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Roman Shvydkoy

Publications and source records attributed to Roman Shvydkoy.

53 records · Page 3Linked to original sources
arXiv2015

Global Well-Posedness Of A Non-Local Burgers Equation: The Periodic Case

This paper is concerned with the study of a non-local Burgers equation for positive bounded periodic initial data. The equation reads $$ u_t - u |\nabla| u + |\nabla|(u^2) = 0. $$ We construct global classical solutions starting from smooth positive data, and global weak solutions starting from data in $L^\infty$. We show that any weak solution is instantaneously regularized into $C^\infty$. We also describe the long-time behavior of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations.

math.AP↗
arXiv2014

On the energy behavior of locally self-similar blowup for the Euler equation

In this note we study locally self-similar blow up for the Euler equation. The main result states that under a mild $L^p$-growth assumption on the profile $v$, namely, $\int_{|y| \sim L} |v|^p dy \lesssim L^{\g}$ for some $\g <p-2$, the self-similar solution carries a positive amount of energy up to the time of blow-up $T$, namely, $\int_{|y| \sim L} |v|^2 dy \sim L^{N-2\a}$. The result implies and extends several previously known exclusion criteria. It also supports a general conjecture relating fractal local dimensions of the energy measure with the rate of velocity growth at the time of possible blowup.

math.AP↗
arXiv2014

2D homogeneous solutions to the Euler equation

In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form $u = \nabla^\perp Ψ$, $Ψ(r,θ) = r^λ ψ(θ)$, for $λ>0$, we show that only trivial solutions exist in the range $0<λ<1/2$, i.e. parallel shear and rotational flows. In other cases many new solutions are exhibited that have hyperbolic, parabolic and elliptic structure of streamlines. In particular, for $λ>9/2$ the number of different non-trivial elliptic solutions is equal to the cardinality of the set $(2,\sqrt{2λ}) \cap \mathbb{N}$. The case $λ= 2/3$ is relevant to Onsager's conjecture. We underline the reasons why no anomalous dissipation of energy occurs for such solutions despite their critical Besov regularity 1/3.

math.AP↗
arXiv2012

A study of energy concentration and drain in incompressible fluids

In this paper we examine two opposite scenarios of energy behavior for solutions of the Euler equation. We show that if $u$ is a regular solution on a time interval $[0,T)$ and if $u \in L^rL^\infty$ for some $r\geq \frac{2}{N}+1$, where $N$ is the dimension of the fluid, then the energy at the time $T$ cannot concentrate on a set of Hausdorff dimension samller than $N - \frac{2}{r-1}$. The same holds for solutions of the three-dimensional Navier-Stokes equation in the range $5/3 0$. The results are applied to find new exclusions of locally self-similar blow-up in cases not covered previously in the literature.

math.AP↗
arXiv2012

On formation of a locally self-similar collapse in the incompressible Euler equations

The paper addresses the question of existence of a locally self-similar blow-up for the incompressible Euler equations. Several exclusion results are proved based on the $L^p$-condition for velocity or vorticity and for a range of scaling exponents. In particular, in $N$ dimensions if in self-similar variables $u \in L^p$ and $u \sim \frac{1}{t^{\a/(1+\a)}}$, then the blow-up does not occur provided $\a >N/2$ or $-1<\a\leq N/p$. This includes the $L^3$ case natural for the Navier-Stokes equations. For $\a = N/2$ we exclude profiles with an asymptotic power bounds of the form $ |y|^{-N-1+\d} \lesssim |u(y)| \lesssim |y|^{1-\d}$. Homogeneous near infinity solutions are eliminated as well except when homogeneity is scaling invariant.

math.AP↗
arXiv2011

A continuous model for turbulent energy cascade

In this paper we introduce a new PDE model in frequency space for the inertial energy cascade that reproduces the classical scaling laws of Kolmogorov's theory of turbulence. Our point of view is based upon studying the energy flux through a continuous range of scales rather than the discrete set of dyadic scales. The resulting model is a variant of Burgers equation on the half line with a boundary condition which represents a constant energy input at integral scales. The viscous dissipation is modeled via a damping term. We show existence of a unique stationary solution, both in the viscous and inviscid cases, which replicates the classical dissipation anomaly in the limit of vanishing viscosity. A survey of recent developments in the deterministic approach to the laws of turbulence, and in particular, to Onsager's conjecture is given.

math.AP↗
arXiv2011

A unified approach to regularity problems for the 3D Navier-Stokes and Euler equations: the use of Kolmogorov's dissipation range

