arXiv subjects

R. Shvydkoy

Publications and source records attributed to R. Shvydkoy.

8 recordsLinked to original sources

arXiv2026

Global well-posedness and relaxation for solutions of the Fokker-Planck-Alignment equations

In this paper we prove global existence of weak solutions, their regularization, and relaxation for large data for a broad class of Fokker-Planck-Alignment models which appear in collective dynamics. The main feature of these results, as opposed to previously known ones, is the lack of regularity or no-vacuum requirements on the initial data. With a particular application to the classical kinetic Cucker-Smale model, we demonstrate that any bounded data with finite higher moment, $f_0 \in L^1(1+ |v|^q) \cap L^\infty$, $q \geq n+4$, gives rise to a global instantly smooth solution, satisfying entropy equality and relaxing exponentially fast. The results are achieved through the use of a new thickness-based renormalization procedure, which circumvents the problem of degenerate diffusion in non-perturbative regime.

math.AP↗

arXiv2017

Eulerian dynamics with a commutator forcing II: flocking

We continue our study of one-dimensional class of Euler equations, introduced in \cite{ST2016}, driven by a forcing with a commutator structure of the form $[\aL_ϕ,u](ρ)=ϕ*(ρu)- (ϕ*ρ)u$, where $u$ is the velocity field and $ϕ$ belongs to a rather general class of \emph{influence} or interaction kernels. In this paper we quantify the large-time behavior of such systems in terms of \emph{fast flocking} for two prototypical sub-classes of kernels: bounded positive $ϕ$'s, and singular $ϕ(r) = r^{-(1+\a)}$ of order $α\in [1,2)$ associated with the action of the fractional Laplacian $\aL_ϕ=-(-\partial_{xx})^{α/2}$. Specifically, we prove fast velocity alignment as the velocity $u(\cdot,t)$ approaches a constant state, $u \to \bar{u}$, with exponentially decaying slope and curvature bounds $|u_x(\cdot,t)|_{\infty}+ |u_{xx}(\cdot,t)|_{\infty}\lesssim e^{-\d t}$. The alignment is accompanied by exponentially fast flocking of the density towards a fixed traveling state $ρ(\cdot,t) - {ρ_{\infty}}(x - \bar{u} t) \rightarrow 0$.

math.AP↗

arXiv2016

Euler equations and turbulence: analytical approach to intermittency

Physical models of intermittency in fully developed turbulence employ many phenomenological concepts such as active volume, region, eddy, energy accumulation set, etc, used to describe non-uniformity of the energy cascade. In this paper we give those notions a precise mathematical meaning in the language of the Littlewood-Paley analysis. We further use our definitions to recover scaling laws for the energy spectrum and second order structure function with proper intermittency correction.

math.AP↗

arXiv2015

On the Onsager conjecture in two dimensions

This note addresses the question of energy conservation for the 2D Euler system with an $L^p$-control on vorticity. We provide a direct argument, based on a mollification in physical space, to show that the energy of a weak solution is conserved if $ω= \nabla \times u \in L^{\frac32}$. An example of a 2D field in the class $ω\in L^{\frac32 - ε}$ for any $ε>0$, and $u\in B^{1/3}_{3,\infty}$ (Onsager critical space) is constructed with non-vanishing energy flux. This demonstrates sharpness of the kinematic argument. Finally we prove that any solution to the Euler equation produced via a vanishing viscosity limit from Navier-Stokes, with $ω\in L^p$, for $p>1$, conserves energy. This is an Onsager-supercritical condition under which the energy is still conserved, pointing to a new mechanism of energy balance restoration.

math.AP↗

arXiv2009

Ill-posedness of basic equations of fluid dynamics in Besov spaces

We give a construction of a divergence-free vector field $u_0 \in H^s \cap B^{-1}_{\infty,\infty}$, for all $s<1/2$, such that any Leray-Hopf solution to the Navier-Stokes equation starting from $u_0$ is discontinuous at $t=0$ in the metric of $B^{-1}_{\infty,\infty}$. For the Euler equation a similar result is proved in all Besov spaces $B^s_{r,\infty}$ where $s>0$ if $r>2$, and $s>n(2/r-1)$ if $1 \leq r \leq 2$.

math.AP↗

arXiv2007

On the energy equality for weak solutions of the 3D Navier-Stokes equations

We prove that the energy equality holds for weak solutions of the 3D Navier-Stokes equations in the functional class $L^3([0,T);V^{5/6})$, where $V^{5/6}$ is the domain of the fractional power of the Stokes operator $A^{5/12}$.

math.AP↗

arXiv2007

Cocycles and Mañe sequences with an application to ideal fluids

Exponential dichotomy of a strongly continuous cocycle $\bFi$ is proved to be equivalent to existence of a Mañe sequence either for $\bFi$ or for its adjoint. As a consequence we extend some of the classical results to general Banach bundles. The dynamical spectrum of a product of two cocycles, one of which is scalar, is investigated and applied to describe the essential spectrum of the Euler equation in an arbitrary spacial dimension.

math.DS↗

arXiv2007

Energy conservation and Onsager's conjecture for the Euler equations

Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in 3D conserve energy only if they have a certain minimal smoothness, (of order of 1/3 fractional derivatives) and that they dissipate energy if they are rougher. In this paper we prove that energy is conserved for velocities in the function space $B^{1/3}_{3,c(\NN)}$. We show that this space is sharp in a natural sense. We phrase the energy spectrum in terms of the Littlewood-Paley decomposition and show that the energy flux is controlled by local interactions. This locality is shown to hold also for the helicity flux; moreover, every weak solution of the Euler equations that belongs to $B^{2/3}_{3,c(\NN)}$ conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.

math.AP↗