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Roberta Bianchini

Publications and source records attributed to Roberta Bianchini.

At least 19 recordsLinked to original sources
arXiv2026

Conservation Laws with Discontinuous Flux: A First Application to Traffic Flow

This work provides a traffic flow interpretation of the model recently introduced in \cite{ABS2025}. We adapt the conservation law with discontinuous gradient-dependent flux to a macroscopic traffic setting, assuming concave flux functions $f(u)$ and $g(u)$ defined on the density domain $[0,u_{\max}]$. To prevent violations of the maximal density constraint, we introduce modified Riemann Solvers tailored to this setting and generalize the existence theory for weak solutions developed in \cite{ABS2025}. Finally, we construct Riemann Solvers for a broader two-phase model incorporating a free-flow regime, in which $f(u)=g(u)$ on a subset of $[0,u_{\max}]$, and establish existence of weak solutions in this more general framework.

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arXiv2024

Adapting Priority Riemann Solver for GSOM on road networks

In this paper, we present an extension of the Generic Second Order Models (GSOM) for traffic flow on road networks. We define a Riemann solver at the junction based on a priority rule and provide an iterative algorithm to construct solutions at junctions with n incoming and m outgoing roads. The logic underlying our solver is as follows: the flow is maximized while respecting the priority rule, which can be adjusted if the supply of an outgoing road exceeds the demand of a higher-priority incoming road. Approximate solutions for Cauchy problems are constructed using wave-front tracking. We establish bounds on the total variation of waves interacting with the junction and present explicit calculations for junctions with two incoming and two outgoing roads. A key novelty of this work is the detailed analysis of returning waves - waves generated at the junction that return to the junction after interacting along the roads - which, in contrast to first-order models such as LWR, can increase flux variation.

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arXiv2024

Non Existence and Strong Ill-Posedness in $H^2$ for the Stable IPM Equation

We prove the non-existence and strong ill-posedness of the Incompressible Porous Media (IPM) equation for initial data that are small $H^2(\mathbb{R}^2)$ perturbations of the linearly stable profile $-x_2$. A remarkable novelty of the proof is the construction of an $H^2$ perturbation, which solves the IPM equation and neutralizes the stabilizing effect of the background profile near the origin, where a strong deformation leading to non-existence in $H^2$ is created. This strong deformation is achieved through an iterative procedure inspired by the work of C\'{o}rdoba and Mart\'{\i}nez-Zoroa (Adv. Math. 2022). However, several differences - beyond purely technical aspects - arise due to the anisotropic and, more importantly, to the partially dissipative nature of the equation, adding further challenges to the analysis.

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arXiv2024

Finite-time singularity formation for scalar stretching equations

We consider equations of the type: \[\partial_t \omega = \omega R(\omega),\] for general linear operators $R$ in any spatial dimension. We prove that such equations almost always exhibit finite-time singularities for smooth and localized solutions. Singularities can even form in settings where solutions dissipate an energy. Such equations arise naturally as models in various physical settings such as inviscid and complex fluids.

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arXiv2024

Large amplitude quasi-periodic traveling waves in two dimensional forced rotating fluids

We establish the existence of quasi-periodic traveling wave solutions for the $\beta$-plane equation on $\mathbb{T}^2$ with a large quasi-periodic traveling wave external force. These solutions exhibit large sizes, which depend on the frequency of oscillations of the external force. Due to the presence of small divisors, the proof relies on a nonlinear Nash-Moser scheme tailored to construct nonlinear waves of large size. To our knowledge, this is the first instance of constructing quasi-periodic solutions for a quasilinear PDE in dimensions greater than one, with a 1-smoothing dispersion relation that is highly degenerate - indicating an infinite-dimensional kernel for the linear principal operator. This degeneracy challenge is overcome by preserving the traveling-wave structure, the conservation of momentum and by implementing normal form methods for the linearized system with sublinear dispersion relation in higher space dimension.

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arXiv2024

Relaxing the sharp density stratification and columnar motion assumptions in layered shallow water systems

We rigorously justify the bilayer shallow-water system as an approximation to the hydrostatic Euler equations in situations where the flow is density-stratified with close-to-piecewise constant density profiles, and close-to-columnar velocity profiles. Our theory accommodates with continuous stratification, so that admissible deviations from bilayer profiles are not pointwise small. This leads us to define refined approximate solutions that are able to describe at first order the flow in the pycnocline. Because the hydrostatic Euler equations are not known to enjoy suitable stability estimates, we rely on thickness-diffusivity contributions proposed by Gent and McWilliams. Our strategy also applies to one-layer and multilayer frameworks.