Motivated by Kolmogorov's theory of turbulence we present a unified approach to the regularity problems for the 3D Navier-Stokes and Euler equations. We introduce a dissipation wavenumber $Λ(t)$ that separates low modes where the Euler dynamics is predominant from the high modes where the viscous forces take over. Then using an indifferent to the viscosity technique we obtain a new regularity criterion which is weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case, and reduces to the Beale-Kato-Majda criterion in the inviscid case. In the viscous case we also we prove that Leray-Hopf solutions are regular provided $Λ\in L^{5/2}$, which improves our previous $Λ\in L^\infty$ condition. We also show that $Λ\in L^1$ for all Leray-Hopf solutions. Finally, we prove that Leray-Hopf solutions are regular when the time-averaged spatial intermittency is small, i.e., close to Kolmogorov's regime.

math.AP↗
arXiv2010

Convex integration for a class of active scalar equations

We show that a general class of active scalar equations, including porous media and certain magnetostrophic turbulence models, admit non-unique weak solutions in the class of bounded functions. The proof is based upon the method of convex integration recently implemented for equations of fluid dynamics in [2,3].

math.AP↗
arXiv2010

Continuous spectrum of the 3D Euler equation is a solid annulus

In this note we give a description of the continuous spectrum of the linearized Euler equations in three dimensions. Namely, for all but countably many times $t\in \R$, the continuous spectrum of the evolution operator $G_t$ is given by a solid annulus with radii $e^{tμ}$ and $e^{t M}$, where $μ$ and $M$ are the smallest and largest, respectively, Lyapunov exponents of the corresponding bicharacteristic-amplitude system of ODEs.

math.AP↗
arXiv2008

On the energy of inviscid singular flows

It is known that the energy of a weak solution to the Euler equation is conserved if it is slightly more regular than the Besov space $B^{1/3}_{3,\infty}$. When the singular set of the solution is (or belongs to) a smooth manifold, we derive various $L^p$-space regularity criteria dimensionally equivalent to the critical one. In particular, if the singular set is a hypersurface the energy of $u$ is conserved provided the one sided non-tangential limits to the surface exist and the non-tangential maximal function is $L^3$ integrable, while the maximal function of the pressure is $L^{3/2}$ integrable. The results directly apply to prove energy conservation of the classical vortex sheets in both 2D and 3D at least in those cases where the energy is finite.

math.AP↗
arXiv2007

On the regularity of weak solutions of the 3D Navier-Stokes equations in $B^{-1}_{\infty,\infty}$

We show that if a Leray-Hopf solution $u$ to the 3D Navier-Stokes equation belongs to $C((0,T]; B^{-1}_{\infty,\infty})$ or its jumps in the $B^{-1}_{\infty,\infty}$-norm do not exceed a constant multiple of viscosity, then $u$ is regular on $(0,T]$. Our method uses frequency local estimates on the nonlinear term, and yields an extension of the classical Ladyzhenskaya-Prodi-Serrin criterion.

math.AP↗
arXiv2007

The essential spectrum of advective equations

A description of the essential spectrum is given for a general class of linear advective PDE with pseudodifferential bounded perturbation. We prove that every point in the Sacker-Sell spectrum of the corresponding bicharacteristic-amplitude system exponentiates into the spectrum of PDE. Exact spectral pictures are found in various cases. Applications to instability are presented.

math-ph↗
arXiv2005

The unstable spectrum of the Navier-Stokes operator in the limit of vanishing viscosity

A general class of linear advective PDEs, whose leading order term is of viscous dissipative type, is considered. It is proved that beyond the limit of the essential spectrum of the underlying inviscid operator, the eigenvalues of the viscous operator, in the limit of vanishing viscosity, converge precisely to those of the inviscid operator. The general class of PDEs includes the equations of incompressible fluid dynamics. Hence eigenvalues of the Navier-Stokes operator converge in the inviscid limit to the eigenvalues of the Euler operator beyond the essential spectrum.

math.AP↗
arXiv2005

Nonlinear instability for the Navier-Stokes equations

It is proved, using a bootstrap argument, that linear instability implies nonlinear instability for the incompressible Navier-Stokes equations in $L^p$ for all $p \in (1,\infty)$ and any finite or infinite domain in any dimension $n$.

math.AP↗
arXiv2003

Essential spectrum of the linearized 2D Euler equation and Lyapunov-Oseledets exponents

The linear stability of a steady state solution of 2D Euler equations of an ideal fluid is being studied. We give an explicit geometric construction of approximate eigenfunctions for the linearized Euler operator $L$ in vorticity form acting on Sobolev spaces on two dimensional torus. We show that each nonzero Lyapunov-Oseledets exponent for the flow induced by the steady state contributes a vertical line to the essential spectrum of $L$. Also, we compute the spectral and growth bounds for the group generated by $L$ via the maximal Lyapunov-Oseledets exponent. When the flow has arbitrarily long orbits, we show that the essential spectrum of $L$ on $L_2$ is the imaginary

math-ph↗