Finer regularity of an entropy solution for 1-d scalar conservation laws with non uniform convex flux
Summary (1 min read)
This proves (2).
- P 0; Where jI(t)j denotes the Lebesgue measure of I(t).
- Then from the Besicovitch differentiation Theorem (see [13] ) it follows that there exists a set E t & [a(t); b(t)] with the property:.
PROOF. From (3.4), D(t) & [a(t); b(t)] and hence
- For proving (3) of the theorem (1.3), the authors need the following backward construction which was proved in [1] in obtaining the solution of control problems for conservation laws.
- Using the backward construction the authors can prove that it is also necessary.
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Cites background from "Finer regularity of an entropy solu..."
...We ask the question that can one expect the similar result when f 6= g? When f = g, it has been noticed in [3] that the assumption of uniformly convexity is optimal in order to prove the BV regularity without the the assumption that u0 / ∈ BV ....
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References
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...Then from the hypothesis of f , it follows that f ∗ ∈ C(1)(IR), strictly convex and having super linear growth (see [4]) ....
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...We follow the same the notations as in [5], for t > 0, x ∈ IR, θ ∈ [0, t] define m±(x, t) = x−y±(x,t) t γ±(θ) = x+m±(x, t)(θ − t) I(x, t) = (y−(x, t), y+(x, t)) C(x, t) = {(ξ, θ); 0 < θ < t, γ−(θ) < ξ < γ+(θ)} I(t) = ⋃ x∈IR I(x, t) c(t) = ⋃ x∈IR C(x, t) D(t) = {x ∈ IR; y+(x, t) has jump at x} = {set of discontinuities of y+(....
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...Under the suitable condition on f, Ambrosio- Dellis [5] proved the following....
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...Assume that for all t > 0, x 7→ u(x, t) ∈ BVloc(IR), then is it in SBVloc(IR)? Under the suitable condition on f, Ambrosio- Dellis [5] proved the following....
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...For (1), we follow the similar path as in [5]....
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