arXiv subjects

Vlad Vicol

Publications and source records attributed to Vlad Vicol.

At least 19 recordsLinked to original sources

arXiv2026

On rotated backwards self-similar solutions of the incompressible 3D Navier-Stokes equations

We consider backwards globally self-similar solutions of the 3D incompressible Navier-Stokes equations which are invariant under the joint action of scaling (the natural parabolic scaling) and rotation about a given axis, at a constant angular speed $\alpha$ in self-similar time. For these so-called rotated self-similar solutions (RSS), we prove that if they satisfy a Type~I upper bound, and if the rotation parameter $\alpha$ is either too small, or too large, then they must be trivial. This Liouville-type result extends the classical works of Ne\v{c}as-R\r{u}\v{z}i\v{c}ka-\v{S}ver\'ak ('96) and Tsai ('98), which only consider $\alpha=0$, to the case of similarity profiles which experience nontrivial rotation. Our results partially answer a question posed by Perelman. For backwards globally self-similar solutions which are invariant under the discrete action of scaling and rotation, the so-called rotated discretely self-similar solutions (RDSS), we obtain similar Liouville-type results under a Type~I upper bound, assuming extreme values of the rotation parameter $\alpha$, and if the scaling factor $\lambda$ is sufficiently close to $1$. We also establish a new regularity criterion for 3D Navier-Stokes which is local in nature: if the solution satisfies a Type~I upper bound in a unit parabolic cylinder, and there is a single time-slice at which the solution is locally approximately self-similar, then the top-center of the parabolic cylinder is a regular point of the Navier-Stokes flow. The proof of all these results rests on the introduction of a robust weighted-$L^2$ framework. In particular, our method is quantitative and is not sensitive to whether the Bernoulli head pressure satisfies a maximum principle, which was a key obstruction in previous works.

math.AP↗

arXiv2026

A new class of Euler explosions

We study the global-in-time continuation, past the singularity, of the smooth, non-isentropic, radially symmetric imploding solutions of the compressible Euler equations recently constructed by Chen, Shkoller, and Vicol. In three space dimensions, for all physically relevant adiabatic exponents $\gamma>1$, we consider the Euler solution that evolves smoothly until an implosion singularity forms at the origin at time $t=0$. We then prove that this solution can be uniquely continued for $t>0$ as a reflected outward-propagating shock, sometimes called a reflected blast wave. For $t>0$, the continuation is a globally forward self-similar weak solution of the Euler equations, selected by the Rankine--Hugoniot conditions and the Lax entropy inequality; it is smooth away from the expanding shock sphere and the spatial origin. The structure at the center of symmetry distinguishes these explosions from the classical Guderley reflected shock. In Guderley's continuation, the reflected blast wave leaves a point vacuum at the origin, where the density vanishes. The solutions constructed here exhibit the opposite behavior: for every fixed $t>0$ the density is unbounded at $r=0$ (though it remains locally integrable), while the pressure stays bounded and the temperature vanishes there.

math.AP↗

arXiv2026

Smooth and stable Euler implosions

We construct a new class of self-similar implosion profiles for the multi-dimensional compressible Euler equations. These profiles are smooth, genuinely non-isentropic, radially/spherically symmetric, and have explicit (closed-form) similarity exponents. We prove that the exact Euler solution corresponding to the ground state implosion profile is stable to radially symmetric perturbations, as a solution to the full nonlinear compressible Euler equations, modulo a one-dimensional compatibility condition on the initial data. For perturbations of the Euler solution corresponding to the ground state implosion profile of a monatomic or diatomic gas, that do not obey any symmetry assumptions, we provide a complete characterization of the set of initial data that yield nonlinear stability.

math.AP↗

arXiv2026

On putative self-similarity for incompressible 3D Euler

We consider hypothetical solutions of 3D Euler which blow up in finite time in a self-similar fashion. We prove that if the initial data has finite kinetic energy, then the similarity exponent $\gamma$ which governs the rate of zooming in must be at least $2/5$. If a smooth globally self-similar blowup profile exists, and this profile satisfies an outgoing property, we prove that $\gamma \geq 1/2$. For axisymmetric solutions, we establish the bound $\gamma\geq 1/2$ under the sole assumption that the velocity profile is $C^2$ smooth.

