Quadratic Interaction Functional for General Systems of Conservation Laws

Abstract

For the Glimm scheme approximation ${u_\varepsilon}$ to the solution of the system of conservation laws in one space dimension

$$u_t + f(u)_x = 0, \qquad u(0, x) = u_0(x) \in \mathbb{R}^n,$$

with initial data u ₀ with small total variation, we prove a quadratic (w.r.t. Tot. Var. (u ₀)) interaction estimate, which has been used in the literature for stability and convergence results. No assumptions on the structure of the flux f are made (apart from 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 ${\mathtt{x}(t, w)}$ 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 ${\mathfrak{Q}}$ 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 ${D_{t} \hat \sigma_k(t,w)}$ is a measure with total mass ${\leq \mathcal{O}(1) {\rm Tot. Var.} (u_0)^2}$ , where ${\hat{\sigma}_k(t, w)}$ is the speed given to the wave w by the Riemann problem at the grid point ${(i\varepsilon, \mathtt{x}(i\varepsilon, w)), t \in [i\varepsilon, (i + 1)\varepsilon)}$.