Laura Caravenna
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Laura Caravenna has a total of 30 co-authors. Here are the top co-authors by number of shared publications:
- Fabio Ancona
Università di Padova
5 shared publications - Andrea Marson
Dipartimento di Matematica Pura ed Applicata, Universitá di Padova, Padova, Italy
3 shared publications - Francesco Rossi
Università Iuav di Venezia, Dipartimento di Culture del Progetto Ca’ Tron, S. Croce, 1957 30135, Venezia, Italy
3 shared publications - Gianluca Crippa
Departement Mathematik und Informatik, Universität Basel, Basel, Switzerland
3 shared publications - Giovanni Alberti
Dipartimento di Matematica, Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, Italy
3 shared publications - Laura V. Spinolo
IMATI-CNR, Pavia, Italy
3 shared publications
Latest Publications
- 1 January 2027
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems
- Vol. 59, 72-122
Local asymptotic patterns for viscous approximations of conservation laws
Solutions to hyperbolic conservation laws can be approximated in many different ways: by vanishing viscosity, relaxations, discrete or semi-discrete numerical schemes, approximation with a nonlocal flux, etc$ \ldots $ For some of these methods, general $ {\bf L}^1 $ convergence results are available. The aim of this paper is to understand the local behavior of these approximations, in a neighborhood of a point where the hyperbolic solution has a singularity; specifically, a point along a shock, or where two shocks interact, or where a new shock is formed. Given a sequence of $ \epsilon $-approximate solutions, a general expectation is that, by a suitable local rescaling of coordinates, as $ \epsilon\to 0 $ a well-defined limit is obtained. This corresponds to a specific "eternal solution" (globally defined both in space and in time) to the approximating equation. Precise results in this direction are here given, in the case of vanishing viscosity.- 1 January 2027
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems
- Vol. 59, 72-122
Local asymptotic patterns for viscous approximations of conservation laws
Solutions to hyperbolic conservation laws can be approximated in many different ways: by vanishing viscosity, relaxations, discrete or semi-discrete numerical schemes, approximation with a nonlocal flux, etc$ \ldots $ For some of these methods, general $ {\bf L}^1 $ convergence results are available. The aim of this paper is to understand the local behavior of these approximations, in a neighborhood of a point where the hyperbolic solution has a singularity; specifically, a point along a shock, or where two shocks interact, or where a new shock is formed. Given a sequence of $ \epsilon $-approximate solutions, a general expectation is that, by a suitable local rescaling of coordinates, as $ \epsilon\to 0 $ a well-defined limit is obtained. This corresponds to a specific "eternal solution" (globally defined both in space and in time) to the approximating equation. Precise results in this direction are here given, in the case of vanishing viscosity.- 27 May 2026
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Control and Optimization
- Vol. 64 (3) , 1522-1544
Consensus of Multiagent Systems under Communication Failure
We consider multiagent systems with cooperative interactions and study the convergence to consensus in the case of time-dependent connections, with possible communication failure. We prove a new condition ensuring consensus: we define a graph in which directed arrows correspond to connection functions that converge (in the weak sense) to some function with a positive integral on all intervals of the form [Formula: see text]. If the graph has a node reachable from all other indices, i.e., “globally reachable”, then the system converges to consensus. We show that this requirement generalizes some known sufficient conditions for convergence, such as Moreau’s or the Persistent Excitation one. We also give a second new condition, transversal to the known ones: total connectedness of the undirected graph formed by the nonvanishing of limiting functions.
- 27 May 2026
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Control and Optimization
- Vol. 64 (3) , 1522-1544
Consensus of Multiagent Systems under Communication Failure
We consider multiagent systems with cooperative interactions and study the convergence to consensus in the case of time-dependent connections, with possible communication failure. We prove a new condition ensuring consensus: we define a graph in which directed arrows correspond to connection functions that converge (in the weak sense) to some function with a positive integral on all intervals of the form [Formula: see text]. If the graph has a node reachable from all other indices, i.e., “globally reachable”, then the system converges to consensus. We show that this requirement generalizes some known sufficient conditions for convergence, such as Moreau’s or the Persistent Excitation one. We also give a second new condition, transversal to the known ones: total connectedness of the undirected graph formed by the nonvanishing of limiting functions.
- Open Access
On the regularity of continuous solutions to multidimensional scalar conservation laws with bounded source
We prove the Hölder regularity of continuous isentropic solutions to multi-dimensional scalar balance laws when the source term is bounded and the flux satisfies general assumptions of nonlinearity. The results are achieved by exploiting the kinetic formulation of the balance law.
Open AccessOn the regularity of continuous solutions to multidimensional scalar conservation laws with bounded source
We prove the Hölder regularity of continuous isentropic solutions to multi-dimensional scalar balance laws when the source term is bounded and the flux satisfies general assumptions of nonlinearity. The results are achieved by exploiting the kinetic formulation of the balance law.
