arXiv subjects

Ricardo M. S. Rosa

Publications and source records attributed to Ricardo M. S. Rosa.

11 recordsLinked to original sources

arXiv2026

Energy cascade for cross-shear length scales in free-shear three-dimensional incompressible viscous flows

The phenomenon of energy cascade is addressed in the case of free-shear flows, modeled with the equations for incompressible Newtonian fluids with mixed periodic and free-slip boundary conditions driven by an imposed mean shear profile. The rigorous results are proved for ensemble averages with respect to stationary statistical solutions in the sense of Foias and Prodi. The flow is decomposed into high and low horizontal wavenumber components, corresponding to scales perpendicular to the mean shear gradient, and the corresponding energy budget balances viscous dissipation, production by the mean shear, and transfer between scales. Exploiting the orthogonality of horizontal Fourier modes to control the shear-production term, we look for a cascade in which energy flows from the large scales, where it is fed by the mean shear, down to the small scales, where viscosity dissipates it. For any given wavenumber, we define an associated horizontal Taylor wavenumber of the high-wavenumber band above the given wavenumber. We prove two energy cascade results, for the energy flux and for a restricted energy flux that discounts possible energy loss to flow singularities, both valid for wavenumbers whose associated horizontal Taylor wavenumber lies above the viscous shear wavenumber and whose low-wavenumber energy dissipation rate is negligible compared to the total energy dissipation rate. On heuristic grounds, these rigorous conditions correspond precisely to an energy cascade in the range from the classical Corrsin scale down to the Kolmogorov dissipation scale, as expected in the conventional theory of turbulent shear flows.

math.AP↗

arXiv2026

The Effect of Stochasticity in Score-Based Diffusion Sampling: a KL Divergence Analysis

Sampling in score-based diffusion models can be performed by solving either a reverse-time stochastic differential equation (SDE) parameterized by an arbitrary stochasticity function or a probability flow ODE, corresponding to setting this stochasticity function to zero. In this work, we investigate the effect of this stochasticity on the generation process through the evolution of Kullback-Leibler (KL) divergences, obtaining general KL divergence bounds and a novel analysis of the impact of the time-profile of the score error on model performance. For exact score functions, stochasticity has a contractive effect, decreasing KL divergence along the sampling trajectory. For approximate scores, however, a trade-off arises between correcting accumulated errors and amplifying current score errors, meaning stochasticity can either improve or degrade generation performance. Theoretical considerations indicate that the gain from stochasticity depends on the time-localization of the trained model error. We test this in experiments on both toy and benchmark data sets, also comparing the KL divergence evolution with the obtained bounds. We also present a fully analytical example, where all the relevant quantities can be computed, and the optimal stochasticity function can be characterized via an optimal control analysis.

cs.LG↗

arXiv2026

Optimal minimax formula for bounds on ensemble averages of statistically stationary three-dimensional Navier-Stokes flows

We establish an optimal upper bound formula for ensemble averages of flow quantities associated with the three-dimensional incompressible Navier-Stokes equations. The formula takes the form of a minimax problem, extending the framework developed by Tobasco, Goluskin, and Doering (2018) for finite-dimensional systems and by Rosa and Temam (2022) for the two-dimensional Navier-Stokes equations. The lack of global well-posedness for the 3D case presents a significant challenge, which we overcome by working within the space of Foias-Prodi stationary statistical solutions. The minimax formula is derived by exploiting suitable compactness and continuity properties of specific subspaces of probability measures under the weak topology of the phase space. A distinguishing feature of our result is the characterization of the maximizing measures: unlike the previous cases, the optimal bounds in 3D are achieved on extreme points that are specific convex combinations of at most two Dirac delta measures, instead of exactly one, a structure that naturally appears from the constraints given by the mean energy dissipation inequalities in the characterization of the Foias-Prodi stationary statistical solutions.

math.AP↗

arXiv2025

Regularity of the global attractor for the 2D incompressible Navier-Stokes equations on channel-like domains

The regularity of the global attractor of the incompressible Navier-Stokes equations for flows on two-dimensional domains is considered. It is assumed that domain $Ω$ is a channel-like domain, i.e an arbitrary bounded or unbounded domain, at first without any regularity assumption on its boundary, with the only assumption that the Poincaré inequality holds on it. The phase space $H$ for the system is the usual closure in the $L^2(Ω)^2$ norm of the space of smooth divergent-free vector-fields with compact support in $Ω.$ The corresponding space obtained as the closure with respect to the $H^1(Ω)^2$ norm is denoted by $V.$ The forcing term is assumed to belong to dual space $V'.$ It is known in this case that the global attractor exists in the phase space $H.$ It is shown in this work that the global attractor is also a compact set in $V$, and that due to the regularization effect of the equations, the solutions converge to the attractor in the norm of $V$, uniformly for initial conditions bounded in $H$. Moreover, it is shown that if the forcing term belongs to $D(A^{-s})$, for some $0<s\leq 1/2$, where $A$ is the Stokes operator, then the global attractor is compact in $D(A^{-s+1})$. If the forcing term is in $H,$ corresponding to the limit case $s=0,$ then it is further assumed that the domain is either a uniformly $\Ccal^{1,1}$ smooth domain or a bounded Lipschitz domain in order to obtain that the global attractor is compact in $D(A).$

math.AP↗

arXiv2025

Strong order-one convergence of the Euler method for random ordinary differential equations driven by semi-martingale noises

