arXiv subjects

A. Cheskidov

Publications and source records attributed to A. Cheskidov.

6 recordsLinked to original sources

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

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↗

arXiv2001

Turbulent boundary layer equations

We study a boundary layer problem for the Navier-Stokes-alpha model obtaining a generalization of the Prandtl equations conjectured to represent the averaged flow in a turbulent boundary layer. We solve the equations for the semi-infinite plate, both theoretically and numerically. The latter solutions agree with some experimental data in the turbulent boundary layer.

nlin.CD↗