math.AP↗

arXiv2026

Finite time singularities in the Landau equation with very hard potentials

We consider the inhomogeneous Landau equation with $\gamma \in (\sqrt{3},2]$ and construct smooth, strictly positive initial data that develop a finite time singularity. The $C^{\alpha}$-norm of the distribution function blows up for every $\alpha>0$, whereas its $L^{\infty}$-norm remains uniformly bounded. In self-similar variables, the solution becomes asymptotically hydrodynamic - the distribution function converges to a local Maxwellian, while the hydrodynamic fields develop an asymptotically self-similar implosion whose profile coincides with a smooth imploding profile of the compressible Euler equations. To our knowledge, this provides the first example of a collisional kinetic model which is globally well-posed in the homogeneous setting, but admits finite time singularities for inhomogeneous data.

math.AP↗

arXiv2025

Classical Euler flows generate the strong Guderley imploding shock wave

We prove that Guderley's self-similar imploding shock solution for the compressible Euler equations with ideal--gas law ($\gamma>1$) arises from classical, radially symmetric, shock--free data. For such data prescribed at initial time $\mathrm{T_{in}} < 0$, we prove that the flow remains smooth up to a first singular time $t=\mathrm{T}_* \in (\mathrm{T_{in}}, 0)$, where a preshock forms with a $C^{1/3}$ cusp in the fast acoustic variable. From this preshock a unique, initially weak, regular shock is born, whose strength can be made arbitrarily large on a controlled time interval; the front then deforms onto the Guderley shock and implodes at the origin at the collapse time $t=0$. There exists a matching time $t=\mathrm{T_{fin}} \in (\mathrm{T}_*,0)$ such that on $[\mathrm{T_{fin}},0)$ the solution coincides exactly with the classical Guderley self--similar profile, and at $t=\mathrm{T_{fin}}$ the shock trajectory matches the self--similar front to all orders. As $t \to 0^-$, the Euler solution implodes at the center, and continues for $t>0$ as a reflected blast wave, providing a global-in-time unique Euler solution which evolves from regular initial conditions.

math.AP↗

arXiv2025

Anomalous diffusion via iterative quantitative homogenization: an overview of the main ideas

Anomalous diffusion is the fundamental ansatz of phenomenological theories of passive scalar turbulence, and has been confirmed numerically and experimentally to an extraordinary extent. The purpose of this survey is to discuss our recent result, in which we construct a class of incompressible vector fields that have many of the properties observed in a fully turbulent velocity field, and for which the associated scalar advection-diffusion equation generically displays anomalous diffusion. Our main contribution is to propose an analytical framework in which to study anomalous diffusion via a backward cascade of renormalized eddy viscosities.

math.AP↗

arXiv2024

Gradient catastrophes and an infinite hierarchy of H\"older cusp-singularities for 1D Euler

We establish an infinite hierarchy of finite-time gradient catastrophes for smooth solutions of the 1D Euler equations of gas dynamics with non-constant entropy. Specifically, for all integers $n\geq 1$, we prove that there exist classical solutions, emanating from smooth, compressive, and non-vacuous initial data, which form a cusp-type gradient singularity in finite time, in which the gradient of the solution has precisely $C^{0,\frac{1}{2n+1}}$ H\"older-regularity. We show that such Euler solutions are codimension-$(2n-2)$ stable in the Sobolev space $W^{2n+2,\infty}$.

math.AP↗

arXiv2024

Vorticity blowup in 2D compressible Euler equations

We prove finite-time vorticity blowup for smooth solutions of the 2D compressible Euler equations with smooth, localized, and non-vacuous initial data. The vorticity blowup occurs at the time of the first singularity, and is accompanied by an axisymmetric implosion in which the swirl velocity enjoys full stability, as opposed to finite co-dimension stability.

math.AP↗

arXiv2024

The geometry of maximal development and shock formation for the Euler equations in multiple space dimensions

