arXiv subjects

Roman Shvydkoy

Publications and source records attributed to Roman Shvydkoy.

At least 19 recordsLinked to original sources

arXiv2026

Finite-energy weak solutions and relaxation for a compressible kinetic--fluid system with locally averaged Brinkman force

We study a kinetic--fluid system in which a Vlasov or Vlasov--Fokker--Planck equation is coupled to the compressible Navier--Stokes equations with density-dependent viscosities through a locally averaged Brinkman force. The averaging is chosen in a conservative form so that the coupled system preserves the total momentum and satisfies a natural energy-dissipation balance, while avoiding the pointwise evaluation of a possibly rough fluid velocity. The first main result is a conditional exponential relaxation estimate for sufficiently regular solutions. The estimate is proved under positive and negative density moment bounds and a Muckenhoupt $\calA_2$ condition on a power of the density, which replace uniform pointwise upper and lower bounds. The proof combines a modulated energy and hypocoercivity analysis with a compensating functional for the density fluctuation. A key ingredient is a weighted Calderón--Zygmund estimate for the associated elliptic corrector, which allows us to control the viscous contribution under the $\calA_2$ condition. The second main result concerns global finite-energy Bresch--Desjardins entropy weak solutions in admissible low-dimensional regimes. Using the additional entropy structure associated with the density-dependent viscosities, we verify the density assumptions required by the conditional relaxation theorem and obtain exponential relaxation for the weak solutions. In the diffusionless case, the particle distribution aligns toward a mono-kinetic state, while in the diffusive case the relaxation is towards a Maxwellian equilibrium.

math.AP↗

arXiv2025

Mean-Field Limits of Deterministic and Stochastic Flocking Models with Nonlinear Velocity Alignment

We study the mean-field limit for a class of agent-based models describing flocking with nonlinear velocity alignment. Each agent interacts through a communication protocol $ϕ$ and a non-linear coupling of velocities given by the power law $A(\bv) = |\bv|^{p-2}\bv$, $p > 2$. The mean-field limit is proved in two settings -- deterministic and stochastic. We then provide quantitative estimates on propagation of chaos for deterministic case in the case of the classical fat-tailed kernels, showing an improved convergence rate of the $k$-particle marginals to a solution of the corresponding Vlasov equation. The stochastic version is addressed with multiplicative noise depending on the local interaction intensity, which leads to the associated Fokker-Planck-Alignment equation. Our results extend the classical Cucker-Smale theory to the nonlinear framework which has received considerable attention in the literature recently.

math.AP↗

arXiv2025

Generic alignment conjecture for systems of Cucker-Smale type

The generic alignment conjecture states that for almost every initial data on the torus solutions to the Cucker-Smale system with a strictly local communication align to the common mean velocity. In this note we present a partial resolution of this conjecture using a statistical mechanics approach. First, the conjecture holds in full for the sticky particle model representing, formally, infinitely strong local communication. In the classical case the conjecture is proved when $N$, the number of agents, is equal to $2$. It follows from a more general result stating that for a system of any size for almost every data at least two agents align. The analysis is extended to the open space $\R^n$ in the presence of confinement and potential interaction forces. In particular, it is shown that almost every non-oscillatory pair of solutions aligns and aggregates in the potential well.

math.DS↗

arXiv2025

Unconditional alignment of solutions to the Fokker-Planck-Navier-Stokes system with locally averaged Brinkman force

We study a coupled Fokker-Planck--Navier-Stokes (FPNS) system modeling the dynamics of interacting particles suspended in a viscous incompressible fluid, where the coupling occurs through a locally averaged Brinkman drag force. Our main result is the unconditional alignment and synchronization of particle and fluid velocities for all weak solutions, in any dimension, on the periodic domain. The proof leverages a new entropy method and a hypocoercivity framework, which together yield quantitative decay estimates and prevent density concentration, a key obstacle in previous analyses. Our approach applies to a broad class of nonlocal alignment protocols, including the Cucker-Smale model. We also prove develop well-posedness theory for global weak solutions with hypoelliptic regularization in any dimension, and global strong solutions in two dimensions.

math.AP↗

arXiv2025

Modulation of the Monokinetic Limit for Models of Collective Dynamics

In this work, we perform modulation analysis of monokinetic limits from the kinetic Cucker- Smale model to the pressureless Euler alignment system. Two regimes are considered -- a strong Fokker- Planck force with vanishing noise and Knudsen number, and a pure noiseless Vlasov scheme. In the former case, we demonstrate convergence of the modulated profile to the standard Gaussian distribution, while in the latter case, the distribution converges to a profile satisfying an explicit transport equation along limiting characteristics.

math.AP↗

arXiv2025

Microscopic, mesoscopic, and macroscopic descriptions of the Euler alignment system with adaptive communication strength

