arXiv subjects

Laura V. Spinolo

Publications and source records attributed to Laura V. Spinolo.

At least 19 recordsLinked to original sources

arXiv2026

Stability of dispersive boundary layers for scalar conservation laws in one space dimension

We study the zero-dispersion limit for a class of Korteweg--de Vries (KdV)-type initial-boundary value problems on the half-line, with Dirichlet boundary conditions assigned at $x=0$. We focus on the outflow regime, where the solution of the limiting scalar conservation law does not attain the boundary condition imposed on the dispersive problem. We construct a boundary layer profile, depending on the fast variable, which is uniquely determined, through the associated stationary third-order boundary layer equation, by the mismatch between the boundary conditions, and by the exponential decay at infinity in the fast variable. Our main result shows that, under suitable regularity and compatibility assumptions on the data, the dispersive solution is well approximated by a WKB expansion given by the sum of the smooth solution of the conservation law and the boundary layer profile. In particular, we establish stability of the boundary layer profile by proving quantitative estimates for the remainder term in a weighted energy norm, and show that it converges to $0$ in $H^1$, uniformly in time and up to the lifespan of the smooth solution of the conservation law. The proof is based on the analysis of a linearized energy functional and does not rely on complete integrability or inverse scattering techniques. It applies to general fluxes and requires no smallness assumption on the amplitude of the boundary layer. To the best of our knowledge, this is the first stability result for boundary layers of KdV-type equation on the half line.

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arXiv2025

Nonlocal Generalized Aw-Rascle-Zhang model: well-posedness and singular limit

We discuss a nonlocal version of the Generalized Aw-Rascle-Zhang model, a second-order vehicular traffic model where the empty road velocity is a Lagrangian marker governed by a transport equation. The evolution of the car density is described by a continuity equation where the drivers' velocity depends on both the empty road velocity and the convolution of the car density with an anisotropic kernel. We establish existence and uniqueness results. When the convolution kernel is replaced by a Dirac Delta, the nonlocal model formally boils down to the classical (local) Generalized Aw-Rascle-Zhang model, which consists of a conservation law coupled with a transport equation. In the case of exponential kernels, we establish convergence in the nonlocal-to-local limit by proving an Oleinik-type estimate for the convolution term. To the best of our knowledge, this is the first nonlocal-to-local limit result for a system of two non-decoupling equations with a nonlocal flux function.

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arXiv2025

Well-posedness results for the Generalized Aw-Rascle-Zhang model

We establish existence, uniqueness and stability results for the so-called Generalized Aw-Rascle-Zhang model, a second order traffic model introduced by Fan, Herty and Seibold. Our analysis is motivated by the companion paper 'Nonlocal Generalized Aw-Rascle-Zhang model: well-posedness and singular limit'.

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arXiv2025

On multidimensional nonlocal conservation laws with BV kernels

We establish local-in-time existence and uniqueness results for nonlocal conservation laws in several space dimensions under weak (that is, Sobolev or BV) differentiability assumptions on the convolution kernel. In contrast to the case of a smooth kernel, in general the solution experiences finite-time blow-up. We provide an explicit example showing that solutions corresponding to different smooth approximations of the convolution kernel in general converge to different measures after the blow-up time. This rules out a fairly natural strategy for extending the notion of solution of the nonlocal conservation law after the blow-up time.

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arXiv2024

An informal account of recent results on initial-boundary value problems for systems of conservation laws

This note aims at providing a rather informal and hopefully accessible overview of the fairly long and technical work [4]. In that paper, the authors established new global-in-time existence results for admissible solutions of nonlinear systems of conservation laws defined in domains with boundaries. The main novelty in [4] is that the solution is constructed by taking into account the underlying viscous mechanism, which is relevant because, in the case of initial-boundary value problems, different viscous approximations yield in general different limits. This note will frame the analysis of [4] in the relevant context, compare the main result with the previous existing literature, and touch upon the most innnovative technical points of the proof.

