we introduce into the subject ideas from homogenization theory to exhibit a new class of operators for which the elliptic measure is well-behaved, featuring the coefficients violating the DKP condition, and on the contrary, oscillating so quickly, that the homogenization takes place.
The present paper establishes that the Robin harmonic measure is quantitatively mutually absolutely continuous with respect to the surface measure on any Ahlfors regular set in any (quantifiably) connected domain for any elliptic operator. This stands in contrast with analogous results for the Dirichlet boundary value problem and also contradicts the expectation, supported by simulations in the physics literature, that the dimension of the Robin harmonic measure in rough domains exhibits a phase transition as the boundary condition interpolates between completely reflecting and completely absorbing. In the adopted traditional language, the corresponding harmonic measure exhibits no dimension drop, and the absolute continuity necessitates neither rectifiability of the boundary nor control of the oscillations of the coefficients of the equation. The expected phase transition is rather exhibited through the detailed non-scale-invariant weight estimates.
The location of the mobility edge is a long standing problem in Anderson localization. In this paper, we show that the effective confining potential introduced in the localization landscape (LL) theory predicts the onset of delocalization in 3D tight-binding models, in a large part of the energy-disorder diagram. Near the edge of the spectrum, the eigenstates are confined inside the basins of the LL-based potential. The delocalization transition corresponds to the progressive merging of these basins resulting in the percolation of this classically-allowed region throughout the system. This approach, shown to be valid both in the cases of uniform and binary disorders despite their very different phase diagrams, allows us to reinterpret the Anderson transition in the tight-binding model: the mobility edge appears to be composed of two parts, one being understood as a percolation transition.
Many geometric and analytic properties of sets hinge on the properties of harmonic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let $Γ\subset \mathbb R^n$ be an Ahlfors regular set of dimension $d<n-1$ (not necessarily integer) and $Ω= \mathbb R^n \setminus Γ$. Let $L = - {\rm div} A\nabla$ be a degenerate elliptic operator with measurable coefficients such that the ellipticity constants of the matrix $A$ are bounded from above and below by a multiple of ${\rm dist}(\cdot, Γ)^{d+1-n}$. We define weak solutions; prove trace and extension theorems in suitable weighted Sobolev spaces; establish the maximum principle, De Giorgi-Nash-Moser estimates, the Harnack inequality, the Hölder continuity of solutions (inside and at the boundary). We define the Green function and provide the basic set of pointwise and/or $L^p$ estimates for the Green function and for its gradient. With this at hand, we define harmonic measure associated to $L$, establish its doubling property, non-degeneracy, change-of-the-pole formulas, and, finally, the comparison principle for local solutions. In another article to appear, we will prove that when $Γ$ is the graph of a Lipschitz function with small Lipschitz constant, we can find an elliptic operator $L$ for which the harmonic measure given here is absolutely continuous with respect to the $d$-Hausdorff measure on $Γ$ and vice versa. It thus extends Dahlberg's theorem to some sets of codimension higher than 1.