We construct a fundamental piece of the boundary of the maximal globally hyperbolic development (MGHD) of Cauchy data for the multi-dimensional compressible Euler equations, which is necessary for the local shock development problem. For an open set of compressive and generic $H^7$ initial data, we construct unique $H^7$ solutions to the Euler equations in the maximal spacetime region below a given time-slice, beyond the time of the first singularity; at any point in this spacetime, the solution can be smoothly and uniquely computed by tracing both the fast and slow acoustic characteristic surfaces backward-in-time, until reaching the Cauchy data prescribed along the initial time-slice. The future temporal boundary of this spacetime region is a singular hypersurface, containing the union of three sets: first, a co-dimension-$2$ surface of ``first singularities'' called the pre-shock; second, a downstream hypersurface called the singular set emanating from the pre-shock, on which the Euler solution experiences a continuum of gradient catastrophes; third, an upstream hypersurface consisting of a Cauchy horizon emanating from the pre-shock, which the Euler solution cannot reach. We develop a new geometric framework for the description of the acoustic characteristic surfaces which is based on the Arbitrary Lagrangian Eulerian (ALE) framework, and combine this with a new type of differentiated Riemann variables which are linear combinations of gradients of velocity, sound speed, and the curvature of the fast acoustic characteristic surfaces. With these new variables, we establish uniform $H^7$ Sobolev bounds for solutions to the Euler equations without derivative loss and with optimal regularity.

math.AP↗

arXiv2023

Anomalous diffusion by fractal homogenization

For every $\alpha < \frac13$, we construct an explicit divergence-free vector field $\mathbf{b}(t,x)$ which is periodic in space and time and belongs to $C^0_t C^{\alpha}_x \cap C^{\alpha}_t C^0_x$ such that the corresponding scalar advection-diffusion equation $$\partial_t \theta^\kappa + \mathbf{b} \cdot \nabla \theta^\kappa - \kappa \Delta \theta^\kappa = 0$$ exhibits anomalous dissipation of scalar variance for arbitrary $H^1$ initial data: $$\limsup_{\kappa \to 0} \int_0^{1} \int_{\mathbb{T}^d} \kappa \| \nabla \theta^\kappa (t,x) \|^2 \,dx\,dt >0.$$ The vector field is deterministic and has a fractal structure, with periodic shear flows alternating in time between different directions serving as the base fractal. These shear flows are repeatedly inserted at infinitely many scales in suitable Lagrangian coordinates. Using an argument based on ideas from quantitative homogenization, the corresponding advection-diffusion equation with small $\kappa$ is progressively renormalized, one scale at a time, starting from the (very small) length scale determined by the molecular diffusivity up to the macroscopic (unit) scale. At each renormalization step, the effective diffusivity is enhanced by the influence of advection on that scale. By iterating this procedure across many scales, the effective diffusivity on the macroscopic scale is shown to be of order one.

math.AP↗

arXiv2023

An Intermittent Onsager Theorem

For any regularity exponent $β<\frac 12$, we construct non-conservative weak solutions to the 3D incompressible Euler equations in the class $C^0_t (H^β \cap L^{\frac{1}{(1-2β)}})$. By interpolation, such solutions belong to $C^0_tB^{s}_{3,\infty}$ for $s$ approaching $\frac 13$ as $β$ approaches $\frac 12$. Hence this result provides a new proof of the flexible side of the $L^3$-based Onsager conjecture. Of equal importance is that the intermittent nature of our solutions matches that of turbulent flows, which are observed to possess an $L^2$-based regularity index exceeding $\frac 13$. Thus our result does not imply, and is not implied by, the work of Isett [A proof of Onsager's conjecture, Annals of Mathematics, 188(3):871, 2018], who gave a proof of the Hölder-based Onsager conjecture. Our proof builds on the authors' previous joint work with Buckmaster et al. (Intermittent convex integration for the 3D Euler equations: (AMS-217), Princeton University Press, 2023), in which an intermittent convex integration scheme is developed for the 3D incompressible Euler equations. We employ a scheme with higher-order Reynolds stresses, which are corrected via a combinatorial placement of intermittent pipe flows of optimal relative intermittency.

math.AP↗

arXiv2023

A new type of stable shock formation in gas dynamics

From an open set of initial data, we construct a family of classical solutions to the 1D nonisentropic compressible Euler equations which form $C^{0,ν}$ cusps as a first singularity, for any $ν\in [1/2,1)$. For this range of $ν$, this is the first result demonstrating the stable formation of such $C^{0,ν}$ cusp-type singularities, also known as pre-shocks. The proof uses a new formulation of the differentiated Euler equations along the fast acoustic characteristic, and relies on a novel set of $L^p$ energy estimates for all $1 < p < \infty$, which may be of independent interest.