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arXiv2024

Ill-posedness of the hydrostatic Euler-Boussinesq equations and failure of hydrostatic limit

We investigate the hydrostatic approximation for inviscid stratified fluids, described by the two-dimensional Euler-Boussinesq equations in a periodic channel. Through a perturbative analysis of the hydrostatic homogeneous setting, we exhibit a stratified steady state violating the Miles-Howard criterion and generating a growing mode, both for the linearized hydrostatic and non-hydrostatic equations. By leveraging long-wave nonlinear instability for the original Euler-Boussinesq system, we demonstrate the breakdown of the hydrostatic limit around such unstable profiles. Finally, we establish the generic nonlinear ill-posedness of the limiting hydrostatic system in Sobolev spaces.

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arXiv2024

A new look at the controllability cost of linear evolution systems with a long gaze at localized data

We revisit the classical issue of the controllability/observability cost of linear first order evolution systems, starting with ODEs, before turning to some linear first order evolution PDEs in several space dimensions, including hyperbolic systems and pseudo-differential systems obtained by linearization in fluid mechanics. In particular we investigate the cost of localized initial data, and in the dispersive case, of initial data which are semi-classically microlocalized.

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arXiv2023

Strong ill-posedness in $L^{\infty}$ of the 2d Boussinesq equations in vorticity form and application to the 3d axisymmetric Euler Equations

We prove the strong ill-posedness in the sense of Hadamard of the two-dimensional Boussinesq equations in $W^{1, \infty}(\mathbb{R}^2)$ without boundary, extending to the case of systems the method that Shikh Khalil \& Elgindi arXiv:2207.04556v1 developed for scalar equations. We provide a large class of initial data with velocity and density of small $W^{1, \infty}(\mathbb{R}^2)$ size, for which the horizontal density gradient has a strong $L^{\infty}(\mathbb{R}^2)$-norm inflation in infinitesimal time, while the vorticity and the vertical density gradient remain bounded. Furthermore, exploiting the three-dimensional version of Elgindi's decomposition of the Biot-Savart law, we apply the method to the three-dimensional axisymmetric Euler equations with swirl and away from the vertical axis, showing that a large class of initial data with velocity field uniformly bounded in $W^{1, \infty}(\mathbb{R}^2)$ provides a solution whose swirl component has a strong $W^{1, \infty}(\mathbb{R}^2)$-norm inflation in infinitesimal time, while the potential vorticity remains bounded at least for small times. Finally, the $W^{1,\infty}$-norm inflation of the swirl (and the $L^{\infty}$-norm inflation of the vorticity field) is quantified from below by an explicit lower bound which depends on time, the size of the data and it is valid for small times.

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arXiv2022

Hard-congestion limit of the p-system in the BV setting

This note is concerned with the rigorous justification of the so-called hard congestion limit from a compressible system with singular pressure towards a mixed compressible-incompressible system modeling partially congested dynamics, for small data in the framework of BV solutions. We present a first convergence result for perturbations of a reference state represented by a single propagating large interface front, while the study of a more general framework where the reference state is constituted by multiple interface fronts is announced in the conclusion and will be the subject of a forthcoming paper. A key element of the proof is the use of a suitably weighted Glimm functional that allows to obtain precise estimates on the BV norm of the front-tracking approximation.

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arXiv2022

Relaxation approximation and asymptotic stability of stratified solutions to the IPM equation

We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in $\dot H^{1-\tau}(\mathbb{R}^2) \cap \dot H^s(\mathbb{R}^2)$ with $s > 3$ and for any $0 < \tau <1$. Such result improves the existing literature, where the asymptotic stability is proved for initial perturbations belonging at least to $H^{20}(\mathbb{R}^2)$. More precisely, the aim of the article is threefold. First, we provide a simplified and improved proof of global-in-time well-posedness of the Boussinesq equations with strongly damped vorticity in $H^{1-\tau}(\mathbb{R}^2) \cap \dot H^s(\mathbb{R}^2)$ with $s > 3$ and $0 < \tau <1$. Next, we prove the strong convergence of the Boussinesq system with damped vorticity towards (IPM) under a suitable scaling. Lastly, the asymptotic stability of stratified solutions to (IPM) follows as a byproduct. A symmetrization of the approximating system and a careful study of the anisotropic properties of the equations via anisotropic Littlewood-Paley decomposition play key roles to obtain uniform energy estimates. Finally, one of the main new and crucial points is the integrable time decay of the vertical velocity $\|u_2(t)\|_{L^\infty (\mathbb{R}^2)}$ for initial data only in $\dot H^{1-\tau}(\mathbb{R}^2) \cap \dot H^s(\mathbb{R}^2)$ with $s >3$.