This is a continuation of our previous joint work on the $\st$-model in[\textit{Well-posedness and long time behavior of the Euler Alignment System with adaptive communication strength}, accepted at the Abel Symposium Proceedings, also arXiv:2310.00269, 2023]. The $\st$-model, introduced by the first author in [\textit{Environmental averaging}. EMS Surv. Math. Sci., 11 (2024), no. 2, 277413],is an alignment model with the property that the strength of the alignment force, $\st$, is transported along an averaged velocity field. The transport of the strength is designed so that it admits an $e$-quantity, $e = \partial_x u + \st$, which controls regularity in 1D similarly to the classical Cucker-Smale case. The utility of the $\st$-model is that it has the versatility to behave qualitatively like the Motsch-Tadmor model, for which global regularity theory is not known. This paper aims to put the $\st$-model on firmer physical grounds by formulating and justifying the microscopic and mesoscopic descriptions from which it arises. A distinctive feature of the microscopic system is that it is a discrete-continuous system: the position and velocity of the particles are discrete objects, while the strength is an active continuum scalar function. We establish a rigorous passage from the microscopic to the mesoscopic description via the Mean Field Limit and from the mesoscopic to the macroscopic description in the monokinetic and Maxwellian limiting regimes. We also address the long-time behavior of the kinetic Fokker-Planck-Alignment equation by establishing the relaxation to the Maxwellian in 1D when the velocity averaging is given by the Favre filtration. As a supplement to the numerical results already presented in our previous work, we provide additional numerical evidence, via a particle simulation, that the $\st$-model behaves qualitatively like the Motsch-Tadmor model.

math.AP↗

arXiv2024

On the inviscid instability of the 2D Taylor-Green vortex

We consider Euler flows on two-dimensional (2D) periodic domain and are interested in the stability, both linear and nonlinear, of a simple equilibrium given by the 2D Taylor-Green vortex. As the first main result, numerical evidence is provided for the fact that such flows possess unstable eigenvalues embedded in the band of the essential spectrum of the linearized operator. However, the unstable eigenfunction is discontinuous at the hyperbolic stagnation points of the base flow and its regularity is consistent with the prediction of Lin (2004). This eigenfunction gives rise to an exponential transient growth with the rate given by the real part of the eigenvalue followed by passage to a nonlinear instability. As the second main result, we illustrate a fundamentally different, non-modal, growth mechanism involving a continuous family of uncorrelated functions, instead of an eigenfunction of the linearized operator. Constructed by solving a suitable PDE optimization problem, the resulting flows saturate the known estimates on the growth of the semigroup related to the essential spectrum of the linearized Euler operator as the numerical resolution is refined. These findings are contrasted with the results of earlier studies of a similar problem conducted in a slightly viscous setting where only the modal growth of instabilities was observed. This highlights the special stability properties of equilibria in inviscid flows.

physics.flu-dyn↗

arXiv2024

Well-posedness and Long Time Behavior of the Euler Alignment System with Adaptive Communication Strength

We study a new flocking model which has the versatility to capture the physically realistic qualitative behavior of the Motsch-Tadmor model, while also retaining the entropy law, which lends to a similar 1D global well-posedness analysis to the Cucker-Smale model. This is an improvement to the situation in the Cucker-Smale case, which may display the physically unrealistic behavior that large flocks overpower the dynamics of small, far away flocks; and it is an improvement in the situation in the Motsch-Tadmor case, where 1D global well-posedness is not known. The new model was proposed in arXiv:2211.00117v3 and has a similar structure to the Cucker-Smale and Motsch-Tadmor hydrodynamic systems, but with a new feature: the communication strength is not fixed, but evolves in time according to its own transport equation along the Favre-filtered velocity field. This transport of the communication strength is precisely what preserves the entropy law. A variety of phenomenological behavior can be obtained from various choices of the initial communication strength, including the aforementioned Motsch-Tadmor-like behavior. We develop the general well-posedness theory for the new model and study the long time behavior -- including alignment, strong flocking in 1D, and entropy estimates to estimate the distribution of the limiting flock, all of which extend the classical results of the Cucker-Smale case. In addition, we provide numerical evidence to show the similar qualitative behavior

math.AP↗

arXiv2024

Environmental Averaging

Many classical examples of models of self-organized dynamics, including the Cucker-Smale, Motsch-Tadmor, multi-species, and several others, include an alignment force that is based upon density-weighted averaging protocol. Those protocols can be viewed as special cases of `environmental averaging'. In this paper we formalize this concept and introduce a unified framework for systematic analysis of alignment models. A series of studies are presented including the mean-field limit in deterministic and stochastic settings, hydrodynamic limits in the monokinetic and Maxwellian regimes, hypocoercivity and global relaxation for dissipative kinetic models, several general alignment results based on chain connectivity and spectral gap analysis. These studies cover many of the known results and reveal new ones, which include asymptotic alignment criteria based on connectivity conditions, new estimates on the spectral gap of the alignment force that do not rely on the upper bound of the macroscopic density, uniform gain of positivity for solutions of the Fokker-Planck-Alignment model based on smooth environmental averaging. As a consequence, we establish unconditional relaxation result for global solutions to the Fokker-Planck-Alignment model, which presents a substantial improvement over previously known perturbative results.

