arXiv subjects

Stefano Bianchini

Publications and source records attributed to Stefano Bianchini.

At least 19 recordsLinked to original sources

arXiv2026

Traveling profiles and control cost for a PDE describing the evolution of invasive species

We develop a detailed analysis of optimal traveling waves $U(t,x) = U(x - \beta t)$ for a model of invasive-species control proposed in [Bressan, Chiri, and Salehi, Math. Models Methods Appl. Sci., 2022] : the relative density $U \in [0,1]$ of the invasive species satisfies the following reaction-diffusion equation with a positive control \begin{equation} \label{Equa:PDE_abstract} U_t = U_{xx} + f(U) - \tilde \alpha(t,x) U, \quad U \in [0,1], \ \tilde \alpha \geq 0. \end{equation} The control $\tilde \alpha(t,x)$ represents the fraction of the population removed at $(t,x)$: the minimal control effort $E(\beta,f)$ required to sustain a traveling invasion front with prescribed speed $\beta$ is defined as the minimal $L^1$-norm of $\tilde \alpha$ for a traveling wave solution $U(x-\beta t)$ to the PDE. In order to study large scale dynamics $(t,x) \mapsto (\epsilon t,\epsilon x)$, a fundamental role is played by the structure of traveling waves and the convexity and regularity properties of $E$. The main results of this paper are the following: 1)In the phase plane $(U,P=U_x)$, there exists a unique optimal profile $P_\beta(U)$ minimizing the effort; 2) It satisfies explicit first-order conditions, which are both necessary and sufficient; 3) The associated control is acting on an open subset of the set $\{U : P_\beta(U) = \sqrt{U f(U)}\}$, in particular it is uniformly integrable, and it depends smoothly on $(\beta,f)$ on a dense open set; 4)The effort function $E(\beta,f)$ is only $C^1$ w.r.t. $\beta$ and Lipschitz w.r.t. $f$ in the $C^2$-topology, and is asymptocally linear for $\beta \to \infty$; 5)$\beta \mapsto E(\beta,f)$ is in general neither convex nor subadditive.

math.OC↗

arXiv2026

Measure preserving maps with bounded total variation

Consider a piecewise affine Lipschitz map $\phi : \Omega \to \mathbb R$, where $\Omega \subset \mathbb R^d$ is an open set, and assume that $x \mapsto x + t \nabla \phi(x)$ is injective for almost every $t > 0$. In (J.-G. Liu, R.~L. Pego, \emph{Rigidly breaking potential flows and a countable Alexandrov theorem for polytopes}, Pure Appl. Anal., \textbf{7}(4), 2025) the authors conjecture that every such $\phi$ must be locally convex. We prove the result assuming additionally $\nabla \phi \in BV_{loc}(\Omega)$, for a more general class of measure preserving maps.

math.AP↗

arXiv2025

Scientific Discovery in the Age of AI and Supercomputing

Artificial intelligence (AI) and high-performance computing (HPC) are transforming scientific capabilities and the way science is conducted. Yet their combined impact on scientific discovery remains poorly understood, as do inequalities in access to these capabilities across countries and institutions. Drawing on metadata from more than five million scientific publications (2000-2024) across 27 fields, we examine how the convergence of AI and HPC correlates with scientific breakthroughs. Our results show that this computational synergy is most pronounced at the scientific frontier: research combining AI and HPC is more likely to introduce novel ideas and achieve top-cited status than either conventional work or research using AI or HPC in isolation. We also document growing disparities in access to supercomputing resources and AI expertise, which are increasingly concentrated in a small number of regions (dominated by the United States and China, though the EU27 aggregate maintains high competitiveness in combined AI+HPC output). The future of discovery will depend not only on advances in algorithms and computing power, but also on enacting policies that democratise these capabilities across the global scientific ecosystem.

cs.CY↗

arXiv2025

Existence of spiral strategies for blocking fire spreading

In this paper we address the problem for blocking fire by constructing a wall $\zeta$ whose shape is spiral-like. This is supposed to be the best strategy when a single firefighter is constructing the wall with a finite construction speed $\sigma$: the barriers which satisfy this bound on the construction speed are called admissible. We prove a sharp version of Bressan's Fire Conjecture in this case, i.e. when admissible barriers are spiral-like curves: namely, there exists a spiral-like barrier confining the fire in a bounded region of $\mathbb R^2$ if and only if the speed of construction of the barrier $\sigma$ is strictly larger than a critical speed $\bar \sigma = 2.614...$. The existence of confining spiral barriers for $\sigma > \bar \sigma$ is already known [Bressan A. et al., 2008, Klein R. et al., 2019], while we concentrate on the negative side, i.e. if $\sigma \leq \bar \sigma$ no admissible spiral blocks the fire. The proof of these results relies on: 1) the precise definition of spiral barrier and its representation; 2) the analysis of saturated spiral barriers as a Retarded Differential Equation (RDE) in the spirit of [Klein R. et al., 2019]; 3) the equivalent reformulation of the conjecture as a minimum problem for a prescribed functional; 4) the construction of the optimal closing spiral; 5) the analysis of a differentiable path of admissible spirals along which the functional is differentiable, and in particular increasing when moving from the optimal spiral to any other one (homotopy argument). Due to the complexity of the solution, the evaluation of the quantities needed to prove that the functional is increasing is performed numerically.

