arXiv subjects

Vlad Vicol

Publications and source records attributed to Vlad Vicol.

At least 55 records · Page 3Linked to original sources

arXiv2017

Onsager's conjecture for admissible weak solutions

We prove that given any $β<1/3$, a time interval $[0,T]$, and given any smooth energy profile $e \colon [0,T] \to (0,\infty)$, there exists a weak solution $v$ of the three-dimensional Euler equations such that $v \in C^β([0,T]\times \mathbb{T}^3)$, with $e(t) = \int_{\mathbb{T}^3} |v(x,t)|^2 dx$ for all $t\in [0,T]$. Moreover, we show that a suitable $h$-principle holds in the regularity class $C^β_{t,x}$, for any $β<1/3$. The implication of this is that the dissipative solutions we construct are in a sense typical in the appropriate space of subsolutions as opposed to just isolated examples.

math.AP↗

arXiv2016

Nonuniqueness of weak solutions to the SQG equation

We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 in the survey arXiv:1111.2700 by De Lellis and Székelyhidi Jr. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian.

math.AP↗

arXiv2016

The Sobolev stability threshold for 2D shear flows near Couette

We consider the 2D Navier-Stokes equation on $\mathbb T \times \mathbb R$, with initial datum that is $\varepsilon$-close in $H^N$ to a shear flow $(U(y),0)$, where $\| U(y) - y\|_{H^{N+4}} \ll 1$ and $N>1$. We prove that if $\varepsilon \ll ν^{1/2}$, where $ν$ denotes the inverse Reynolds number, then the solution of the Navier-Stokes equation remains $\varepsilon$-close in $H^1$ to $(e^{t ν\partial_{yy}}U(y),0)$ for all $t>0$. Moreover, the solution converges to a decaying shear flow for times $t \gg ν^{-1/3}$ by a mixing-enhanced dissipation effect, and experiences a transient growth of gradients. In particular, this shows that the stability threshold in finite regularity scales no worse than $ν^{1/2}$ for 2D shear flows close to the Couette flow.

math.AP↗

arXiv2016

Remarks on the inviscid limit for the Navier-Stokes equations for uniformly bounded velocity fields

We consider the vanishing viscosity limit of the Navier-Stokes equations in a half space, with Dirichlet boundary conditions. We prove that the inviscid limit holds in the energy norm if the product of the components of the Navier-Stokes solutions are equicontinuous at $x_2=0$. A sufficient condition for this to hold is that the tangential Navier-Stokes velocity remains uniformly bounded and has a uniformly integrable tangential gradient near the boundary.

math.AP↗

arXiv2015

The van Dommelen and Shen singularity in the Prandtl equations

In 1980, van Dommelen and Shen provided a numerical simulation that predicts the spontaneous generation of a singularity in the Prandtl boundary layer equations from a smooth initial datum, for a nontrivial Euler background. In this paper we provide a proof of this numerical conjecture by rigorously establishing the finite time blowup of the boundary layer thickness.

math.AP↗

arXiv2015

On some electroconvection models

We consider a model of electroconvection motivated by studies of the motion of a two dimensional annular suspended smectic film under the influence of an electric potential maintained at the boundary by two cylindrical electrodes. We prove that this electroconvection model has global in time unique smooth solutions.

math.AP↗

arXiv2015

Almost global existence for the Prandtl boundary layer equations

We consider the Prandtl boundary layer equations on the half plane, with initial datum that lies in a weighted $H^1$ space with respect to the normal variable, and is real-analytic with respect to the tangential variable. The boundary trace of the horizontal Euler flow is taken to be a constant. We prove that if the Prandtl datum lies within $\varepsilon$ of a stable profile, then the unique solution of the Cauchy problem can be extended at least up to time $T_\varepsilon \geq \exp(\varepsilon^{-1}/ \log(\varepsilon^{-1}))$.

math.AP↗

arXiv2015

Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations

We consider the incompressible Euler equations on ${\mathbb R}^d$, where $d \in \{ 2,3 \}$. We prove that: (a) In Lagrangian coordinates the equations are locally well-posed in spaces with fixed real-analyticity radius (more generally, a fixed Gevrey-class radius). (b) In Lagrangian coordinates the equations are well-posed in highly anisotropic spaces, e.g.~Gevrey-class regularity in the label $a_1$ and Sobolev regularity in the labels $a_2,...,a_d$. (c) In Eulerian coordinates both results (a) and (b) above are false.

math.AP↗

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.

math.AP↗

arXiv2015

Uniformly attracting limit sets for the critically dissipative SQG equation

We consider the global attractor of the critical SQG semigroup $S(t)$ on the scale-invariant space $H^1(\mathbb{T}^2)$. It was shown in~\cite{CTV13} that this attractor is finite dimensional, and that it attracts uniformly bounded sets in $H^{1+δ}(\mathbb{T}^2)$ for any $δ>0$, leaving open the question of uniform attraction in $H^1(\mathbb{T}^2)$. In this paper we prove the uniform attraction in $H^1(\mathbb{T}^2)$, by combining ideas from DeGiorgi iteration and nonlinear maximum principles.