math.AP↗

arXiv2023

A characteristics approach to shock formation in 2D Euler with azimuthal symmetry and entropy

We provide a detailed analysis of the shock formation process for the non-isentropic 2d Euler equations in azimuthal symmetry. We prove that from an open set of smooth and generic initial data, solutions of Euler form a first singularity or gradient blow-up or shock. This first singularity is termed a Hölder $C^{\frac{1}{3}}$ pre-shock, and our analysis provides the first detailed description of this cusp solution. The novelty of this work relative to [Buckmaster-Drivas-Shkoller-Vicol, 2022] is that we herein consider a much larger class of initial data, allow for a non-constant initial entropy, allow for a non-trivial sub-dominant Riemann variable, and introduce a host of new identities to avoid apparent derivative loss due to entropy gradients. The method of proof is also new and robust, exploring the transversality of the three different characteristic families to transform space derivatives into time derivatives. Our main result provides a fractional series expansion of the Euler solution about the pre-shock, whose coefficients are computed from the initial data.

math.AP↗

arXiv2022

Non-conservative $H^{\frac 12-}$ weak solutions of the incompressible 3D Euler equations

For any positive regularity parameter $β< \frac 12$, we construct non-conservative weak solutions of the 3D incompressible Euler equations which lie in $H^β$ uniformly in time. In particular, we construct solutions which have an $L^2$-based regularity index \emph{strictly larger} than $\frac 13$, thus deviating from the $H^{\frac{1}{3}}$-regularity corresponding to the Kolmogorov-Obhukov $\frac 53$ power spectrum in the inertial range.

math.AP↗

arXiv2021

Exact Boundary Controllability for the Ideal Magneto-hydrodynamic Equations

We address the problem of controllability of the MHD system in a rectangular domain with a control prescribed on the side boundary. We identify a necessary and sufficient condition on the data to be null controllable, i.e., can be driven to the zero state. We also show that the validity of this condition allows the states to be stirred to each other. If the condition is not satisfied, one can move from one state to another with the help of a simple shear external magnetic force.

math.AP↗

arXiv2021

On the Euler+Prandtl expansion for the Navier-Stokes equations

We establish the validity of the Euler$+$Prandtl approximation for solutions of the Navier-Stokes equations in the half plane with Dirichlet boundary conditions, in the vanishing viscosity limit, for initial data which are analytic only near the boundary, and Sobolev smooth away from the boundary. Our proof does not require higher order correctors, and works directly by estimating an $L^{1}$-type norm for the vorticity of the error term in the expansion Navier-Stokes$-($Euler$+$Prandtl$)$. An important ingredient in the proof is the propagation of local analyticity for the Euler equation, a result of independent interest.

math.AP↗

arXiv2021

Simultaneous development of shocks and cusps for 2D Euler with azimuthal symmetry from smooth data

A fundamental question in fluid dynamics concerns the formation of discontinuous shock waves from smooth initial data. We prove that from smooth initial data, smooth solutions to the 2d Euler equations in azimuthal symmetry form a first singularity, the so-called $C^{\frac{1}{3}} $ pre-shock. The solution in the vicinity of this pre-shock is shown to have a fractional series expansion with coefficients computed from the data. Using this precise description of the pre-shock, we prove that a discontinuous shock instantaneously develops after the pre-shock. This regular shock solution is shown to be unique in a class of entropy solutions with azimuthal symmetry and regularity determined by the pre-shock expansion. Simultaneous to the development of the shock front, two other characteristic surfaces of cusp-type singularities emerge from the pre-shock. These surfaces have been termed weak discontinuities by Landau & Lifschitz [Chapter IX, §96], who conjectured some type of singular behavior of derivatives along such surfaces. We prove that along the slowest surface, all fluid variables except the entropy have $C^{1, {\frac{1}{2}} }$ one-sided cusps from the shock side, and that the normal velocity is decreasing in the direction of its motion; we thus term this surface a weak rarefaction wave. Along the surface moving with the fluid velocity, density and entropy form $C^{1, {\frac{1}{2}} }$ one-sided cusps while the pressure and normal velocity remain $C^2$; as such, we term this surface a weak contact discontinuity.

math.AP↗