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arXiv2024

Existence of vanishing physical viscosity solutions of characteristic initial-boundary value problems for systems of conservation laws

We consider initial boundary-value problems for nonlinear systems of conservation laws in one space variable. It is known that in general different viscous mechanisms yield different solutions in the zero-viscosity limit. Here we focus on the most technically demanding case, known as boundary characteristic case, which occurs when one of the characteristic velocities of the system vanishes. We work in small total variation regimes and assume that every characteristic field is either genuinely nonlinear or linearly degenerate. We establish existence of admissible solutions satisfying a boundary condition consistent with the vanishing viscosity approximation given by a large class of physical (that is, mixed hyperbolic-parabolic) systems. In particular, our results apply to the zero-viscosity limit of the Navier-Stokes and viscous MHD equations, written in both Eulerian and Lagrangian coordinates. Our analysis relies on a fine boundary layers analysis and is based on the introduction of a new wave front-tracking algorithm. From the technical viewpoint, the most innovative elements are i) a new class of interaction estimates for boundary layers and boundary characteristic wave fronts hitting the boundary, which yields the introduction of a new Glimm-type functional; ii) a detailed analysis of the behavior of the wave front-tracking algorithm close to the boundary, which in turn yields relevant information on the limit.

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arXiv2023

An overview on the local limit of non-local conservation laws, and a new proof of a compactness estimate

Consider a non-local (i.e., involving a convolution term) conservation law: when the convolution term converges to a Dirac delta, in the limit we formally recover a classical (or "local") conservation law. In this note we overview recent progress on this so-called non-local to local limit and in particular we discuss the case of anistropic kernels, which is extremely relevant in view of applications to traffic models. We also provide a new proof of a related compactness estimate.

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arXiv2023

On the continuum limit of epidemiological models on graphs: convergence and approximation results

We focus on an epidemiological model (the archetypical SIR system) defined on graphs and study the asymptotic behavior of the solutions as the number of vertices in the graph diverges. By relying on the theory of so called graphons we provide a characterization of the limit and establish convergence results. We also provide approximation results for both deterministic and random discretizations.

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arXiv2023

Ole\uınik-type estimates for nonlocal conservation laws and applications to the nonlocal-to-local limit

We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term $W:=\mathbb{1}_{(-\infty,0]}(\cdot)\exp(\cdot) \ast ρ$ satisfy an Ole\uınik-type entropy condition. More precisely, under different sets of assumptions on the velocity function $V$, we prove that $W$ satisfies a one-sided Lipschitz condition and that $V'(W) W \partial_x W$ satisfies a one-sided bound, respectively. As a byproduct, we deduce that, as the exponential kernel is rescaled to converge to a Dirac delta distribution, the weak solution of the nonlocal problem converges to the unique entropy-admissible solution of the corresponding local conservation law, under the only assumption that the initial datum is essentially bounded and not necessarily of bounded variation.

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arXiv2022

Nonlocal traffic models with general kernels: singular limit, entropy admissibility, and convergence rate

Nonlocal conservation laws (the signature feature being that the flux function depends on the solution through the convolution with a given kernel) are extensively used in the modeling of vehicular traffic. In this work we discuss the singular local limit, namely the convergence of the nonlocal solutions to the entropy admissible solution of the conservation law obtained by replacing the convolution kernel with a Dirac delta. Albeit recent counter-examples rule out convergence in the general case, in the specific framework of traffic models (with anisotropic convolution kernels) the singular limit has been established under rigid assumptions, i.e. in the case of the exponential kernel (which entails algebraic identities between the kernel and its derivatives) or under fairly restrictive requirements on the initial datum. In this work we obtain general convergence results under assumptions that are entirely natural in view of applications to traffic models, plus a convexity requirement on the convolution kernels. We then provide a general criterion for entropy admissibility of the limit and a convergence rate. We also exhibit a counter-example showing that the convexity assumption is necessary for our main compactness estimate.

