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Laura Caravenna

Publications and source records attributed to Laura Caravenna.

17 recordsLinked to original sources
arXiv2026

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. Aim of this paper is to understand the local behavior of these approximations, in a neighborhood of 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 $ε$-approximate solutions, a general expectation is that, by a suitable local rescaling of coordinates, as $ε\to 0$ 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 this direction are here given, in the case of vanishing viscosity.

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arXiv2025

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. For more than one equation, this is new also in the case of systems of conservation laws out of the context of genuine nonlinearity. For general smooth strictly hyperbolic systems of balance laws, this regularity fails, as known for systems of balance laws: we generalize the SBV-like regularity of the eigenvalue functions of the Jacobian matrix of flux from conservation to balance laws. Proofs are based on extending Oleinink-type balance estimates, with the introduction of new source measures, classical localization arguments, and observations in real analysis.

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arXiv2025

Consensus in Multiagent Systems under communication failure

We consider multi-agent 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 $[t,+\infty)$. 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 non-vanishing of limiting functions.

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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.

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arXiv2024

Hölder regularity of continuous solutions to balance laws and applications in the Heisenberg group

We prove Hölder regularity of any continuous solution $u$ to a $1$-D scalar balance law $u_t + [f(u)]_x = g$, when the source term $g$ is bounded and the flux $f$ is nonlinear of order $\ell \in \mathbb{N}$ with $\ell \ge 2$. For example, $\ell = 3$ if $f(u) = u^3$. Moreover, we prove that at almost every point $(t,x)$, it holds $u(t,x+h) - u(t,x) = o(|h|^{\frac{1}{\ell}})$ as $h \to 0$. Due to Lipschitz regularity along characteristics, this implies that at almost every point $(t,x)$, it holds $u(t+k,x+h) - u(t,x) = o((|h|+|k|)^{\frac{1}{\ell}})$ as $|(h,k)|\to 0$. We apply the results to provide a new proof of the Rademacher theorem for intrinsic Lipschitz functions in the first Heisenberg group.

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arXiv2024

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.

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arXiv2022

Exponential Stability of Large BV Solutions in a Model of Granular flow

We consider a $2\times 2$ system of hyperbolic balance laws, in one-space dimension, that describes the evolution of a granular material with slow erosion and deposition. The dynamics is expressed in terms of the thickness of a moving layer on top and of a standing layer at the bottom. The system is linearly degenerate along two straight lines in the phase plane and genuinely nonlinear in the subdomains confined by such lines. In particular, the characteristic speed of the first characteristic family is strictly increasing in the region above the line of linear degeneracy and strictly decreasing in the region below such a line. The non dissipative source term is the product of two quantities that are transported with the two different characteristic speeds. The global existence of entropy weak solutions of the Cauchy problem for such a system was established by Amadori and Shen for initial data with bounded but possibly large total variation, under the assumption that the initial height of the moving layer be sufficiently small. In this paper we establish the Lipschitz ${\bf L^1}$-continuous dependence of the solutions on the initial data with a Lipschitz constant that grows exponentially in time. The proof of the ${\bf L^1}$-stability of solutions is based on the construction of a Lyapunov like functional equivalent to the ${\bf L^1}$-distance, in the same spirit of the functional introduced by Liu and Yang and then developed by Bressan, Liu, Yang for systems of conservation laws with genuinely nonlinear or linearly degenerate characteristic fields.

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arXiv2021

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.

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arXiv2021

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 non-equivalent 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 $W^{1,p}$, 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.

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arXiv2018

On the structure of solutions for general hyperbolic systems of balance laws

The paper describes the qualitative structure of BV entropy solutions of a strictly hyperbolic system of balance laws with characteristic fields either piecewise genuinely nonlinear or linearly degenerate. In particular, we provide an accurate description of the local and global wave-front structure of a BV solution generated by a fractional step scheme combined with a wave-front tracking algorithm. This extends the corresponding results by Binahcini and Yu for strictly hyperbolic systems of conservation laws.