- Open Access
- 10 June 2025
- Published by Springer Nature in Nonlinear Differential Equations and Applications NoDEA
- Vol. 32 (4) , 1-59
SBV regularity of Entropy Solutions for Hyperbolic Systems of Balance Laws with General Flux function
We prove that vanishing viscosity solutions to smooth non-degenerate systems of balance laws, having small bounded variation, in one space dimension, must be functions of special bounded variation ( $$\textrm{SBV}$$ ). For more than one equation, this $$\textrm{SBV}$$ -regularity for non-degenerate fluxes is new also in the case of systems of conservation laws outside the context of genuine nonlinearity. For general smooth strictly hyperbolic systems of balance laws, this regularity fails, as known for systems of conservation laws: in such case we generalize the $$\textrm{SBV}$$ -like regularity of the eigenvalue functions of the Jacobian matrix of the flux from conservation to balance laws. Proofs are based on extending Oleinink-type balance estimates, with the introduction of new source measure, a localization argument from [14, 40], and observations in real analysis.- 10 June 2025
- Published by Springer Nature in Nonlinear Differential Equations and Applications NoDEA
- Vol. 32 (4) , 1-59
Open AccessSBV regularity of Entropy Solutions for Hyperbolic Systems of Balance Laws with General Flux function
We prove that vanishing viscosity solutions to smooth non-degenerate systems of balance laws, having small bounded variation, in one space dimension, must be functions of special bounded variation ( $$\textrm{SBV}$$ ). For more than one equation, this $$\textrm{SBV}$$ -regularity for non-degenerate fluxes is new also in the case of systems of conservation laws outside the context of genuine nonlinearity. For general smooth strictly hyperbolic systems of balance laws, this regularity fails, as known for systems of conservation laws: in such case we generalize the $$\textrm{SBV}$$ -like regularity of the eigenvalue functions of the Jacobian matrix of the flux from conservation to balance laws. Proofs are based on extending Oleinink-type balance estimates, with the introduction of new source measure, a localization argument from [14, 40], and observations in real analysis. - 10 February 2025
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Mathematical Analysis
- Vol. 57 (1) , 979-995
Hölder Regularity of Continuous Solutions to Balance Laws and Applications in the Heisenberg Group
We prove Hölder regularity of any continuous solution [Formula: see text] to a one-dimensional scalar balance law when the source term is bounded and the flux is nonlinear of order [Formula: see text] with [Formula: see text]. Moreover, we prove that at almost every point [Formula: see text], it holds that [Formula: see text] as [Formula: see text]. Due to Lipschitz regularity along characteristics, this implies that at almost every point [Formula: see text], it holds that [Formula: see text] as [Formula: see text]. We apply the results to provide a new proof of the Rademacher theorem for intrinsic Lipschitz functions in the first Heisenberg group.
- 10 February 2025
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Mathematical Analysis
- Vol. 57 (1) , 979-995
Hölder Regularity of Continuous Solutions to Balance Laws and Applications in the Heisenberg Group
We prove Hölder regularity of any continuous solution [Formula: see text] to a one-dimensional scalar balance law when the source term is bounded and the flux is nonlinear of order [Formula: see text] with [Formula: see text]. Moreover, we prove that at almost every point [Formula: see text], it holds that [Formula: see text] as [Formula: see text]. Due to Lipschitz regularity along characteristics, this implies that at almost every point [Formula: see text], it holds that [Formula: see text] as [Formula: see text]. We apply the results to provide a new proof of the Rademacher theorem for intrinsic Lipschitz functions in the first Heisenberg group.
- Open Access
SBV-like regularity of Entropy Solutions for a Scalar Balance Law
In this note we discuss the SBV-regularity for a scalar balance law in one space dimension as a case study in order to explain the strategy that we apply in a separate paper to general hyperbolic systems of balance laws in one space dimension. While for a single balance law the more general work by Robyr is already available, the case of 1d-systems presents new behaviors that require a different strategy. This is why in this note we make the effort to introduce the notation and tools that are required for the case of more equations. When the flux presents linear degeneracies, it is know that entropy solutions can present nasty fractal Cantor-like behaviors, although f'(u) is still SBV: we thus discuss SBV-like regularity generalizing the work by Bianchini-Yu as SBV-regularity fails.
Open AccessSBV-like regularity of Entropy Solutions for a Scalar Balance Law
In this note we discuss the SBV-regularity for a scalar balance law in one space dimension as a case study in order to explain the strategy that we apply in a separate paper to general hyperbolic systems of balance laws in one space dimension. While for a single balance law the more general work by Robyr is already available, the case of 1d-systems presents new behaviors that require a different strategy. This is why in this note we make the effort to introduce the notation and tools that are required for the case of more equations. When the flux presents linear degeneracies, it is know that entropy solutions can present nasty fractal Cantor-like behaviors, although f'(u) is still SBV: we thus discuss SBV-like regularity generalizing the work by Bianchini-Yu as SBV-regularity fails.
