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Stefano Scrobogna

Publications and source records attributed to Stefano Scrobogna.

At least 19 recordsLinked to original sources
arXiv2025

Long-time dynamics for the Kelvin-Helmholtz equations close to circular vortex sheets

We consider the Kelvin-Helmholtz system describing the evolution of a vortex-sheet near the circular stationary solution. Answering previous numerical conjectures in the 90s physics literature, we prove an almost global existence result for small-amplitude solutions. We first establish the existence of a linear stability threshold for the Weber number, which represents the ratio between the square of the background velocity jump and the surface tension. Then, we prove that for almost all values of the Weber number below this threshold any small solution lives for almost all times, remaining close to the equilibrium. Our analysis reveals a remarkable stabilization phenomenon: the presence of both non-zero background velocity jump and capillarity effects enables to prevent nonlinear instability phenomena, despite the inherently unstable nature of the classical Kelvin-Helmholtz problem. This long-time existence would not be achievable in a setting where capillarity alone provides linear stabilization, without the richer modulation induced by the velocity jump. Our proof exploits the Hamiltonian nature of the equations. Specifically, we employ Hamiltonian Birkhoff normal form techniques for quasi-linear systems together with a general approach for paralinearization of non-linear singular integral operators. This approach allows us to control resonances and quasi-resonances at arbitrary order, ensuring the desired long-time stability result.

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arXiv2024

Steady vortex sheets in presence of surface tension

We prove a bifurcation result of uniformly-rotating/stationary non-trivial vortex sheets near the circular distribution for a model of two irrotational fluids with same density taking into account surface tension effects. As bifurcation parameters, we play with either the speed of rotation, the surface tension coefficient or the mean vorticity.

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arXiv2024

Quadratic lifespan for the sublinear $\alpha$-SQG sharp front problem

In this paper we consider the generalized surface quasi-geostrophic $\alpha$-SQG equations, in the "sublinear regime" $\alpha \in (0, 1)$ and we study the stability of vortex patches close to vortex discs. We shall prove that for regular, Sobolev initial vortex patches $\varepsilon$-close to a vortex disc, the solutions stay $\varepsilon$-close to a vortex disc for a time interval of order $O(\varepsilon^{- 2})$. The proof is based on a paradifferential Birkhoff normal form reduction, implemented in the case where the dispersion relation is sublinear.

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arXiv2023

Paralinearization and extended lifespan for solutions of the $ \alpha $-SQG sharp front equation

In this paper we paralinearize the contour dynamics equation for sharp-fronts of $\alpha$-SQG, for any $ \alpha \in (0,1) \cup (1,2) $, close to a circular vortex. This turns out to be a quasi-linear Hamiltonian PDE. After deriving the asymptotic expansion of the linear frequencies of oscillations at the vortex disk and verifying the absence of three wave interactions, we prove that, in the most singular cases $ \alpha \in (1,2) $, any initial vortex patch which is $ \varepsilon $-close to the disk exists for a time interval of size at least $ \sim \varepsilon^{-2} $. This quadratic lifespan result relies on a paradifferential Birkhoff normal form reduction and exploits cancellations arising from the Hamiltonian nature of the equation. This is the first normal form long time existence result of sharp fronts.

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arXiv2023

Quantitative aspects on the ill-posedness of the Prandtl and hyperbolic Prandtl equations

We address a physically-meaningful extension of the Prandtl system, also known as hyperbolic Prandtl equations. We show that the linearised model around a non-monotonic shear flow is ill-posed in any Sobolev spaces. Indeed, shortly in time, we generate solutions that experience a dispersion relation of order k^(1/3) in the frequencies of the tangential direction, akin the pioneering result of Gerard-Varet and Dormy in [10] for Prandtl (where the dispersion was of order k^(1/2)). We emphasise however that this growth rate does not imply ill-posedness in Gevrey-class m, with m > 3 and we relate also these aspects to the original Prandtl equations in Gevrey-class m, with m > 2. By relaxing certain assumptions on the shear flow and on the solutions, namely allowing for unbounded flows in the vertical direction, we however determine a suitable shear flow for the mentioned ill-posedness in Gevrey spaces.

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arXiv2022

Gevrey-class-3 regularity of the linearised hyperbolic Prandtl system on a strip

In the present paper, we address a physically-meaningful extension of the linearised Prandtl equations around a shear flow. Without any structural assumption, it is well-known that the optimal regularity of Prandtl is given by the class Gevrey 2 along the horizontal direction. The goal of this paper is to overcome this barrier, by dealing with the linearisation of the so-called hyperbolic Prandtl equations in a strip domain. We prove that the local well-posedness around a general shear flow holds true, with solutions that are Gevrey class 3 in the horizontal direction.

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arXiv2022

Well-posedness theory for non-homogeneous incompressible fluids with odd viscosity

Several fluid systems are characterised by time reversal and parity breaking. Examples of such phenomena arise both in quantum and classical hydrodynamics. In these situations, the viscosity tensor, often dubbed ``odd viscosity'', becomes non-dissipative. At the mathematical level, this fact translates into a loss of derivatives at the level of \textsl{a priori} estimates: while the odd viscosity term depends on derivatives of the velocity field, no parabolic smoothing effect can be expected. In the present paper, we establish a well-posedness theory in Sobolev spaces for a system of incompressible non-homogeneous fluids with odd viscosity. The crucial point of the analysis is the introduction of a set of \emph{good unknowns}, which allow for the emerging of a hidden hyperbolic structure underlying the system of equations. It is exactly this hyperbolic structure which makes it possible to circumvent the derivative loss and propagate high enough Sobolev norms of the solution. The well-posedness result is local in time; two continuation criteria are also established.

