IRIS Università degli Studi di Padovahttps://www.research.unipd.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Mon, 18 Jan 2021 16:49:37 GMT2021-01-18T16:49:37Z10261Analysis and control on networks: Trends and perspectiveshttp://hdl.handle.net/11577/3253891Titolo: Analysis and control on networks: Trends and perspectives
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/11577/32538912017-01-01T00:00:00ZA note on regularity and failure of regularity for systems of conservation laws via Lagrangian formulationhttp://hdl.handle.net/11577/3145928Titolo: A note on regularity and failure of regularity for systems of conservation laws via Lagrangian formulation
Abstract: 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 Ole ̆ınik-Ambroso-De Lellis SBV estimate: it provides bounds on ∂xu when u is an entropy solution of the Cauchy problem for Burgers’ equation with L∞-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 Ck-data which are generic, in a Baire category sense, 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 rigorous proofs.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/11577/31459282016-01-01T00:00:00ZA Directional Lipschitz Extension Lemma, with Applications to Uniqueness and Lagrangianity for the Continuity Equationhttp://hdl.handle.net/11577/3286423Titolo: A Directional Lipschitz Extension Lemma, with Applications to Uniqueness and Lagrangianity for the Continuity Equation
Abstract: 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 W1,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.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11577/32864232019-01-01T00:00:00ZRecent developments related to conservation laws and Hamilton-Jacobi equationshttp://hdl.handle.net/11577/3271794Titolo: Recent developments related to conservation laws and Hamilton-Jacobi equations
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11577/32717942018-01-01T00:00:00ZSchaeffer's regularity theorem for scalar conservation laws does not extend to systemshttp://hdl.handle.net/11577/3087299Titolo: Schaeffer's regularity theorem for scalar conservation laws does not extend to systems
Abstract: Several regularity results hold for the Cauchy problem involving one scalar conservation law having convex flux. Among these, Schaeffer's theorem guarantees that if the initial datum is smooth and is generic, in the Baire sense, the entropy admissible solution develops at most finitely many shocks, locally, and stays smooth out of them. We rule out with the present paper the possibility of extending Schaeffer's regularity result to the class of genuinely nonlinear, strictly hyperbolic systems of conservation laws. The analysis relies on careful interaction estimates and uses fine properties of the wave-front tracking approximation.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/11577/30872992017-01-01T00:00:00ZEulerian, Lagrangian and Broad continuous solutions to a balance law with non-convex flux Ihttp://hdl.handle.net/11577/3189644Titolo: Eulerian, Lagrangian and Broad continuous solutions to a balance law with non-convex flux I
Abstract: We discuss different notions of continuous solutions to the balance law with source term g bounded, flux f twice continuously differentiable. We extend previous works relative to the quadratic flux. We establish the equivalence among distributional solutions and a suitable notion of Lagrangian solutions for general smooth fluxes. We eventually find that continuous solutions are Kruzkov iso-entropy solutions, which yields uniqueness for the Cauchy problem. We also establish the ODE reduction on any characteristics under the sharp assumption that the set of inflection points of the flux f is negligible. The correspondence of the source terms in the two settings is matter of a companion work, where we also provide counterexamples when the negligibility on inflection points fails.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/11577/31896442016-01-01T00:00:00ZNew interaction estimates for the Baiti-Jenssen systemhttp://hdl.handle.net/11577/3145927Titolo: New interaction estimates for the Baiti-Jenssen system
Abstract: 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.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/11577/31459272016-01-01T00:00:00ZOn L1-stability of BV solutions for a model of granular flowhttp://hdl.handle.net/11577/3300535Titolo: On L1-stability of BV solutions for a model of granular flow
Abstract: We are concerned with the well-posedness of a model of granular flow that consists of a hyperbolic system of two balance laws in one-space dimension, which is linearly degenerate along two straight lines in the phase plane and genuinely nonlinear in the subdomains confined by such lines. This note provides a survey of recent results on the Lipschitz L1-continuous dependence of the entropy weak solutions on the initial data, with a Lipschitz constant that grows exponentially in time. Our analysis relies on the extension of a Lyapunov like functional
and provide the first construction of a Lipschitz semigroup of entropy weak solutions to the regime of hyperbolic systems of balance laws (i) with characteristic families that are neither genuinely nonlinear nor linearly degenerate and (ii) initial data of arbitrarily large total variation.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11577/33005352020-01-01T00:00:00ZExponential stability of large BV solutions in a model of granular flow,http://hdl.handle.net/11577/3300566Titolo: Exponential stability of large BV solutions in a model of granular flow,
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11577/33005662019-01-01T00:00:00ZA Directional Lipschitz Extension Lemma, with Applications to Uniqueness and Lagrangianity for the Continuity Equationhttp://hdl.handle.net/11577/3286426.1Titolo: A Directional Lipschitz Extension Lemma, with Applications to Uniqueness and Lagrangianity for the Continuity Equation
Abstract: 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 W1,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.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11577/3286426.12018-01-01T00:00:00Z