01648nas a2200121 4500008004100000245007800041210006900119260001000188520123500198100002301433700002001456856005001476 2014 en d00aQuadratic interaction functional for general systems of conservation laws0 aQuadratic interaction functional for general systems of conserva bSISSA3 aFor 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)2.1 aBianchini, Stefano1 aModena, Stefano uhttp://urania.sissa.it/xmlui/handle/1963/7440