Browsing by Author "Modena, Stefano"

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    Quadratic interaction estimate for conservation laws: motivations, techniques and open problems
    (2015) Modena, Stefano; ; Mathematics
    In a series joint works with S. Bianchini [3, 4, 5], we proved a quadratic interaction estimate for general systems of conservation laws. Aim of this article is to present the results obtained in the three cited papers, setting them in the context of the theory of Hyperbolic Conservation Laws. To this purpose, first we explain why we considered this quadratic estimate interesting, then we give a brief overview of the techniques we used to prove it and finally we present some related open problems.
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    Quadratic interaction functional for general systems of conservation laws
    (SISSA, 2014) Bianchini, Stefano; Modena, Stefano
    For the Glimm scheme approximation u" to the solution of the system of conservation laws in one space dimension ut + f(u)x = 0; u(0; x) = u0(x) 2 Rn; with initial data u0 with small total variation, we prove a quadratic (w.r.t. Tot.Var.(u0)) interaction estimate, which has been used in the literature for stability and convergence results. No assumptions on the structure of the ux f are made (apart smoothness), and this estimate is the natural extension of the Glimm type interaction estimate for genuinely nonlinear systems. More precisely we obtain the following results: a new analysis of the interaction estimates of simple waves; a Lagrangian representation of the derivative of the solution, i.e. a map x(t;w) which follows the trajectory of each wave w from its creation to its cancellation; the introduction of the characteristic interval and partition for couples of waves, representing the common history of the two waves; a new functional Q controlling the variation in speed of the waves w.r.t. time. This last functional is the natural extension of the Glimm functional for genuinely nonlinear systems. The main result is that the distribution Dttx(t;w) is a measure with total mass O(1)Tot.Var.(u0)2.