math.AP↗

arXiv2014

On the global regularity for the supercritical SQG equation

We consider the initial value problem for the fractionally dissipative quasi-geostrophic equation \[ \partial_t θ+ \mathcal{R}^\perp θ\cdot \nabla θ+ Λ^γθ= 0, \qquad θ(\cdot,0) =θ_0 \] on $\mathbb{T}^2 = [0,1]^2$, with $γ\in (0,1)$. The coefficient in front of the dissipative term $Λ^γ= (-Δ)^{γ/2}$ is normalized to $1$. We show that given a smooth initial datum with $\|θ_0\|_{L^2}^{γ/2} \|θ_0\|_{\dot{H}^2}^{1-γ/2}\leq R$, where {\em $R$ is arbitrarily large}, there exists $γ_1 = γ_1(R) \in (0,1)$ such that for $γ\geq γ_1$, the solution of the supercritical SQG equation with dissipation $Λ^γ$ does not blow up in finite time. The main ingredient in the proof is a new concise proof of eventual regularity for the supercritical SQG equation, that relies solely on nonlinear lower bounds for the fractional Laplacian and the maximum principle.

math.AP↗

arXiv2014

Enhanced dissipation and inviscid damping in the inviscid limit of the Navier-Stokes equations near the 2D Couette flow

In this work we study the long time, inviscid limit of the 2D Navier-Stokes equations near the periodic Couette flow, and in particular, we confirm at the nonlinear level the qualitative behavior predicted by Kelvin's 1887 linear analysis. At high Reynolds number Re, we prove that the solution behaves qualitatively like 2D Euler for times t \lesssim Re^(1/3), and in particular exhibits inviscid damping (e.g. the vorticity weakly approaches a shear flow). For times t \gtrsim Re^(1/3), which is sooner than the natural dissipative time scale O(Re), the viscosity becomes dominant and the streamwise dependence of the vorticity is rapidly eliminated by an enhanced dissipation effect. Afterward, the remaining shear flow decays on very long time scales t \gtrsim Re back to the Couette flow. When properly defined, the dissipative length-scale in this setting is L_D \sim Re^(-1/3), larger than the scale L_D \sim Re^(-1/2) predicted in classical Batchelor-Kraichnan 2D turbulence theory. The class of initial data we study is the sum of a sufficiently smooth function and a small (with respect to Re^(-1)) $L^2$ function.

math.AP↗

arXiv2014

On a transport equation with nonlocal drift

In \cite{CordobaCordobaFontelos05}, Córdoba, Córdoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions \[ \partial_t θ+u \; \partial_x θ= 0, \qquad u = H θ, \] where $H$ is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to the equation with diffusion \[ \partial_t θ+ u \; \partial_x θ+ Λ^γθ= 0, \qquad u = H θ, \] where $Λ= (-Δ)^{1/2}$, and $1/2 \leq γ<1$. Our results also apply to the model with velocity field $u = Λ^s H θ$, where $s \in (-1,1)$. We conjecture that solutions which arise as limits from vanishing viscosity approximations are bounded in the Hölder class in $C^{(s+1)/2}$, for all positive time.

math.AP↗

arXiv2014

Holder Continuous Solutions of Active Scalar Equations

We consider active scalar equations $\partial_t θ+ \nabla \cdot (u \, θ) = 0$, where $u = T[θ]$ is a divergence-free velocity field, and $T$ is a Fourier multiplier operator with symbol $m$. We prove that when $m$ is not an odd function of frequency, there are nontrivial, compactly supported solutions weak solutions, with Hölder regularity $C^{1/9-}_{t,x}$. In fact, every integral conserving scalar field can be approximated in ${\cal D}'$ by such solutions, and these weak solutions may be obtained from arbitrary initial data. We also show that when the multiplier $m$ is odd, weak limits of solutions are solutions, so that the $h$-principle for odd active scalars may not be expected.

math.AP↗

arXiv2014

Analyticity of Lagrangian trajectories for well posed inviscid incompressible fluid models

We discuss general incompressible inviscid models, including the Euler equations, the surface quasi-geostrophic equation, incompressible porous medium equation, and Boussinesq equations. All these models have classical unique solutions, at least for short time. We show that they have real analytic Lagrangian paths. More precisely, we show that as long as a solution of any of these equations is in a class of regularity that assures Hölder continuous gradients of velocity, the corresponding Lagrangian paths are real analytic functions of time. The method of proof is conceptually straightforward and general, and we address the combinatorial issues head-on.

math.AP↗

arXiv2014

Inviscid Limits for a Stochastically Forced Shell Model of Turbulent Flow

We establish the anomalous mean dissipation rate of energy in the inviscid limit for a stochastic shell model of turbulent fluid flow. The proof relies on viscosity independent bounds for stationary solutions and on establishing ergodic and mixing properties for the viscous model. The shell model is subject to a degenerate stochastic forcing in the sense that noise acts directly only through one wavenumber. We show that it is hypo-elliptic (in the sense of Hormander) and use this property to prove a gradient bound on the Markov semigroup.

math-ph↗

arXiv2014

On the inviscid limit of the Navier-Stokes equations

We consider the convergence in the $L^2$ norm, uniformly in time, of the Navier-Stokes equations with Dirichlet boundary conditions to the Euler equations with slip boundary conditions. We prove that if the Oleinik conditions of no back-flow in the trace of the Euler flow, and of a lower bound for the Navier-Stokes vorticity is assumed in a Kato-like boundary layer, then the inviscid limit holds.

math.AP↗

arXiv2014

On the local well-posedness of the Prandtl and the hydrostatic Euler equations with multiple monotonicity regions

We find a new class of data for which the Prandtl boundary layer equations and the hydrostatic Euler equations are locally in time well-posed. In the case of the Prandtl equations, we assume that the initial datum $u_0$ is monotone on a number of intervals (on some strictly increasing on some strictly decreasing) and analytic on the complement and show that the local existence and uniqueness hold. The same is true for the hydrostatic Euler equations except that we assume this for the vorticity $ω_0=\partial_y u_0$.

math.AP↗