math.AP↗

arXiv2024

Eulerian, Lagrangian and broad continuous solutions to a balance law with non convex flux II

We consider a *continuous* solution $u$ of the balance law \[ \partial_{\mathit t} u + \partial_{\mathit x} (f(u)) = g\] in one space dimension, where the flux function $f$ is of class $C^2$ and the source term $g$ is bounded. This equation admits an Eulerian intepretation (namely the distributional one) and a Lagrangian intepretation (which can be further specified). Since $u$ is only continuous, these interpretations do not necessessarily agree; moreover each interpretation naturally entails a different equivalence class for the source term $g$. In this paper we complete the comparison between these notions of solutions started in the companion paper [Alberti-Bianchini-Caravenna I], and analize in detail the relations between the corresponding notions of source term.

math.AP↗

arXiv2023

Drivers and Barriers of AI Adoption and Use in Scientific Research

New technologies have the power to revolutionize science. It has happened in the past and is happening again with the emergence of new computational tools, such as artificial intelligence and machine learning. Despite the documented impact of these technologies, there remains a significant gap in understanding the process of their adoption within the scientific community. In this paper, we draw on theories of scientific and technical human capital to study the integration of AI in scientific research, focusing on the human capital of scientists and the external resources available within their network of collaborators and institutions. We validate our hypotheses on a large sample of publications from OpenAlex, covering all sciences from 1980 to 2020, and identify a set key drivers and inhibitors of AI adoption and use in science. Our results suggest that AI is pioneered by domain scientists with a `taste for exploration' and who are embedded in a network rich of computer scientists, experienced AI scientists and early-career researchers; they come from institutions with high citation impact and a relatively strong publication history on AI. The access to computing resources only matters for a few scientific disciplines, such as chemistry and medical sciences. Once AI is integrated into research, most adoption factors continue to influence its subsequent reuse. Implications for the organization and management of science in the evolving era of AI-driven discovery are discussed.

cs.CY↗

arXiv2023

Existence and Blow-up for Non-autonomous Scalar Conservation Laws With Viscosity

We consider a question posed in [GSZZ22], namely the blow-up of the PDE $$u_t + (b(t,x) u^{1+k})_x = u_{xx}$$ when $b$ is uniformly bounded, Lipschitz and $k = 2$. We give a complete answer to the behavior of solutions when $b$ belongs to the Lorentz spaces $b \in L^{p,\infty}$, $p \in (2,\infty]$, or $b_x \in L^{p,\infty}$, $p \in (1,\infty]$.

math.AP↗

arXiv2023

Optimal Solutions for a Class of Set-Valued Evolution Problems

The paper is concerned with a class of optimization problems for moving sets $t\mapstoΩ(t)\subset\mathbb{R}^2$, motivated by the control of invasive biological populations. Assuming that the initial contaminated set $Ω_0$ is convex, we prove that a strategy is optimal if an only if at each given time $t\in [0,T]$ the control is active along the portion of the boundary $\partial Ω(t)$ where the curvature is maximal. In particular, this implies that $Ω(t)$ is convex for all $t\geq 0$. The proof relies on the analysis of a one-step constrained optimization problem, obtained by a time discretization.

math.OC↗

arXiv2023

Questioning the impact of AI and interdisciplinarity in science: Lessons from COVID-19

Artificial intelligence (AI) has emerged as one of the most promising technologies to support COVID-19 research, with interdisciplinary collaborations between medical professionals and AI specialists being actively encouraged since the early stages of the pandemic. Yet, our analysis of more than 10,000 papers at the intersection of COVID-19 and AI suggest that these collaborations have largely resulted in science of low visibility and impact. We show that scientific impact was not determined by the overall interdisciplinarity of author teams, but rather by the diversity of knowledge they actually harnessed in their research. Our results provide insights into the ways in which team and knowledge structure may influence the successful integration of new computational technologies in the sciences.

cs.CY↗

arXiv2022

Metric entropy for Hamilton-Jacobi equation with uniformly directionally convex Hamiltonian

