Searcharxiv⌕ Search

arXiv subjects

Vladimir Sverak

Publications and source records attributed to Vladimir Sverak.

At least 19 recordsLinked to original sources
arXiv2026

The three-vortex system: Hopf fibration, symplectic reduction, and near-collisions

We study the dynamics of collisions and near-collisions in the system of three point vortices in the plane. The dynamics is known to be completely integrable and can be reduced to the study of Hamiltonian systems on two-dimensional symplectic leaves. This still allows for interesting behavior of the solutions, various aspects of which have been investigated in a large body of work by many authors. Our focus will be on collisions and the possibility of their regularization through perturbations of suitable parameters of the system. We will use the reduced coordinate given by $ζ=(z_2-z_1)/(z_3-z_1)$, where $z_1,z_2,z_3$ are the complex numbers representing the positions of the vortices. This choice seems very natural, although we have not come across it in the existing literature on the topic. Some of our results concerning the near-collision behavior appear to be new, and we also prove inequalities between critical energy values that we have not found in the literature. The calculations we do to analyze the collisions and near-collisions also reproduce various classical results that were previously obtained by other authors using different coordinates, and parts of the paper therefore have an expository character.

math.DS↗
arXiv2024

Vanishing viscosity limit for axisymmetric vortex rings

For the incompressible Navier-Stokes equations in $R^3$ with low viscosity $ν>0$, we consider the Cauchy problem with initial vorticity $ω_0$ that represents an infinitely thin vortex filament of arbitrary given strength $Γ$ supported on a circle. The vorticity field $ω(x,t)$ of the solution is smooth at any positive time and corresponds to a vortex ring of thickness $\sqrt{νt}$ that is translated along its symmetry axis due to self-induction, an effect anticipated by Helmholtz in 1858 and quantified by Kelvin in 1867. For small viscosities, we show that $ω(x,t)$ is well-approximated on a large time interval by $ω_{lin}(x-a(t),t)$, where $ω_{lin}(\cdot,t)=\exp(νtΔ)ω_0$ is the solution of the heat equation with initial data $ω_0$, and $\dot a(t)$ is the instantaneous velocity given by Kelvin's formula. This gives a rigorous justification of the binormal motion for circular vortex filaments in weakly viscous fluids. The proof relies on the construction of a precise approximate solution, using a perturbative expansion in self-similar variables. To verify the stability of this approximation, one needs to rule out potential instabilities coming from very large advection terms in the linearized operator. This is done by adapting V. I. Arnold's geometric stability methods developed in the inviscid case $ν=0$ to the slightly viscous situation. It turns out that although the geometric structures behind Arnold's approach are no longer preserved by the equation for $ν> 0$, the relevant quadratic forms behave well on larger subspaces than those originally used in Arnold's theory and interact favorably with the viscous terms.

math.AP↗
arXiv2021

Arnold's variational principle and its application to the stability of planar vortices

We consider variational principles related to V. I. Arnold's stability criteria for steady-state solutions of the two-dimensional incompressible Euler equation. Our goal is to investigate under which conditions the quadratic forms defined by the second variation of the associated functionals can be used in the stability analysis, both for the Euler evolution and for the the Navier-Stokes equation at low viscosity. In particular, we revisit the classical example of Oseen's vortex, providing a new stability proof with stronger geometric flavor. Our analysis involves a fairly detailed functional-analytic study of the inviscid case, which may be of independent interest, and a careful investigation of the influence of the viscous term in the particular example of the Gaussian vortex.

math.AP↗
arXiv2017

On the De Gregorio modification of the Constantin-Lax-Majda Model

We study a modification due to De Gregorio of the Constantin-Lax-Majda (CLM) model $ω_t = ωHω$ on the unit circle. The De Gregorio equation is $ω_t+u ω_x-u_xω=0, u_x = Hω.$ In contrast with the CLM model, numerical simulations suggest that the solutions of the De Gregorio model with smooth initial data exist globally for all time, and generically converge to equilibria when $t\to\pm\infty$, in a way resembling inviscid damping. We prove that such a behavior takes place near a manifold of equilibria.

math.AP↗
arXiv2016

Uniqueness of axisymmetric viscous flows originating from circular vortex filaments

The incompressible Navier-Stokes equations in R^3 are shown to admit a unique axisymmetric solution without swirl if the initial vorticity is a circular vortex filament with arbitrarily large circulation Reynolds number. The emphasis is on uniqueness, as existence has already been established in [10]. The main difficulty which has to be overcome is that the nonlinear regime for such flows is outside of applicability of standard perturbation theory, even for short times. The solutions we consider are archetypal examples of viscous vortex rings, and can be thought of as axisymmetric analogues of the self-similar Lamb-Oseen vortices in two-dimensional flows. Our method provides the leading term in a fixed-viscosity short-time asymptotic expansion of the solution, and may in principle be extended so as to give a rigorous justification, in the axisymmetric situation, of higher-order formal asymptotic expansions that can be found in the literature [7].

