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Michele Dolce

Publications and source records attributed to Michele Dolce.

At least 19 recordsLinked to original sources
arXiv2026

Initial layer instability of the kinetic Lamb-Oseen Vortex

It is well-known that solutions to the incompressible Navier-Stokes system are limits in various contexts (weak or strong), of solutions to the Boltzmann equation, when the Mach and the Knudsen numbers go to zero. In particular the case of smooth solutions is by now rather well understood. Recent works have aimed at choosing initial data in function spaces as close as possible to those corresponding to well-posedness for the incompressible Navier-Stokes system. This paper tackles the case of measurevalued initial vorticity, in two space dimensions: in the special case when the initial vorticity is a Dirac mass, it is known that the unique solution to the Navier-Stokes system is the solution to the heat equation. We prove that the kinetic emanation of this initial vorticity (slightly smoothed out) leads to a solution of the Boltzmann equation which diverges in a strong way and in a very small time layer, from the expected hydrodynamic limit.

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arXiv2026

Viscous vortex crystals

We study the solution to the two-dimensional incompressible Navier-Stokes equations arising from a sum of Dirac masses in a particular co-rotating configuration. This configuration consists of a polygonal vortex crystal with or without a central vortex. By exploiting the symmetries and stability properties of the system, we describe and control the solution up to sub-diffusive time scales, prior to the expected onset of vortex merging.

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arXiv2026

Instability of two-dimensional Taylor-Green Vortices

For a wide class of linear Hamiltonian operators we develop a general criterion that characterizes the unstable eigenvalues as the zeros of a holomorphic function given by the determinant of a finite-dimensional matrix. We apply the latter result to prove the spectral instability of the Taylor-Green vortex in two-dimensional ideal fluids. The linearized Euler operator at this steady state possesses different invariant subspaces, within which we apply our criterion to rule out or detect instabilities. We show linear stability of odd perturbations, for which the unstable spectrum can appear only on the real axis. We exclude this possibility by applying our stability criterion. Real instabilities, instead, exist and can be detected with the same criterion if we consider suitable rescalings of the Taylor-Green vortex. In the subspace of functions even in both variables, the problem is reduced to finding a single complex root of our stability function. We successfully locate this value by combining our general criterion with a rigorous computer-assisted argument. As a consequence, we fully characterize the unstable spectrum of the Taylor-Green vortex.

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arXiv2025

Long-wave instability of periodic shear flows for the 2D Navier-Stokes equations

In 1959, Kolmogorov proposed to study the instability of the shear flow $(\sin(y),0)$ in the vanishing viscosity regime in tori $\mathbb{T}_{\alpha}\times \mathbb{T}$. This question was later resolved by Meshalkin and Sinai. We extend the problem to general shear flows $(U(y),0)$ and show that every $U(y)$ exhibits long-wave instability whenever $\|\partial_y^{-1} U\|_{L^2} > \nu$ and $\alpha\ll \nu$, with $\nu$ being the kinematic viscosity. This instability mechanism confirms previous findings by Yudovich in 1966, supported also by several numerical results, and is established through two independent approaches: one via the construction of Kato's isomorphism and one via normal forms. Unlike in many other applications of the latter methods, both proofs deal with the presence of a delicate term in the linearized operator that becomes singular as $\alpha$ approaches $0$.

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arXiv2024

Self-similar instability and forced nonuniqueness: an application to the 2D Euler equations

Building on an approach introduced by Golovkin in the '60s, we show that nonuniqueness in some forced PDEs is a direct consequence of the existence of a self-similar linearly unstable eigenvalue: the key point is a clever choice of the forcing term removing complicated nonlinear interactions. We use this method to give a short and self-contained proof of nonuniqueness in 2D perfect fluids, first obtained in Vishik's groundbreaking result. In particular, we present a direct construction of a forced self-similar unstable vortex, where we treat perturbatively the self-similar operator in a new and more quantitative way.

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arXiv2024

Large norm inflation of the current in the viscous, non-resistive magnetohydrodynamics equations

We consider the ideally conducting, viscous magnetohydrodynamics (MHD) equations in two dimensions. Specifically, we study the nonlinear dynamics near a combination of Couette flow and a constant magnetic field in a periodic infinite channel. In contrast to the Navier-Stokes equations this system is shown to exhibit algebraic instability and large norm inflation of the magnetic current on non-perturbative time scales.

math.AP↗
arXiv2024

The long way of a viscous vortex dipole

We consider the evolution of a viscous vortex dipole in $R^2$ originating from a pair of point vortices with opposite circulations. At high Reynolds number $Re >> 1$, the dipole can travel a very long way, compared to the distance between the vortex centers, before being slowed down and eventually destroyed by diffusion. In this regime we construct an accurate approximation of the solution in the form of a two-parameter asymptotic expansion involving the aspect ratio of the dipole and the inverse Reynolds number. We then show that the exact solution of the Navier-Stokes equations remains close to the approximation on a time interval of length $O(Re^\sigma)$, where $\sigma < 1$ is arbitrary. This improves upon previous results which were essentially restricted to $\sigma = 0$. As an application, we provide a rigorous justification of an existing formula which gives the leading order correction to the translation speed of the dipole due to finite size effects.

