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Roman Shvydkoy

Publications and source records attributed to Roman Shvydkoy.

At least 37 records · Page 2Linked to original sources
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.

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arXiv2020

Unidirectional flocks in hydrodynamic Euler Alignment system II: Singular models

In this note we continue our study of unidirectional solutions to hydrodynamic Euler alignment systems with strongly singular communication kernels $ϕ(x):=|x|^{-(n+α)}$ for $α\in(0,2)$. The solutions describe unidirectional parallel motion of agents governing multi-dimensional collective behavior of flocks. Here, we consider the range $1<α<2$ and establish the global regularity of smooth solutions, together with a full description of their long time dynamics. Specifically, we develop the flocking theory of these solutions and show long time convergence to traveling wave with rapidly aligned velocity field.

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arXiv2019

Existence and stability of unidirectional flocks in hydrodynamic Euler Alignment systems

In this note we reveal new classes of solutions to hydrodynamic Euler alignment systems governing collective behavior of flocks. The solutions describe unidirectional parallel motion of agents, and are globally well-posed in multi-dimensional settings subject to a threshold condition similar to the one dimensional case. We develop the flocking and stability theory of these solutions and show long time convergence to traveling wave with rapidly aligned velocity field. In the context of multi-scale models introduced in \cite{ST-multi} our solutions can be superimposed into Mikado formations -- clusters of unidirectional flocks pointing in various directions. Such formations exhibit multiscale alignment phenomena and resemble realistic behavior of interacting large flocks.

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arXiv2019

Local well-posedness of the topological Euler alignment models of collective behavior

In this paper we address the problem of well-posedness of multi-dimensional topological Euler-alignment models introduced in \cite{ST-topo}. The main result demonstrates local existence and uniqueness of classical solutions in class $(ρ,u) \in H^{m+α} \times H^{m+1}$ on the periodic domain $\mathbb{T}^n$, where $0<α<2$ is the order of singularity of the topological communication kernel $ϕ(x,y)$, and $m = m(n,α)$ is large. Our approach is based on new sharp coercivity estimates for the topological alignment operator \[ \mathcal{L}_ϕf(x) = \int_{\mathbb{T}^n} ϕ(x,y) (f(y) - f(x) ) dy, \] which render proper a priori estimates and help stabilize viscous approximation of the system. In dimension 1, this result, in conjunction with the technique developed in \cite{ST-topo} gives global well-posendess in the natural space of data mentioned above.

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arXiv2019

On the Structure of Limiting Flocks in Hydrodynamic Euler Alignment Models

The goal of this note is to study limiting behavior of a self-organized continuous flock evolving according to the 1D hydrodynamic Euler Alignment model. We provide a series of quantitative estimates that show how far the density of the limiting flock is from a uniform distribution. The key quantity that controls density distortion is the entropy $\mathcal{H} = \int ρ\log ρ\,\mbox{d}x$, and the measure of deviation from uniformity is given by a well-known conserved quantity $e = u' + \mathcal{L}_ψρ$, where $u$ is velocity and $\mathcal{L}_ψ$ is the communication operator with kernel $ψ$. The cases of Lipschitz, singular geometric, and topological kernels are covered in the study.

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arXiv2019

Entropy Hierarchies for equations of compressible fluids and self-organized dynamics

We develop a method of obtaining a hierarchy of new higher-order entropies in the context of compressible models with local and non-local diffusion and isentropic pressure. The local viscosity is allowed to degenerate as the density approaches vacuum. The method provides a tool to propagate initial regularity of classical solutions provided no vacuum has formed and serves as an alternative to the classical energy method. We obtain a series of global well-posedness results for state laws in previously uncovered cases including $p(ρ) = c_p ρ$. As an application we prove global well-posedness of collective behavior models with pressure arising from agent-based Cucker-Smale system.