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arXiv2021

New regularity results for scalar conservation laws, and applications to a source-destination model for traffic flows on networks

We focus on entropy admissible solutions of scalar conservation laws in one space dimension and establish new regularity results with respect to time. First, we assume that the flux function $f$ is strictly convex and show that, for every $ x \in \mathbb{R}$, the total variation of the composite function $f \circ u(\cdot, x)$ is controlled by the total variation of the initial datum. Next, we assume that $f$ is monotone and, under no convexity assumption, we show that, for every $x$, the total variation of the left and right trace $u(\cdot, x^\pm)$ is controlled by the total variation of the initial datum. We also exhibit a counter-example showing that in the first result the total variation bound does not extend to the function $u$, or equivalently that in the second result we cannot drop the monotonicity assumption. We then discuss applications to a source-destination model for traffic flows on road networks. We introduce a new approach, based on the analysis of transport equations with irregular coefficients, and, under the assumption that the network only contains so-called T-junctions, we establish existence and uniqueness results for merely bounded data in the class of solutions where the traffic is not congested. Our assumptions on the network and the traffic congestion are basically necessary to obtain well-posedness in view of a counter-example due to Bressan and Yu. We also establish stability and propagation of $BV$ regularity, and this is again interesting in view of recent counter-examples.

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arXiv2021

Initial-boundary value problems for merely bounded nearly incompressible vector fields in one space dimension

We establish existence and uniqueness results for initial-boundary value problems for transport equations in one space dimension with nearly incompressible velocity fields, under the sole assumption that the fields are bounded. In the case where the velocity field is either nonnegative or nonpositive, one can rely on similar techniques as in the case of the Cauchy problem. Conversely, in the general case we introduce a new and more technically demanding construction, which heuristically speaking relies on a "lagrangian formulation" of the problem, albeit in a highly irregular setting. We also establish stability of the solution in weak and strong topologies, and propagation of the $BV$ regularity. In the case of either nonnegative or nonpositive velocity fields we also establish a $BV$-in-time regularity result, and we exhibit a counterexample showing that the result is false in the case of sign-changing vector fields. To conclude, we establish a trace renormalization property.

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arXiv2019

Local limit of nonlocal traffic models: convergence results and total variation blow-up

Consider a nonlocal conservation where the flux function depends on the convolution of the solution with a given kernel. In the singular local limit obtained by letting the convolution kernel converge to the Dirac delta one formally recovers a conservation law. However, recent counter-examples show that in general the solutions of the nonlocal equations do not converge to a solution of the conservation law. In this work we focus on nonlocal conservation laws modeling vehicular traffic: in this case, the convolution kernel is anisotropic. We show that, under fairly general assumptions on the (anisotropic) convolution kernel, the nonlocal-to-local limit can be rigorously justified provided the initial datum satisfies a one-sided Lipschitz condition and is bounded away from $0$. We also exhibit a counter-example showing that, if the initial datum attains the value $0$, then there are severe obstructions to a convergence proof.

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arXiv2019

On the singular local limit for conservation laws with nonlocal fluxes

We give an answer to a question posed in [P. Amorim, R. Colombo, and A. Teixeira, ESAIM Math. Model. Numerics. Anal. 2015], which can be loosely speaking formulated as follows. Consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law: can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one in general does not have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law.

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arXiv2019

On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws

We deal with the numerical investigation of the local limit of nonlocal conservation laws. Previous numerical experiments suggest convergence in the local limit. However, recent analytic results state that (i) in general convergence does not hold because one can exhibit counterexamples; (ii) convergence can be recovered provided viscosity is added to both the local and the nonlocal equations. Motivated by these analytic results, we investigate the role of numerical viscosity in the numerical study of the local limit of nonlocal conservation laws. In particular, we show that the numerical viscosity of Lax-Friedrichs type schemes jeopardizes the reliability of the numerical scheme and erroneously detects convergence in cases where convergence is ruled out by analytic results. We also test Godunov type schemes, less affected by numerical viscosity, and show that in some cases they provide more reliable results.

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arXiv2019

Recent results on the singular local limit for nonlocal conservation laws

We provide an informal overview of recent developments concerning the singular local limit of nonlocal conservation laws. In particular, we discuss some counterexamples to convergence and we highlight the role of numerical viscosity in the numerical investigation of the nonlocal-to-local limit. We also state some open questions and describe recent related progress.

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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.

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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.

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