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arXiv2024

Dissipation enhancing properties for a class of Hamiltonian flows with closed streamlines

We study the evolution of a passive scalar subject to molecular diffusion and advected by an incompressible velocity field on a 2D bounded domain. The velocity field is $u = \nabla^\perp H$, where H is an autonomous Hamiltonian whose level sets are Jordan curves foliating the domain. We focus on the high P\'eclet number regime ($Pe := \nu^{-1} \gg 1$), where two distinct processes unfold on well separated time-scales: streamline averaging and standard diffusion. For a specific class of Hamiltonians with one non-degenerate elliptic point (including perturbed radial flows), we prove exponential convergence of the solution to its streamline average on a subdiffusive time-scale $T_\nu \ll \nu^{-1}$ ,up to a small correction related to the shape of the streamlines. The time-scale $T_\nu$ is determined by the behavior of the period function around the elliptic point. To establish this result, we introduce a model problem arising naturally from the difference between the solution and its streamline average. We use pseudospectral estimates to infer decay in the model problem, and, in fact, this analysis extends to a broader class of Hamiltonian flows. Finally, we perform an asymptotic expansion of the full solution, revealing that the leading terms consist of the streamline average and the solution of the model problem.

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arXiv2024

On the convergence rates of discrete solutions to the Wave Kinetic Equation

In this paper, we consider the long-term behavior of some special solutions to the Wave Kinetic Equation (WKE). This equation provides a mesoscopic description of wave systems interacting nonlinearly via the cubic NLS equation. Escobedo and Velázquez showed that, starting with initial data given by countably many Dirac masses, solutions remain a linear combination of countably many Dirac masses at all times. Moreover, there is convergence to a single Dirac mass at long times. The first goal of this paper is to give quantitative rates for the speed of said convergence. In order to study the optimality of the bounds we obtain, we introduce and analyze a toy model accounting only for the leading order quadratic interactions.

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arXiv2023

Stability threshold of the 2D Couette flow in a homogeneous magnetic field using symmetric variables

We consider a 2D incompressible and electrically conducting fluid in the domain $\mathbb{T}\times\mathbb{R}$. The aim is to quantify stability properties of the Couette flow $(y,0)$ with a constant homogenous magnetic field $(β,0)$ when $|β|>1/2$. The focus lies on the regime with small fluid viscosity $ν$, magnetic resistivity $μ$ and we assume that the magnetic Prandtl number satisfies $μ^2\lesssim\mathrm{Pr}_{\mathrm{m}}=ν/μ\leq 1$. We establish that small perturbations around this steady state remain close to it, provided their size is of order $\varepsilon\llν^{2/3}$ in $H^N$ with $N$ large enough. Additionally, the vorticity and current density experience a transient growth of order $ν^{-1/3}$ while converging exponentially fast to an $x$-independent state after a time-scale of order $ν^{-1/3}$. The growth is driven by an inviscid mechanism, while the subsequent exponential decay results from the interplay between transport and diffusion, leading to the dissipation enhancement. A key argument to prove these results is to reformulate the system in terms of symmetric variables, inspired by the study of inhomogeneous fluid, to effectively characterize the system's dynamic behavior.

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arXiv2023

Diffusion enhancement and Taylor dispersion for rotationally symmetric flows in discs and pipes

In this note, we study the long-time dynamics of passive scalars driven by rotationally symmetric flows. We focus on identifying precise conditions on the velocity field in order to prove enhanced dissipation and Taylor dispersion in three-dimensional infinite pipes. As a byproduct of our analysis, we obtain an enhanced decay for circular flows on a disc of arbitrary radius.

math.AP↗
arXiv2022

Taylor dispersion and phase mixing in the non-cutoff Boltzmann equation on the whole space

In this paper, we describe the long-time behavior of the non-cutoff Boltzmann equation with soft potentials near a global Maxwellian background on the whole space in the weakly collisional limit (i.e. infinite Knudsen number $1/ν\to \infty$). Specifically, we prove that for initial data sufficiently small (independent of the Knudsen number), the solution displays several dynamics caused by the phase mixing/dispersive effects of the transport operator $v \cdot \nabla_x$ and its interplay with the singular collision operator. For $x$-wavenumbers $k$ with $|k|\ggν$, one sees an enhanced dissipation effect wherein the characteristic decay time-scale is accelerated to $O(1/ν^{\frac{1}{1+2s}} |k|^{\frac{2s}{1+2s}})$, where $s \in (0,1]$ is the singularity of the kernel ($s=1$ being the Landau collision operator, which is also included in our analysis); for $|k|\ll ν$, one sees Taylor dispersion, wherein the decay is accelerated to $O(ν/|k|^2)$. Additionally, we prove almost-uniform phase mixing estimates. For macroscopic quantities as the density $ρ$, these bounds imply almost-uniform-in-$ν$ decay of $(t\nabla_x)^βρ$ in $L^\infty_x$ due to Landau damping and dispersive decay.