The present paper first aims to study the BV-type regularity for viscosity solutions of the Hamilton-Jacobi equation \[ u_t(t,x)+H\big(D_{x} u(t,x)\big)~=~0\qquad\forall (t,x)\in ]0,\infty[\times\mathbb{R}^d \] with a coercive and uniformly directionally convex Hamiltonian $H\in\mathcal{C}^{1}(\mathbb{R}^d)$. More precisely, we establish a BV bound on the slope of backward characteristics $DH(u(t,\cdot))$ starting at a positive time $t>0$. Relying on the BV bound, we quantify the metric entropy in ${\bf W}^{1,1}_{\mathrm{loc}}(\mathbb{R}^d)$ for the map $S_t$ that associates to every given initial data $u_0\in{\bf Lip}\big(\mathbb{R}^d\big)$, the corresponding solution $S_tu_0$. Finally, a counter example is constructed to show that both $D_xu(t,\cdot)$ and $DH(D_xu(t,\cdot))$ fail to be in $BV_{\mathrm{loc}}$ for a general strictly convex and coercive $H\in\mathcal{C}^2(\mathbb{R}^d)$.

math.AP↗

arXiv2021

Exact integrability conditions for cotangent vector fields

In Quantum Hydro-Dynamics the following problem is relevant: let $(\sqrtρ,Λ) \in H^1(\R^d) \times L^2(\R^d,\R^d)$ be a finite energy hydrodynamics state, i.e. $Λ= 0$ when $ρ= 0$ and \begin{equation*} E = \int_{\R^d} \frac{1}{2} \big| \nabla \sqrtρ \big|^2 + \frac{1}{2} Λ^2 \mathcal L^d < \infty. \end{equation*} The question is under which conditions there exists a wave function $ψ\in H^1(\R^d,\C)$ such that \begin{equation*} \sqrtρ = |ψ|, \quad J = \sqrtρ Λ= \Im \big( \bar ψ\nabla ψ). \end{equation*} The second equation gives for $ψ= \sqrtρ w$ smooth, $|w| = 1$, that $i Λ= \sqrtρ \bar w \nabla w$. Interpreting $ρ\mathcal L^d$ as a measure in the metric space $\R^d$, this question can be stated in generality as follows: given metric measure space $(X,d,μ)$ and a cotangent vector field $v \in L^2(T^* X)$, is there a function $w \in H^1(μ,\mathbb S^1)$ such that \begin{equation*} dw = i w v. \end{equation*} %dw = i w v$? We show that under some assumptions on the metric measure space $(X,d,μ)$ (conditions which are verified on Riemann manifolds with the measure $μ= ρ\mathrm{Vol}$ or more generally on non-branching $MCP(K,N)$), we show that the necessary and sufficient conditions for the existence of $w$ is that (in the case of differentiable manifold) \begin{equation*} \int v(γ(t)) \cdot \dot γ(t) dt \in 2π\Z \end{equation*} for $π$-a.e. $γ$, where $π$ is a test plan supported on closed curves. This condition generalizes the conditions that the vorticity is quantized. We also give a representation of every possible solution. In particular, we deduce that the wave function $ψ= \sqrtρ w$ is in $W^{1,2}(X)$ whenever $\sqrtρ \in W^{1,2}(X)$.

math.FA↗

arXiv2021

Properties of Mixing BV vector fields

We consider the density properties of divergence-free vector fields $ b \in L^1([0,1],\textit{BV}([0,1]^2)) $ which are ergodic/weakly mixing/strongly mixing: this means that their Regular Lagrangian Flow $X_t$ is an ergodic/weakly mixing/strongly mixing measure preserving map when evaluated at $t=1$. Our main result is that there exists a $G_δ$-set $\mathcal U \subset L^1_{t,x}([0,1]^3)$ made of divergence-free vector fields such that $1)$ the map $Φ$ associating $b$ with its RLF $X_t$ can be extended as a continuous function to the $G_δ$-set $\mathcal{U}$; $2)$ ergodic vector fields $b$ are a residual $G_δ$-set in $\mathcal{U}$; $3)$ weakly mixing vector fields $b$ are a residual $G_δ$-set in $\mathcal{U}$; $4)$ strongly mixing vector fields $b$ are a first category set in $\mathcal{U}$; $5)$ exponentially (fast) mixing vector fields are a dense subset of $\mathcal{U}$. The proof of these results is based on the density of BV vector fields such that $X_{t=1}$ is a permutation of subsquares, and suitable perturbations of this flow to achieve the desired ergodic/mixing behavior. These approximation results have an interest of their own. A discussion on the extension of these results to $d \geq 3$ is also presented.

math.DS↗

arXiv2021

Global health science leverages established collaboration network to fight COVID-19