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arXiv2022

Small energy stabilization for 1D Nonlinear Klein Gordon Equations

We give a partial extension to dimension 1 of the result proved by Bambusi and Cuccagna on the absence of small energy real valued periodic solutions for the NLKG in dimension 3. We combine the framework in Kowalczyk and Martel with the notion of "refined profile".

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arXiv2021

Global well-posedness and decay for viscous water wave models

The motion of the free surface of an incompressible fluid is a very active research area. Most of these works examine the case of an inviscid fluid. However, in several practical applications, there are instances where the viscous damping needs to be considered. In this paper we derive and study a new asymptotic model for the motion of unidirectional viscous water waves. In particular, we establish the global well-posedness in Sobolev spaces. Furthermore, we also establish the global well-posedness and decay of a fourth order PDE modelling bidirectional water waves with viscosity moving in deep water with or without surface tension effects.

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arXiv2021

Global existence in the Lipschitz class for the N-Peskin problem

In this paper we study a toy model of the Peskin problem that captures the motion of the full Peskin problem in the normal direction and discards the tangential elastic stretching contributions. This model takes the form of a fully nonlinear scalar contour equation. The Peskin problem is a fluid-structure interaction problem that describes the motion of an elastic rod immersed in an incompressible Stokes fluid. We prove global in time existence of solution for initial data in the critical Lipschitz space. Using a new decomposition together with cancellation properties, pointwise methods allow us to obtain the desired estimates in the Lipschitz class. Moreover, we perform energy estimates in order to obtain that the solution lies in the space $L^2 \left( [0,T];H^{3/2} \right) $ to satisfy the contour equation pointwise.

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arXiv2020

Well-posedness of the water-wave with viscosity problem

In this paper we study the motion of a surface gravity wave with viscosity. In particular we prove two well-posedness results. On the one hand, we establish the local solvability in Sobolev spaces for arbitrary dissipation. On the other hand, we establish the global well-posedness in Wiener spaces for a sufficiently large viscosity. These results are the first rigorous proofs of well-posedness for the Dias, Dyachenko \& Zakharov system ({\em Physics Letters A} 2008) modelling gravity waves with viscosity when surface tension is not taken into account.

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arXiv2020

Well-posedness of a water wave model with viscous effects

Starting from the paper by Dias, Dyachenko and Zakharov (\emph{Physics Letters A, 2008}) on viscous water waves, we derive a model that describes water waves with viscosity moving in deep water with or without surface tension effects. This equation takes the form of a nonlocal fourth order wave equation and retains the main contributions to the dynamics of the free surface. Then, we prove the well-posedness in Sobolev spaces of such equation.

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arXiv2019

Surface tension stabilization of the Rayleigh-Taylor instability for a fluid layer in a porous medium

This paper studies the dynamics of an incompressible fluid driven by gravity and capillarity forces in a porous medium. The main interest is the stabilization of the fluid in Rayleigh-Taylor unstable situations where the fluid lays on top of a dry region. An important feature considered here is that the layer of fluid is under an impervious wall. This physical situation have been widely study by mean of thin film approximations in the case of small characteristic high of the fluid considering its strong interaction with the fixed boundary. Here, instead of considering any simplification leading to asymptotic models, we deal with the complete free boundary problem. We prove that, if the fluid interface is smaller than an explicit constant, the solution is global in time and it becomes instantly analytic. In particular, the fluid does not form drops in finite time. Our results are stated in terms of Wiener spaces for the interface together with some non-standard Wiener-Sobolev anisotropic spaces required to describe the regularity of the fluid pressure and velocity. These Wiener-Sobolev spaces are of independent interest as they can be useful in other problems. Finally, let us remark that our techniques do not rely on the irrotational character of the fluid in the bulk and they can be applied to other free boundary problems.

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arXiv2019

Models for damped water waves

In this paper we derive some new weakly nonlinear asymptotic models describing viscous waves in deep water with or without surface tension effects. These asymptotic models take into account several different dissipative effects and are obtained from the free boundary problems formulated in the works of Dias, Dyachenko and Zakharov (Physics Letters A, 2008), Jiang, Ting, Perlin and Schultz (Journal of Fluid Mechanics,1996) and Wu, Liu and Yue (Journal of Fluid Mechanics, 2006).

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arXiv2019

On an asymptotic model for free boundary Darcy flow in porous media

We provide a rigorous mathematical study of an asymptotic model describing Darcy flow with free boundary in a low amplitude/large wavelength approximation. In particular, we prove several well-posedness results in critical spaces. Furthermore, we also study how the solution decays towards the flat equilibrium.

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arXiv2019

Asymptotic models for free boundary flow in porous media

We provide rigorous asymptotic models for the free boundary Darcy and Forchheimer problem under the assumption of weak nonlinear interaction, in a regime in which the steepness parameter of the interface is considered to be very small. The models we derive capture the nonlinear interaction of the original free boundary Darcy and Forchheimer problem up to quadratic terms. Furthermore, we provide models that consider both the two-dimensional and three-dimensional cases, with and without bottom topography.

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