- 1 September 2024
- Published by World Scientific Pub Co Pte Ltd in Journal of Hyperbolic Differential Equations
- Vol. 21 (03) , 621-657
Eulerian, Lagrangian and broad continuous solutions to a balance law with non convex flux II
We consider a continuous solution [ ] of the balance law [ ] in one space dimension, where the flux function [ ] is of class [ ] and the source term [ ] is bounded. This equation admits an Eulerian interpretation (namely the distributional one) and a Lagrangian interpretation (which can be further specified). Since [ ] is only continuous, these interpretations do not necessarily agree; moreover each interpretation naturally entails a different equivalence class for the source term [ ]. In this paper we complete the comparison between these notions of solutions started in the companion paper [G. Alberti, S. Bianchini and L. Caravenna, Eulerian, lagrangian and broad continuous solutions to a balance law with non convex flux I, J. Differ. Equ. 261 (2016) 4298–4337], and analize in detail the relations between the corresponding notions of source term.
- 1 September 2024
- Published by World Scientific Pub Co Pte Ltd in Journal of Hyperbolic Differential Equations
- Vol. 21 (03) , 621-657
Eulerian, Lagrangian and broad continuous solutions to a balance law with non convex flux II
We consider a continuous solution [ ] of the balance law [ ] in one space dimension, where the flux function [ ] is of class [ ] and the source term [ ] is bounded. This equation admits an Eulerian interpretation (namely the distributional one) and a Lagrangian interpretation (which can be further specified). Since [ ] is only continuous, these interpretations do not necessarily agree; moreover each interpretation naturally entails a different equivalence class for the source term [ ]. In this paper we complete the comparison between these notions of solutions started in the companion paper [G. Alberti, S. Bianchini and L. Caravenna, Eulerian, lagrangian and broad continuous solutions to a balance law with non convex flux I, J. Differ. Equ. 261 (2016) 4298–4337], and analize in detail the relations between the corresponding notions of source term.
- 17 November 2021
- Published by Elsevier in Systems & Control Letters
- Vol. 158, 105078
On the Lebesgue measure of the boundary of the evoluted set
- 17 November 2021
- Published by Elsevier in Systems & Control Letters
- Vol. 158, 105078
On the Lebesgue measure of the boundary of the evoluted set
- Open Access
On the Lebesgue measure of the boundary of the evoluted set
The evoluted set is the set of configurations reached from an initial set via a fixed flow for all times in a fixed interval. We find conditions on the initial set and on the flow ensuring that the evoluted set has negligible boundary (i.e. its Lebesgue measure is zero). We also provide several counterexample showing that the hypotheses of our theorem are close to sharp.
Open AccessOn the Lebesgue measure of the boundary of the evoluted set
The evoluted set is the set of configurations reached from an initial set via a fixed flow for all times in a fixed interval. We find conditions on the initial set and on the flow ensuring that the evoluted set has negligible boundary (i.e. its Lebesgue measure is zero). We also provide several counterexample showing that the hypotheses of our theorem are close to sharp.
- Open Access
- 1 March 2021
- Published by Taylor & Francis in Communications in Partial Differential Equations
- Vol. 46 (8) , 1488-1520
A directional Lipschitz extension lemma, with applications to uniqueness and Lagrangianity for the continuity equation
We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two nonequivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multi-valued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance of the extension can be estimated in terms of the Lipschitz constant for the geodesic distance of the original function. This Lipschitz extension lemma allows us to remove the high integrability assumption on the solution needed for the uniqueness within the DiPerna-Lions theory of continuity equations in the case of vector fields in the Sobolev space where p is larger than the space dimension, under the assumption that the so-called “forward-backward integral curves” associated to the vector field are trivial for almost every starting point. More precisely, for such vector fields we prove uniqueness and Lagrangianity for weak solutions of the continuity equation that are just locally integrable. Additionally, for such vector fields it is possible to prove almost everywhere uniqueness of (standard) integral curves, which also implies uniqueness of positive measure solutions to the continuity equation with absolutely continuous initial datum.- 1 March 2021
- Published by Taylor & Francis in Communications in Partial Differential Equations
- Vol. 46 (8) , 1488-1520
Open AccessA directional Lipschitz extension lemma, with applications to uniqueness and Lagrangianity for the continuity equation
We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two nonequivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multi-valued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance of the extension can be estimated in terms of the Lipschitz constant for the geodesic distance of the original function. This Lipschitz extension lemma allows us to remove the high integrability assumption on the solution needed for the uniqueness within the DiPerna-Lions theory of continuity equations in the case of vector fields in the Sobolev space where p is larger than the space dimension, under the assumption that the so-called “forward-backward integral curves” associated to the vector field are trivial for almost every starting point. More precisely, for such vector fields we prove uniqueness and Lagrangianity for weak solutions of the continuity equation that are just locally integrable. Additionally, for such vector fields it is possible to prove almost everywhere uniqueness of (standard) integral curves, which also implies uniqueness of positive measure solutions to the continuity equation with absolutely continuous initial datum.