Abstract
M.-J. Kang and one of us [2] developed a new version of the relative entropy method, which is efficient in the study of the long-time stability of extreme shocks. When a system of conservation laws is rich, we show that this can be adapted to the case of intermediate shocks.
Highlights
We consider a strictly hyperbolic system of conservation laws in one space dimension ut + f (u)x = 0, u(x, t) ∈ U, where U is a convex open domain in Rn and f : U → Rn is a smooth vector field
Leger & Vasseur showed in [3, 4] that the relative entropy method applied to a scalar conservation law, with a convex flux f, yield the strong property that t → inf h∈R
H is non-increasing for every shock U and every entropy solution u, whenever the initial data u(·, 0) belongs to U + L2(R). We interpret this property as saying that scalar shock waves are attractors up to a shift
Summary
Leger & Vasseur showed in [3, 4] that the relative entropy method applied to a scalar (case n = 1) conservation law, with a convex (or concave) flux f , yield the strong property that t → inf h∈R h. H is non-increasing for every shock U and every entropy solution u, whenever the initial data u(·, 0) belongs to U + L2(R). We interpret this property as saying that scalar shock waves are attractors up to a shift. Our main result (Theorem 3.1) is that given a Lax shock of a rich system, strongly convex entropies η± can be chosen so that the shock be a local attractor up to a shift.
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