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arXiv2016

Uniqueness and Lagrangianity for solutions with lack of integrability of the continuity equation

We deal with the uniqueness of distributional solutions to the continuity equation with a Sobolev vector field and with the property of being a Lagrangian solution, that means transported by a flow of the associated ordinary differential equation. We work in a framework with lack of local integrability of the solution, in which the classical DiPerna-Lions theory of uniqueness and Lagrangianity of distributional solutions does not apply due to the insufficient integrability of the commutator. We introduce a general principle to prove that a solution is Lagrangian: we rely on a disintegration along the unique flow and on a new directional Lipschitz extension lemma, used to construct a large class of test functions in the Lagrangian distributional formulation of the continuity equation.

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arXiv2015

New interaction estimates for the Baiti-Jenssen system

We establish new interaction estimates for a system introduced by Baiti and Jenssen. These estimates are pivotal to the analysis of the wave front-tracking approximation. In a companion paper we use them to construct a counter-example which shows that Schaeffer's Regularity Theorem for scalar conservation laws does not extend to systems. The counter-example we construct shows, furthermore, that a wave-pattern containing infinitely many shocks can be robust with respect to perturbations of the initial data. The proof of the interaction estimates is based on the explicit computation of the wave fan curves and on a perturbation argument.

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arXiv2015

Schaeffer's regularity theorem for scalar conservation laws does not extend to systems

Schaeffer's regularity theorem for scalar conservation laws can be loosely speaking formulated as follows. Assume that the flux is uniformly convex, then for a generic smooth initial datum the admissible solution is smooth outside a locally finite number of curves in the (t,x) plane. Here the term `generic' is to be interpreted in a suitable sense, related to the Baire Category Theorem. Whereas other regularity results valid for scalar conservation laws with convex fluxes have been extended to systems of conservation laws with genuinely nonlinear characteristic fields, in this work we exhibit an explicit counterexample which rules out the possibility of extending Schaeffer's Theorem. The analysis relies on careful interaction estimates and uses fine properties of the wave front-tracking approximation.

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arXiv2015

A note on regularity and failure of regularity for systems of conservation laws via Lagrangian formulation

The paper recalls two of the regularity results for Burgers' equation, and discusses what happens in the case of genuinely nonlinear, strictly hyperbolic systems of conservation laws. The first regularity result which is considered is Oleinik-Ambroso-De Lellis SBV estimate: it provides bounds on the x-derivative of u when u is an entropy solution of the Cauchy problem for Burgers' equation with bounded initial data. Its extensions to the case of systems is then mentioned. The second regularity result of debate is Schaeffer's theorem: entropy solutions to Burgers' equation with smooth and generic, in a Baire category sense, initial data are piecewise smooth. The failure of the same regularity for general genuinely nonlinear systems is next described. The main focus of this paper is indeed including heuristically an original counterexample where a kind of stability of a shock pattern made by infinitely many shocks shows up, referring to [Caravenna-Spinolo] for the rigorous result.

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arXiv2011

SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one space dimension

We prove that if $t \mapsto u(t) \in \mathrm {BV}(\R)$ is the entropy solution to a $N \times N$ strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields \[ u_t + f(u)_x = 0, \] then up to a countable set of times $\{t_n\}_{n \in \mathbb N}$ the function $u(t)$ is in $\mathrm {SBV}$, i.e. its distributional derivative $u_x$ is a measure with no Cantorian part. The proof is based on the decomposition of $u_x(t)$ into waves belonging to the characteristic families \[ u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal M(\R), \ \tilde r_i(t) \in \mathrm R^N, \] and the balance of the continuous/jump part of the measures $v_i$ in regions bounded by characteristics. To this aim, a new interaction measure $μ_{i,\jump}$ is introduced, controlling the creation of atoms in the measure $v_i(t)$. The main argument of the proof is that for all $t$ where the Cantorian part of $v_i$ is not 0, either the Glimm functional has a downward jump, or there is a cancellation of waves or the measure $μ_{i,\mathrm{jump}}$ is positive.

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