math.AP↗

arXiv2022

Continuous model of opinion dynamics with convictions

In this note we study a new kinetic model of opinion dynamics. The model incorporates two forces -- alignment of opinions under all-to-all communication driving the system to a consensus, and Rayleigh type friction force that drives each `player' to its fixed conviction value. The balance between these forces creates a non-trivial limiting outcome. We establish existence of a global mono-opinion state, whereby any initial distribution of opinions for each conviction value aggregates to the Dirac measure concentrated on a single opinion. We identify several cases where such a state is unique and depends continuously on the initial distribution of convictions. Several regularity properties of the limiting distribution of opinions are presented.

math.AP↗

arXiv2022

Volumetric theory of intermittency in fully developed turbulence

This study introduces a new family of volumetric flatness factors which give a rigorous parametric description of the phenomenon of intermittency in fully developed turbulent flows. These quantities gather information about the most "active" part of a velocity field at each scale $\ell$, and allows one to define a dimension function $p \to D_p$ that recovers intermittency correction to the structure exponents $ζ_p$ in an explicit way. In particular, the predictions of the Frisch-Parisi multifractal formalism can be recovered in a systematic and rigorous way. Within this framework we identify active regions that carry the most energetic part of a velocity field at a given scale $\ell$. A threshold for what constitutes to be active is defined explicitly. Active regions have proven to be experimentally observable in our previous joint work \cite{Ph-paper}, and shown to capture concentration of the energy cascade as $\ell \to 0$, in \cite{CS2014}. We present several examples of fields which exhibit arbitrary multifractal spectrum within theoretically permitted limitations. At the same time we demonstrate with the use of a probabilistic argument that a random field is expected to produce the classical K41 spectrum in the limit $\ell\to 0$. Intermittent deviations from K41 theory are estimated at any finite scale also. Lastly, we present a detailed information-theoretic analysis of the introduced objects. In particular, we quantify concentration of a given source-field in terms of the volume factors, thresholds, and active regions.

math.AP↗

arXiv2021

Propagation of Chaos for the Cucker-Smale Systems under heavy tail communication

In this work we study propagation of chaos for solutions of the Liouville equation for the classical discrete Cucker-Smale system. Assuming that the communication kernel satisfies the heavy tail condition -- known to be necessary to induce exponential alignment -- we obtain a linear in time convergence rate of the $k$-th marginals $f^{(k)}$ to the product of $k$ solutions of the corresponding Vlasov-Alignment equation, $f^{\otimes k}$. Specifically, the following estimate holds in terms of Wasserstein-2 metric \begin{equation}\label{e:abst} \mathcal{W}_2(f^{(k)}_t,f^{\otimes k}_t) \leq C k^{1/2} \min\left\{ 1, \frac{t}{\sqrt{N}} \right\}. \end{equation} For systems with the Rayleigh-type friction and self-propulsion force, we obtain a similar result for sectorial solutions. Such solutions are known to align exponentially fast via the method of Grassmannian reduction, \cite{LRS-friction}. We recast the method in the kinetic settings and show that the bound \eqref{e:abst} persists but with the quadratic dependence on time. In both the forceless and forced cases, the result represents an improvement over the exponential bounds established earlier in the work of Natalini and Paul, \cite{NP2021-orig}, although those bounds hold for general kernels. The main message of our work is that flocking dynamics improves the rate considerably.

math.AP↗

arXiv2021

Global hypocoercivity of kinetic Fokker-Planck-Alignment equations

In this note we establish hypocoercivity and exponential relaxation to the Maxwellian for a class of kinetic Fokker-Planck-Alignment equations arising in the studies of collective behavior. Unlike previously known results in this direction that focus on convergence near Maxwellian, our result is global for hydrodynamically dense flocks, which has several consequences. In particular, if communication is long-range, the convergence is unconditional. If communication is local then all nearly aligned flocks quantified by smallness of the Fisher information relax to the Maxwellian. In the latter case the class of initial data is stable under the vanishing noise limit, i.e.\ it reduces to a non-trivial and natural class of traveling wave solutions to the noiseless Vlasov-Alignment equation. The main novelty in our approach is the adaptation of a mollified Favre filtration of the macroscopic momentum into the communication protocol. Such filtration has been used previously in large eddy simulations of compressible turbulence and its new variant appeared in the proof of the Onsager conjecture for inhomogeneous Navier-Stokes system. A rigorous treatment of well-posedness for smooth solutions is provided. Lastly, we prove that in the limit of strong noise and local alignment solutions to the Fokker-Planck-Alignment equation Maxwellialize to solutions of the macroscopic hydrodynamic system with the isothermal pressure.