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arXiv2019

On Cucker-Smale dynamical systems with degenerate communication

This note introduces a new method for establishing alignment in systems of collective behavior with degenerate communication protocol. The communication protocol consists of a kernel defining interaction between pairs of agents. Degeneracy presumes that the kernel vanishes in a region, which creates a zone of indifference. A motivating example is the case of a local kernel. Lapses in communication create a lack of coercivity in the energy estimates. Our approach is the construction of a corrector functional that compensates for this lack of coercivity. We obtain a series of new results: unconditional alignment for systems on $\mathbb{R}^d$ with degeneracy at close range and fat tail in the long range, and for systems on the circle with purely local kernels. The results are proved in the context of both the agent based model and its hydrodynamic counterpart (Euler alignment model). The method covers bounded and singular communication kernels.

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arXiv2018

The Energy Measure for the Euler and Navier-Stokes Equations

The potential failure of energy equality for a solution $u$ of the Euler or Navier-Stokes equations can be quantified using a so-called `energy measure': the weak-$*$ limit of the measures $|u(t)|^2\,\mbox{d}x$ as $t$ approaches the first possible blowup time. We show that membership of $u$ in certain (weak or strong) $L^q L^p$ classes gives a uniform lower bound on the lower local dimension of $\mathcal{E}$; more precisely, it implies uniform boundedness of a certain upper $s$-density of $\mathcal{E}$. We also define and give lower bounds on the `concentration dimension' associated to $\mathcal{E}$, which is the Hausdorff dimension of the smallest set on which energy can concentrate. Both the lower local dimension and the concentration dimension of $\mathcal{E}$ measure the departure from energy equality. As an application of our estimates, we prove that any solution to the $3$-dimensional Navier-Stokes Equations which is Type-I in time must satisfy the energy equality at the first blowup time.

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arXiv2018

Global existence and stability of nearly aligned flocks

We study regularity of a hydrodynamic singular model of collective behavior introduced in \cite{ST1}. In this note we address the question of global well-posedness in multi-dimensional settings. It is shown that any initial data $(u,ρ)$ with small velocity variations $|u(x) - u(y)| < \e$ relative to its higher order norms, gives rise to a unique global regular solution which aligns and flocks exponentially fast. Moreover, we prove that the limiting flocks are stable.

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arXiv2018

Conditions Implying Energy Equality for Weak Solutions of the Navier--Stokes Equations

When a Leray--Hopf weak solution to the NSE has a singularity set $S$ of dimension $d$ less than $3$---for example, a suitable weak solution---we find a family of new $L^q L^p$ conditions that guarantee validity of the energy equality. Our conditions surpass the classical Lions--Ladyženskaja $L^4 L^4$ result in the case $d<1$. Additionally, we establish energy equality in certain cases of Type-I blowup. The results are also extended to the NSE with fractional power of the Laplacian below $1$.

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arXiv2018

On the Rayleigh-Taylor instability in presence of a background shear

In this note we revisit the classical subject of the Rayleigh-Taylor instability in presence of an incompressible background shear flow. We derive a formula for the essential spectral radius of the evolution group generated by the linearization near the steady state and reveal that the velocity variations neutralize shortwave instabilities. The formula is a direct generalization of the result of H. J. Hwang and Y. Guo in the hydrostatic case \cite{HG}. Furthermore, we construct a class of steady states which posses unstable discrete spectrum with neutral essential spectrum. The technique involves the WKB analysis of the evolution equation and contains novel compactness criterion for pseudo-differential operators on unbounded domains.

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arXiv2017

Eulerian dynamics with a commutator forcing III. Fractional diffusion of order $0<α<1$

We continue our study of hydrodynamic models of self-organized evolution of agents with singular interaction kernel $ϕ(x) = |x|^{-(1+α)}$. Following our works \cite{ST2017a,ST2017b} which focused on the range $1\leq α<2$, and Do et. al. \cite{DKRT2017} which covered the range $0<α<1$, in this paper we revisit the latter case and give a short(-er) proof of global in time existence of smooth solutions, together with a full description of their long time dynamics. Specifically, we prove that starting from any initial condition in $(ρ_0,u_0) \in H^{2+α}\times H^3$, the solution approaches exponentially fast to a flocking state solution consisting of a wave $\barρ=ρ_\infty(x-t\bar{u}))$ traveling with a constant velocity determined by the conserved average velocity $\bar{u}$. The convergence is accompanied by exponential decay of all higher order derivatives of $u$.

