A Sharp Decay Estimate for Positive Nonlinear Waves

We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers’ equation with impulsive sources. 1 Introduction Consider a strictly hyperbolic system of n conservation laws ut + f(u)x = 0 (1.1) and assume that all characteristic fields are genuinely nonlinear. Call λ1(u) < · · · < λn(u) the eigenvalues of the Jacobian matrix A(u) . = Df(u). We shall use bases of left and right eigenvectors li(u), ri(u) normalized so that ∇λi(u) ri(u) ≡ 1 , li(u) rj(u) = { 1 if i = j, 0 if i 6= j. (1.2) Given a function u : IR 7→ IR with small total variation, following [BC], [B] one can define the measures μ of i-waves in u as follows. Since u ∈ BV , its distributional derivative Dxu is a Radon measure. We define μ as the measure such that μ . = li(u) ·Dxu (1.3) restricted to the set where u is continuous, while, at each point x where u has a jump, we define μ ( {x} ) . = σi , (1.4) 1 where σi is the strength of the i-wave in the solution of the Riemann problem with data u − = u(x−), u = u(x+). In accordance with (1.2), if the solution of the Riemann problem contains the intermediate states u = ω0, ω1, . . . , ωn = u , the strength of the i-wave is defined as σi . = λi(ωi)− λi(ωi−1). (1.5) Observing that σi = li(u ) · (u − u) +O(1) · |u − u|, we can find a vector li(x) such that