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arXiv2022

On the hydrostatic limit of stably stratified fluids with isopycnal diffusivity

This article is concerned with rigorously justifying the hydrostatic limit for continuously stratified incompressible fluids under the influence of gravity. The main distinction of this work compared to previous studies is the absence of any (regularizing) viscosity contribution added to the fluid-dynamics equations; only thickness diffusivity effects are considered. Motivated by applications to oceanography, the diffusivity effects in this work arise from an additional advection term, the specific form of which was proposed by Gent and McWilliams in the 1990s to model the effective contributions of geostrophic eddy correlations in non-eddy-resolving systems. The results of this paper heavily rely on the assumption of stable stratification. We establish the well-posedness of the hydrostatic equations and the original (non-hydrostatic) equations for stably stratified fluids, along with their convergence in the limit of vanishing shallow-water parameter. These results are obtained in high but finite Sobolev regularity and carefully account for the various parameters involved. A key element of our analysis is the reformulation of the systems using isopycnal coordinates, enabling us to provide meticulous energy estimates that are not readily apparent in the original Eulerian coordinate system.

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arXiv2022

Reflection of internal gravity waves in the form of quasi-axisymmetric beams

Preservation of the angle of reflection when an internal gravity wave hits a sloping boundary generates a focusing mechanism if the angle between the direction of propagation of the incident wave and the horizontal is close to the slope inclination (near-critical reflection). This paper provides an explicit description of the leading approximation of the unique Leray solution to the near-critical reflection of internal waves from a slope in the form of a beam wave. More precisely, our beam wave approach allows to construct a fully consistent and Lyapunov stable approximate solution, $L^2$ -close to the Leray solution, in the form of a beam wave, within a certain (nonlinear) time-scale. To the best of our knowledge, this is the first result wherea mathematical study of internal waves in terms of spatially localized beam waves is performed.\\%A beam wave is a linear superposition of rapidly oscillating plane waves, where the high frequency of oscillation is proportional to the inverse of a power of the small parameter measuring the weak amplitude of waves. \\%Being localized in the physical space thanks to rapid oscillations (and high variations of the modulus of the wavenumber), beams are physically more relevant than plane waves/packets of waves, whose wavenumber is nearly fixed (microloca\-li\-zed). At the mathematical level, this marks a strong difference between the previous plane waves/packets of waves analysis and our approach. \\%The main novelty of this work is to exploit the spatial localization of beam waves to exhibit a spatially localized, physically relevant solution and to improve the previous mathematical results from a twofold perspective: 1) our beam wave approximate solution is the sum of a finite number of terms, each of them is a consistent solution to the system and there is no artificial/non-physical corrector; 2) thanks to the absence of artificial correctors (used in the previous results) and to the special structure of the nonlinear term, we can push the expansion of our solution to next orders, so improving the accuracy and enlarging the consistency time-scale.Finally, our results provide a set of initial conditions localized on rays, for which the Leray solution maintains approximately in $L^2$ the same localization.

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arXiv2022

Linear boundary layer analysis of the near-critical reflection of internal gravity waves with different sizes of viscosity and diffusivity

The aim of this work is to make a further step towards the understanding of the near-critical reflection of internal gravity waves from a slope in the more general and realistic context where the size of viscosity $\nu$ and the size of diffusivity $\kappa$ are different. In particular, we provide a systematic characterization of boundary layers (boundary layer wave packets) decays and sizes depending on the order of magnitude of viscosity and diffusivity. We can construct an $L^2$ stable approximate solution to the linear near-critical reflection problem under the scaling assumption of Dauxois \& Young JFM 1999, where either viscosity of diffusivity satisfies a precise scaling law in terms of the criticality parameter.

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arXiv2021

Asymptotic behavior of 2D stably stratified fluids with a damping term in the velocity equation

This article is concerned with the asymptotic behavior of the two-dimensional inviscid Boussinesq equations with a damping term in the velocity equation. Precisely, we provide the time-decay rates of the smooth solutions to that system. The key ingredient is a careful analysis of the Green kernel of the linearized problem in Fourier space, combined with bilinear estimates and interpolation inequalities for handling the nonlinearity.

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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.

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