Andrea Corli
Hysteresis in traffic flows
Abstract
We discuss two hyperbolic nonlinear systems of partial differential equations modeling traffic flows. The main feature of both of them is the introduction of a hysteretic state variable and a related equation for it, which is designed to model stop-and-go waves. The fundamental diagram is naturally divided into two disconnected regions, one for congested flows and one for free flows. The diffusive counterparts of both models are also studied and have interesting properties, such as maximum principles. We show that the inviscid Riemann problem has a solution, which is unique among solutions that are monotone in the velocity; all the waves exploited in the construction have suitable viscous profiles. We explicitly show how both systems can model the decay, emergence or persistence of stop-and-go waves (a feature that is missing in the famous Lighthill-Whitham-Richards and Aw-Rascle-Zhang models) in the congested zone, and characterize such a behavior by a simple geometric condition. We further show that this behavior may strongly depend on different driving habits. We also study the decay of stop-and-go waves in the free zone, where the hysteresis loops reverse their direction: in that region stop-and-go waves can decay even faster than in the congested zone under the above geometric condition.Giuseppina D’Aguì
Two weak solutions for nonlinear differential problems
Abstract
In this talk, we present a result concerning two critical points for a class of differentiable functionals defined in a Banach space. This result derives from a combination of a recent local minimum theorem and the classical Ambrosetti-Rabinowitz theorem. In particular, our analysis shows that the local minimum is also the global minimum of the functional under a suitable restriction. Consequently, we find that all the paths starting from it have a high level greater than zero, so the second critical point must be non-zero. Furthermore, we obtain that the two non-trivial critical points have opposite energy signs. By applying it to different classes of nonlinear differential problems, we get the existence of two non-trivial weak solutions.Michela Eleuteri
Lipschitz regularity results for energy integrals with slow growth
Abstract
Since the pioneering results by G. Stampacchia [S] (see also [HS]), the bounded slope condition has been a fundamental tool allowing to show existence of weak solutions for non-linear elliptic equations in the class of Lipschitz continuous functions. Our aim is to present a generalization of such classical results to the case of non-uniformly elliptic functionals of the Calculus of Variations of the form \begin{equation} \mathcal{F}(u) = \int_{\Omega} f(D(u(x))) \, dx \tag{1} \end{equation} where the quadratic form of the second derivatives of \(f\), namely \(D^2 f,\) satisfies \begin{equation} g_1 (|\xi|) |\lambda|^2 \le \, \sum_{i, j =1}^n f_{\xi_i \xi_j}(\xi) \lambda_i \lambda_j \le \,g_2 (|\xi|) |\lambda|^2 \tag{2}\end{equation} and where \(g_1, g_2: [0, + \infty) \rightarrow [0, + \infty)\) are given non-negative real functions which allow to control the ellipticity in the minimization problem. The connection with PDEs is established once we observe that a local minimizer of the functional (1) satisfies, under suitable convexity assumptions, a non-linear elliptic PDE in divergence form (namely the Euler-Lagrange equation associated to the integral functional). In this talk we will present two recent results where the bounded slope condition revealed to be crucial to get the Lipschitz regularity for the local minimizers. The first one [EMMP] is a joint project with P. Marcellini, E. Mascolo and S. Perrotta and it is concerned with the case of general slow growth conditions under the ellipticity condition (2), where \(g_2(t)\) is a decreasing function with respect to \(t\); of course in the model case \(g_2(t) = (1 + t^2)^{\frac{q – 2}{2}}\), this corresponds to \(q \le \, 2\). The second one [EPT] is a joint project with S. Perrotta and G. Treu where also lower order terms are included.- [EMMP] M. Eleuteri, P. Marcellini, E. Mascolo, S. Perrotta: Local Lipschitz continuity for energy integrals with slow growth, Annali di Matematica Pura ed Applicata, 201 (3), (2022), 1005-1032.
- [EPT] M. Eleuteri, S. Perrotta, G. Treu: Local Lipschitz continuity for energy integrals with slow growth and lower order terms, preprint arXiv:2309.10727, submitted.
- [HS] P. Hartman, G. Stampacchia: On some nonlinear elliptic differential functional equations, Acta Math., 115 (1966), 271-310.
- [S] G. Stampacchia: On some regular multiple integral problems in the calculus of variations, Comm. Pure Appl. Math., 16 (1963), 383-421.
Guglielmo Feltrin
Periodic perturbations of central force problems and an application to a restricted 3-body problem
Abstract
We deal with a periodic perturbation of a central force problem. Based on the introduction of suitable time-maps (the radial period and the apsidal angle) for the unperturbed problem and of an associated non-degeneracy condition (not requiring the explicit knowledge of the Hamiltonian in action-angle coordinates), we apply a higher-dimensional version of the Poincaré-Birkhoff fixed point theorem to prove the existence of non-circular periodic solutions bifurcating from invariant tori. Next, we show that this non-degeneracy condition is satisfied for some concrete examples of physical interest. At last, an application is given to a restricted 3-body problem with a non-Newtonian interaction. The talk is based on joint work with Alberto Boscaggin and Walter Dambrosio (University of Torino).Graziano Guerra
Unique solutions to hyperbolic conservation laws with a strictly convex entropy
Abstract
Consider a strictly hyperbolic \(n \times n\) system of conservation laws, where each characteristic field is either genuinely nonlinear or linearly degenerate. In this standard setting, it is well known that there exists a Lipschitz semigroup of weak solutions, defined on a domain of functions with small total variation. If the system admits a strictly convex entropy, we give a short proof that every entropy weak solution taking values within the domain of the semigroup coincides with a semigroup trajectory [2]. The result shows that the assumptions of “Tame Variation” or “Tame Oscillation” (see [1] and references therein), previously used to achieve uniqueness, can be removed in the presence of a strictly convex entropy. Combined with a compactness argument, the result yields a uniform convergence rate, \(\rho(\varepsilon)\to0\) as \(\varepsilon\to0\), for a very wide class of approximation algorithms. Some partial estimates on \(\rho(\varepsilon)\) are given.- [1] A. Bressan and P. Goatin Oleinik type estimates and uniqueness for \(n \times n\) conservation laws, J. Differential Equations, 156 (1999), 26-49
- [2] A. Bressan and G. Guerra Unique solutions to hyperbolic conservation laws with a strictly convex entropy, J. Differential Equations, 387 (2024), 432-447.
Eduardo Muñoz Hernández
Multiplicity and uniqueness of coexistence states in a heterogeneous predator-prey model