math.AP↗
arXiv2015

Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations

Motivated by applications to vortex rings, we study the Cauchy problem for the three-dimensional axisymmetric Navier-Stokes equations without swirl, using scale invariant function spaces. If the axisymmetric vorticity is integrable with respect to the two-dimensional measure dr dz, where (r,θ,z) denote the cylindrical coordinates in R^3, we show the existence of a unique global solution, which converges to zero in L^1 norm as time goes to infinity. The proof of local well-posedness follows exactly the same lines as in the two-dimensional case, and our approach emphasizes the similarity between both situations. The solutions we construct have infinite energy in general, so that energy dissipation cannot be invoked to control the long-time behavior. We also treat the more general case where the initial vorticity is a finite measure whose atomic part is small enough compared to viscosity. Such data include point masses, which correspond to vortex filaments in the three-dimensional picture.

math.AP↗
arXiv2013

On Inviscid Limits for the Stochastic Navier-Stokes Equations and Related Models

We study inviscid limits of invariant measures for the 2D Stochastic Navier-Stokes equations. As shown in \cite{Kuksin2004} the noise scaling $\sqrt{ν}$ is the only one which leads to non-trivial limiting measures, which are invariant for the 2D Euler equations. We show that any limiting measure $μ_{0}$ is in fact supported on bounded vorticities. Relationships of $μ_{0}$ to the long term dynamics of Euler in the $L^{\infty}$ with the weak$^{*}$ topology are discussed. In view of the Batchelor-Krainchnan 2D turbulence theory, we also consider inviscid limits for the weakly damped stochastic Navier-Stokes equation. In this setting we show that only an order zero noise (i.e. the noise scaling $ν^0$) leads to a nontrivial limiting measure in the inviscid limit.

math.AP↗
arXiv2013

Backward uniqueness for the heat equation in cones

It is known that a bounded solution of the heat equation in a half-space which becomes zero at some time must be identically zero, even though no assumptions are made on the boundary values of the solutions. In a recent example, Luis Escauriaza showed that this statement fails if the half-space is replaced by cones with opening angle smaller than 90 degrees. Here we show the result remains true for cones with opening angle larger than 110 degrees. The proof covers heat equations having lower-order terms with bounded measurable coefficients.

math.AP↗
arXiv2012

On Landau's Solutions of the Navier-Stokes Equations

In 1944 L.D.Landau calculated a very interesting family of explicit solutions of the steady-state 3d Navier-Stokes equations. The solutions are derived under certain assumptions of symmetry, which reduce the Navier-Stokes equations to a system of ODEs. We investigate what happens when some of the symmetry conditions are dropped (and we have to deal with PDEs). Implications of these calculations for more general classes of solutions are also discussed. We also discuss the situation for general dimension.

math.AP↗
arXiv2010

On Divergence-free Drifts

We investigate the validity and failure of Liouville theorems and Harnack inequalities for parabolic and elliptic operators with low regularity coefficients. We are particularly interested in operators of the form $\partial_t - Δ+b\cdot\nabla$ and $-Δ+b\cdot\nabla$ with a divergence-free drift $b$. We prove the Liouville theorem and Harnack inequality when $b\in L_\infty(BMO^{-1})$ resp. $b\in BMO^{-1}$ and provide a counterexample to such results demonstrating sharpness of our conditions on the drift. Our results generalize to divergence-form operators with an elliptic symmetric part and a BMO skew-symmetric part. We also prove the existence of a modulus of continuity for solutions to the elliptic problem in two dimensions, depending on the non-scale-invariant norm $\|b\|_{L_1}$. In three dimensions, on the other hand, bounded solutions with $L_1$ drifts may be discontinuous.

math.AP↗
arXiv2006

Zeros of complex caloric functions and singularities of complex viscous Burgers equation

We show that the 1d viscous Burgers equation considered for complex valued functions develops finite-time singularities from compactly supported smooth data. By means of the Cole-Hopf transformation, the singularities of the solutions are related to zeros of complex-valued solutions of the heat equation. We prove that such zeros are isolated if they are not present in the initial data.

math.AP↗
arXiv2003

Singular and regular solutions of a non-linear parabolic system

We study a dissipative nonlinear equation modelling certain features of the Navier-Stokes equations. We prove that the evolution of radially symmetric compactly supported initial data does not lead to singularities in dimensions $n\leq 4$. For dimensions $n>4$ we present strong numerical evidence supporting existence of blow-up solutions. Moreover, using the same techniques we numerically confirm a conjecture of Lepin regarding existence of self-similar singular solutions to a semi-linear heat equation.

math.AP↗