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arXiv2016

Eulerian dynamics with a commutator forcing

We study a general class of Euler equations driven by a forcing with a \emph{commutator structure} of the form $[\mathcal{L},\mathbf{u}](ρ)=\mathcal{L}(ρ\mathbf{u})- \mathcal{L}(ρ)\mathbf{u}$, where $\mathbf{u}$ is the velocity field and $\mathcal{L}$ is the "action" which belongs to a rather general class of translation invariant operators. Such systems arise, for example, as the hydrodynamic description of velocity alignment, where action involves convolutions with bounded, positive influence kernels, $\mathcal{L}_ϕ(f)=ϕ*f$. Our interest lies with a much larger class of $\mathcal{L}$'s which are neither bounded nor positive. In this paper we develop a global regularity theory in the one-dimensional setting, considering three prototypical sub-classes of actions. We prove global regularity for \emph{bounded} $ϕ$'s which otherwise are allowed to change sign. Here we derive sharp critical thresholds such that sub-critical initial data $(ρ_0,u_0)$ give rise to global smooth solutions. Next, we study \emph{singular} actions associated with $\mathcal{L}=-(-\partial_{xx})^{α/2}$, which embed the fractional Burgers' equation of order $α$. We prove global regularity for $α\in [1,2)$. Interestingly, the singularity of the fractional kernel $|x|^{-(n+α)}$, avoids an initial threshold restriction. Global regularity of the critical endpoint $α=1$ follows with double-exponential $W^{1,\infty}$-bounds. Finally, for the other endpoint $α=2$, we prove the global regularity of the Navier-Stokes equations with density-dependent viscosity associated with the \emph{local} $\mathcal{L}=Δ$.

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arXiv2016

Addendum: 2D homogeneous solutions to the Euler equation

In this addendum note we fill in the gap left in \cite{ls} in the description of 2D homogeneous solutions to the stationary Euler system with the help of the results of \cite{sd}. This gives a complete classification of all solutions. The note includes updated classification tables, and a reverse application to the results of \cite{sd}.

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arXiv2016

The Energy Balance Relation for Weak solutions of the Density-Dependent Navier-Stokes Equations

We consider the incompressible inhomogeneous Navier-Stokes equations with constant viscosity coefficient and density which is bounded and bounded away from zero. We show that the energy balance relation for this system holds for weak solutions if the velocity, density, and pressure belong to a range Besov spaces of smoothness $1/3$. A density-dependent version of the classical Kármán-Howarth-Monin relation is derived.

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arXiv2015

Homogeneous solutions to the 3D Euler system

We study stationary homogeneous solutions to the 3D Euler equation. The problem is motivated be recent exclusions of self-similar blowup for Euler and its relation to Onsager conjecture and intermittency. We reveal several new classes of solutions and prove rigidity properties in specific categories of genuinely 3D solutions. In particular, irrotational solutions are characterized by vanishing of the Bernoulli function; and tangential flows are necessarily 2D axisymmetric pure rotations. In several cases solutions are excluded altogether. The arguments reveal geodesic features of the Euler flow on the sphere. We further show that in the case when homogeneity corresponds to the Onsager-critical state, the anomalous energy flux at the singularity vanishes, which is suggestive of absence of extreme $0$-dimensional intermittencies in dissipative flows.

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arXiv2015

Global regularity for 2D Muskat equations with finite slope

We consider the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy's law. We establish that as long as the slope of the interface between the two fluids remains bounded and uniformly continuous, the solution remains regular. The proofs exploit the nonlocal nonlinear parabolic nature of the equations through a series of nonlinear lower bounds for nonlocal operators. These are used to deduce that as long as the slope of the interface remains uniformly bounded, the curvature remains bounded. The nonlinear bounds then allow us to obtain local existence for arbitrarily large initial data in the class $W^{2,p}$, $1<p\leq \infty$. We provide furthermore a global regularity result for small initial data: if the initial slope of the interface is sufficiently small, there exists a unique solution for all time.

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