It is well known that the Euler method for a random ordinary differential equation $\mathrm{d}X_t/\mathrm{d}t = f(t, X_t, Y_t)$ driven by a stochastic process $\{Y_t\}_t$ with $θ$-Hölder sample paths is estimated to be of strong order $θ$ with respect to the time step, provided $f=f(t, x, y)$ is sufficiently regular and with suitable bounds. This order is known to increase to $1$ in some special cases. Here, it is proved that, in many more typical cases, further structures on the noise can be exploited so that the strong convergence is of order 1. In fact, we prove so for any semi-martingale noise. This includes Itô diffusion processes, point-process noises, transport-type processes with sample paths of bounded variation, and time-changed Brownian motion. The result follows from estimating the global error as an iterated integral over both large and small mesh scales, and by switching the order of integration to move the critical regularity to the large scale. The work is complemented with numerical simulations showing the optimality of the strong order 1 convergence in those cases, and with an example with fractional Brownian motion noise with Hurst parameter $0 < H < 1/2,$ which is not a semi-martingale and for which the order of convergence is $H + 1/2$, hence lower than the attained order 1 in the semi-martingale case, but still higher than the order $H$ of convergence expected from previous works.

math.PR↗

arXiv2024

On the convergence of trajectory statistical solutions

In this work, a recently introduced general framework for trajectory statistical solutions is considered, and the question of convergence of families of such solutions is addressed. Conditions for the convergence are given which rely on natural assumptions related to a priori estimates for the individual solutions of typical approximating problems. The first main result is based on the assumption that the superior limit of suitable families of compact subsets of carriers of the family of trajectory statistical solutions be included in the set of solutions of the limit problem. The second main result is a version of the former in the case in which the approximating family is associated with a well-posed system. These two results are then applied to the inviscid limit of incompressible Navier-Stokes system in two and three spatial dimensions, showing, in particular, the existence of trajectory statistical solutions to the two- and three-dimensional Euler equations, in the context of weak and dissipative solutions, respectively. Another application of the second main result is on the Galerkin approximations of statistical solutions of the three-dimensional Navier-Stokes equations.

math.AP↗

arXiv2021

Optimal minimax bounds for time and ensemble averages for the incompressible Navier-Stokes equations

Obtaining sharp estimates for quantities involved in a given model is an integral part of the modeling process. For dynamical systems whose orbits display a complicated, perhaps chaotic, behaviour, the aim is usually to estimate time or ensemble averages of given quantities. This is the case, for instance, in turbulent flows. In this work, the aim is to present a minimax optimization formula that yields optimal bounds for time and/or ensemble averages for the two- and three-dimensional Navier-Stokes equations. The results presented here are extensions to the infinite-dimensional setting of a recent result on the finite-dimensional case given by Tobasco, Goluskin, and Doering in 2017. The optimal result occurs in the form of a minimax optimization problem and does not require knowledge of the solutions, only the law of the system. The minimax optimization problem appears in the form of a maximization over a portion of the phase space of the system and a minimization over a family of auxiliary functions made of cylindrical test functionals defined on the phase space. The function to be optimized is the desired quantity plus the duality product between the law of the system and the derivative of the auxiliary function.

math.AP↗

arXiv2016

Properties of stationary statistical solutions of the three-dimensional Navier-Stokes equations

The stationary version of a modified definition of statistical solution for the three-dimensional incompressible Navier-Stokes equations introduced in a previous work is investigated. Particular types of such stationary statistical solutions and their analytical properties are addressed. Results on the support and carriers of these stationary statistical solutions are also given, showing in particular that they are supported on the weak global attractor and are carried by a more regular part of the weak global attractor containing Leray-Hopf weak solutions which are locally strong solutions. Two recurrence-type results related to these measures are also proved.

math.AP↗

arXiv2015

Abstract Framework for the Theory of Statistical Solutions

An abstract framework for the theory of statistical solutions is developed for general evolution equations, extending the theory initially developed for the three-dimensional incompressible Navier-Stokes equations. The motivation for this concept is to model the evolution of uncertainties on the initial conditions for systems which have global solutions that are not known to be unique. Both concepts of statistical solution in trajectory space and in phase space are given, and the corresponding results of existence of statistical solution for the associated initial value problems are proved. The wide applicability of the theory is illustrated with the very incompressible Navier-Stokes equations, a reaction-diffusion equation, and a nonlinear wave equation, all displaying the property of global existence of weak solutions without a known result of global uniqueness.

math.AP↗

arXiv2015

Convergence of time averages of weak solutions of the three-dimensional Navier-Stokes equations

Using the concept of stationary statistical solution, which generalizes the notion of invariant measure, it is proved that, in a suitable sense, time averages of almost every Leray-Hopf weak solution of the three-dimensional incompressible Navier-Stokes equations converge as the averaging time goes to infinity. This system of equations is not known to be globally well-posed, and the above result answers a long-standing problem, extending to this system a classical result from ergodic theory. It is also showed that, from a measure-theoretic point of view, the stationary statistical solution obtained from a generalized limit of time averages is independent of the choice of the generalized limit. Finally, any Borel subset of the phase space with positive measure with respect to a stationary statistical solution is such that for almost all initial conditions in that Borel set and for at least one Leray-Hopf weak solution starting with that initial condition, the corresponding orbit is recurrent to that Borel subset and its mean sojourn time within that Borel subset is strictly positive.

math.AP↗

arXiv2013

On the convergence of statistical solutions of the 3D Navier-Stokes-$α$ model as $α$ vanishes

In this paper statistical solutions of the 3D Navier-Stokes-$α$ model with periodic boundary condition are considered. It is proved that under certain natural conditions statistical solutions of the 3D Navier-Stokes-$α$ model converge to statistical solutions of the exact 3D Navier-Stokes equations as $α$ goes to zero. The statistical solutions considered here arise as families of time-projections of measures in suitable trajectory spaces.

math.AP↗