math.AP↗

arXiv2021

Global solutions to multi-dimensional topological Euler alignment systems

We present a systematic approach to regularity theory of the multi-dimensional Euler alignment systems with topological diffusion introduced in \cite{STtopo}. While these systems exhibit flocking behavior emerging from purely local communication, bearing direct relevance to empirical field studies, global and even local well-posedness has proved to be a major challenge in multi-dimensional settings due to the presence of topological effects. In this paper we reveal two important classes of global smooth solutions -- parallel shear flocks with incompressible velocity and stationary density profile, and nearly aligned flocks with close to constant velocity field but arbitrary density distribution. Existence of such classes is established via an efficient continuation criterion requiring control only on the Lipschitz norm of state quantities, which makes it accessible to the applications of fractional parabolic theory. The criterion presents a major improvement over the existing result of \cite{RS2020}, and is proved with the use of quartic paraproduct estimates.

math.AP↗

arXiv2020

Grassmannian reduction of Cucker-Smale systems and dynamical opinion games

In this note we study a new class of alignment models with self-propulsion and Rayleigh-type friction forces, which describes the collective behavior of agents with individual characteristic parameters. We describe the long time dynamics via a new method which allows to reduce analysis from the multidimensional system to a simpler family of two-dimensional systems parametrized by a proper Grassmannian. With this method we demonstrate exponential alignment for a large (and sharp) class of initial velocity configurations confined to a sector of opening less than $π$. In the case when characteristic parameters remain frozen, the system governs dynamics of opinions for a set of players with constant convictions. Viewed as a dynamical non-cooperative game, the system is shown to possess a unique stable Nash equilibrium, which represents a settlement of opinions most agreeable to all agents. Such an agreement is furthermore shown to be a global attractor for any set of initial opinions.

math.AP↗

arXiv2020

Topologically-based fractional diffusion and emergent dynamics with short-range interactions

We introduce a new class of models for emergent dynamics. It is based on a new communication protocol which incorporates two main features: short-range kernels which restrict the communication to local geometric balls, and anisotropic communication kernels, adapted to the local density in these balls, which form topological neighborhoods. We prove flocking behavior -- the emergence of global alignment for regular, non-vacuous solutions of the $n$-dimensional models based on short-range topological communication. Moreover, global regularity (and hence unconditional flocking) of the one-dimensional model is proved via an application of a De Giorgi-type method. To handle the non-symmetric singular kernels that arise with our topological communication, we develop a new analysis for local fractional elliptic operators, interesting for its own sake, encountered in the construction of our class of models.

math.AP↗

arXiv2020

Geometric Structure of Mass Concentration Sets for Pressureless Euler Alignment Systems

We study the limiting dynamics of the Euler Alignment system with a smooth, heavy-tailed interaction kernel $ϕ$ and unidirectional velocity $\mathbf{u} = (u, 0, \ldots, 0)$. We demonstrate a striking correspondence between the entropy function $e_0 = \partial_1 u_0 + ϕ*ρ_0$ and the limiting 'concentration set', i.e., the support of the singular part of the limiting density measure. In a typical scenario, a flock experiences aggregation toward a union of $C^1$ hypersurfaces: the image of the zero set of $e_0$ under the limiting flow map. This correspondence also allows us to make statements about the fine properties associated to the limiting dynamics, including a sharp upper bound on the dimension of the concentration set, depending only on the smoothness of $e_0$. In order to facilitate and contextualize our analysis of the limiting density measure, we also include an expository discussion of the wellposedness, flocking, and stability of the Euler Alignment system, most of which is new.

math.AP↗

arXiv2020

Multi-flocks: emergent dynamics in systems with multi-scale collective behavior

We study the multi-scale description of large-time collective behavior of agents driven by alignment. The resulting multi-flock dynamics arises naturally with realistic initial configurations consisting of multiple spatial scaling, which in turn peak at different time scales. We derive a `master-equation' which describes a complex multi-flock congregations governed by two ingredients: (i) a fast inner-flock communication; and (ii) a slow(-er) inter-flock communication. The latter is driven by macroscopic observables which feature the up-scaling of the problem. We extend the current mono-flock theory, proving a series of results which describe rates of multi-flocking with natural dependencies on communication strengths. Both agent-based, kinetic, and hydrodynamic descriptions are considered, with particular emphasis placed on the discrete and macroscopic descriptions.

math.AP↗