Invariance property of a conservation law without convexity

Abstract

The main goal of this paper is to investigate the mechanism of a conservation law that gives the N-wave like asymptotics. It turns out that the positivity of the flux function provides a certain invariance of solution which singles out the right asymptotics among two parameter family of N-waves. Two kinds of long time asymptotic convergence orders in L 1-norm to this N-wave are proved using a potential comparison technique. The first one is of the magnitude of the N-wave itself and the second one is of order 1/t. We observe that these asymptotic convergence orders are related to space and time translations of potentials.

Relative Newtonian Potentials of Radial Functions and Asymptotics in Nonlinear Diffusion

“…where d ≥ 3 and the Radon measure v is radial, which justifies the use of the same symbol in (2) and (16).…”

“…Unfortunately, this definition is not well-defined in general. The total mass M should be the limit lim (16) is not defined since the only non-negative function that has the mass concentration of order V(r) = O(r d−2−ϵ ) for r > 0 large is the trivial one. That is why it is forced to choose r 0 = 0 for dimension d = 1.…”

“…Note that, for the case of dimension one, the Newtonian potential of ν := u r is simply the antiderivative of the solution u, which gives the mass concentration of the solution. (This antiderivative successfully played the role of a potential for a convection problem in [16,17].) In fact, for all dimensions d ≥ 1, we take the mass concentration,…”

“…Hence the maximum principle on the unbounded domainR d × [t 0 , T ] concludes the proposition. ⊓ ⊔ Let d ≥ 3.Then, the Newtonian potentials U 1 and U 2 of two solutions u 1 and u 2 given by(16) obviously satisfy E(r, t; u 1 , u 2 ) = U 1 (r, t) − U 2 (r, t).…”