How has the science system reacted to the early stages of the COVID-19 pandemic? Here we compare the (growing) international network for coronavirus research with the broader international health science network. Our findings show that, before the outbreak, coronavirus research realized a relatively small and rather peculiar niche within the global health sciences. As a response to the pandemic, the international network for coronavirus research expanded rapidly along the hierarchical structure laid out by the global health science network. Thus, in face of the crisis, the global health science system proved to be structurally stable yet versatile in research. The observed versatility supports optimistic views on the role of science in meeting future challenges. However, the stability of the global core-periphery structure may be worrying, because it reduces learning opportunities and social capital of scientifically peripheral countries -- not only during this pandemic but also in its "normal" mode of operation.

econ.GN↗

arXiv2020

Deep Learning in Science

Much of the recent success of Artificial Intelligence (AI) has been spurred on by impressive achievements within a broader family of machine learning methods, commonly referred to as Deep Learning (DL). This paper provides insights on the diffusion and impact of DL in science. Through a Natural Language Processing (NLP) approach on the arXiv.org publication corpus, we delineate the emerging DL technology and identify a list of relevant search terms. These search terms allow us to retrieve DL-related publications from Web of Science across all sciences. Based on that sample, we document the DL diffusion process in the scientific system. We find i) an exponential growth in the adoption of DL as a research tool across all sciences and all over the world, ii) regional differentiation in DL application domains, and iii) a transition from interdisciplinary DL applications to disciplinary research within application domains. In a second step, we investigate how the adoption of DL methods affects scientific development. Therefore, we empirically assess how DL adoption relates to re-combinatorial novelty and scientific impact in the health sciences. We find that DL adoption is negatively correlated with re-combinatorial novelty, but positively correlated with expectation as well as variance of citation performance. Our findings suggest that DL does not (yet?) work as an autopilot to navigate complex knowledge landscapes and overthrow their structure. However, the 'DL principle' qualifies for its versatility as the nucleus of a general scientific method that advances science in a measurable way.

cs.CY↗

arXiv2019

Renormalization for autonomous nearly incompressible BV vector fields in 2D

Given a bounded autonomous vector field $b \colon \mathbb R^d \to \mathbb R^d$, we study the uniqueness of bounded solutions to the initial value problem for the related transport equation \begin{equation*} \partial_t u + b \cdot \nabla u= 0. \end{equation*} We are interested in the case where $b$ is of class BV and it is nearly incompressible. Assuming that the ambient space has dimension $d=2$, we prove uniqueness of weak solutions to the transport equation. The starting point of the present work is the result which has been obtained in \cite{BG} (where the \emph{steady} case is treated). Our proof is based on splitting the equation onto a suitable partition of the plane: this technique was introduced in \cite{ABC1}, using the results on the structure of level sets of Lipschitz maps obtained in \cite{ABC2}. Furthermore, in order to construct the partition, we use Ambrosio's superposition principle \cite{ambrosiobv}.

math.AP↗

arXiv2018

Characteristic boundary layers for mixed hyperbolic-parabolic systems in one space dimension, and applications to the Navier-Stokes and MHD equations

We provide a detailed analysis of the boundary layers for mixed hyperbolic-parabolic systems in one space dimension and small amplitude regimes. As an application of our results, we describe the solution of the so-called boundary Riemann problem recovered as the zero viscosity limit of the physical viscous approximation. In particular, we tackle the so called doubly characteristic case, which is considerably more demanding from the technical viewpoint and occurs when the boundary is characteristic for both the mixed hyperbolic-parabolic system and for the hyperbolic system obtained by neglecting the second order terms. Our analysis applies in particular to the compressible Navier-Stokes and MHD equations in Eulerian coordinates, with both positive and null conductivity. In these cases, the doubly characteristic case occurs when the velocity is close to 0. The analysis extends to non-conservative systems.

math.AP↗

arXiv2017

Optimality of integrability estimates for advection-diffusion equations

We discuss $L^p$ integrability estimates for the solution $u$ of the advection-diffusion equation $\partial_t u + \mathrm{div} (bu) = Δu$, where the velocity field $b \in L^r_t L^q_x$. We first summarize some classical results proving such estimates for certain ranges of the exponents $r$ and $q$. Afterwards we prove the optimality of such ranges by means of new original examples.

math.AP↗

arXiv2016

On the structure of $L^\infty$-entropy solutions to scalar conservation laws in one-space dimension

We prove that if $u$ is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. \\ In particular the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a $C^0$-sense up to the degeneracy due to the segments where $f"=0$. We prove also that the initial data is taken in a suitably strong sense and we give some counterexamples which show that these results are sharp.

math.AP↗