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arXiv2022

On maximally mixed equilibria of two-dimensional perfect fluids

The vorticity of a two-dimensional perfect (incompressible and inviscid) fluid is transported by its area preserving flow. Given an initial vorticity distribution $\omega_0$, predicting the long time behavior which can persist is an issue of fundamental importance. In the infinite time limit, some irreversible mixing of $\omega_0$ can occur. Since kinetic energy $\mathsf{E}$ is conserved, not all the mixed states are relevant and it is natural to consider only the ones with energy $\mathsf{E}_0$ corresponding to $\omega_0$. The set of said vorticity fields, denoted by $\overline{\mathcal{O}_{\omega_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$, contains all the possible end states of the fluid motion. A. Shnirelman introduced the concept of maximally mixed states (any further mixing would necessarily change their energy), and proved they are perfect fluid equilibria. We offer a new perspective on this theory by showing that any minimizer of any strictly convex Casimir in $\overline{\mathcal{O}_{\omega_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$ is maximally mixed, as well as discuss its relation to classical statistical hydrodynamics theories. Thus, (weak) convergence to equilibrium cannot be excluded solely on the grounds of vorticity transport and conservation of kinetic energy. On the other hand, on domains with symmetry (e.g. straight channel or annulus), we exploit all the conserved quantities and the characterizations of $\overline{\mathcal{O}_{\omega_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$ to give examples of open sets of initial data which can be arbitrarily close to any shear or radial flow in $L^1$ of vorticity but do not weakly converge to them in the long time limit.

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arXiv2021

Global Existence for the Two-dimensional Kuramoto-Sivashinsky equation with a Shear Flow

We consider the Kuramoto-Sivashinsky equation (KSE) on the two-dimensional torus in the presence of advection by a given background shear flow. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we prove global existence of solutions with data in $L^2$, using a bootstrap argument. The initial data can be taken arbitrarily large.

math.AP↗
arXiv2021

Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations

We investigate the long-time properties of the two-dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size $\varepsilon$. Under the classical Miles-Howard stability condition on the Richardson number, we prove that the system experiences a shear-buoyancy instability: the density variation and velocity undergo an $O(t^{-1/2})$ inviscid damping while the vorticity and density gradient grow as $O(t^{1/2})$. The result holds at least until the natural, nonlinear timescale $t \approx \varepsilon^{-2}$. Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles-Howard spectral stability condition; (B) a variation of the Fourier time-dependent energy method introduced for the inviscid, homogeneous Couette flow problem developed on a toy model adapted to the Boussinesq equations, i.e. tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth.

math.AP↗
arXiv2021

Linear inviscid damping for shear flows near Couette in the 2D stably stratified regime

We investigate the linear stability of shears near the Couette flow for a class of 2D incompressible stably stratified fluids. Our main result consists of nearly optimal decay rates for perturbations of stationary states whose velocities are monotone shear flows $(U(y),0)$ and have an exponential density profile. In the case of the Couette flow $U(y)=y$, we recover the rates predicted by Hartman in 1975, by adopting an explicit point-wise approach in frequency space. As a by-product, this implies optimal decay rates as well as Lyapunov instability in $L^2$ for the vorticity. For the previously unexplored case of more general shear flows close to Couette, the inviscid damping results follow by a weighted energy estimate. Each outcome concerning the stably stratified regime applies to the Boussinesq equations as well. Remarkably, our results hold under the celebrated Miles-Howard criterion for stratified fluids.

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arXiv2021

Linear stability analysis of the homogeneous Couette flow in a 2D isentropic compressible fluid

In this paper, we study the linear stability properties of perturbations around the homogeneous Couette flow for a 2D isentropic compressible fluid in the domain $\mathbb{T}\times \mathbb{R}$. In the inviscid case there is a generic Lyapunov type instability for the density and the irrotational component of the velocity field. More precisely, we prove that their $L^2$ norm grows as $t^{1/2}$ and this confirms previous observations in the physics literature. Instead, the solenoidal component of the velocity field experience inviscid damping, meaning that it decays to zero even in the absence of viscosity. For a viscous compressible fluid, we show that the perturbations may have a transient growth of order $ν^{-1/6}$ (with $ν^{-1}$ being proportional to the Reynolds number) on a time-scale $ν^{-1/3}$, after which it decays exponentially fast. This phenomenon is also called enhanced dissipation and our result appears to be the first to detect this mechanism for a compressible fluid, where an exponential decay for the density is not a priori trivial given the absence of dissipation in the continuity equation.

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