Entropy formulation for fractal conservation laws.
Summary (1 min read)
1 Introduction
- All the methods used in this paper would apply, but this would lead to more technical difficulties and for the sake of clarity, the authors have chosen to present only the framework above.
- For λ ∈]0, 1], the order of the diffusive part is lower than the order of the hyperbolic part; hence, the authors do not expect any regularizing effect, since it is natural to think that (1.1) could behaves as (1.2).
2 Entropy formulation
- To present their formulation for (1.1), the authors have to recall the following result on g. Theorem 2.1 (Droniou, Imbert 2005).
- Here is their entropy formulation for (1.1).
- Hence, these proofs are left to the reader.
- Let us conclude this section with giving the proof of i), which explains how the authors obtained their formulation (see also Remark 4.2 about the treatment of the fractal part).
3 Main results
- Here is their existence and uniqueness result for (1.1).
- The uniqueness derives from a more precise result which generalizes the finite propagation speed for pure scalar conservation laws.
- The most important property of K is its non-negativity, which gives a maximum principle for the preceding equation.
- Here are other properties of entropy solutions to (1.1), that will be seen in the course of their study.
- Let us conclude this section with their convergence result for (1.3).
4.1 Doubling variables technique
- Consider u and v as functions of the (t, x)- and the (s, y)-variables, respectively.
- The limit n→ +∞ in (2.2), thanks again to the dominated convergence theorem, then implies that the entropy-flux pair (ηk,φk) can be used in Definition 2.1.
- Moreover, Taylor’s Formula and Fubini’s Theorem applied to (2.1) give ||g[φ].
4.2 Conclusion
- From a technical viewpoint, (4.17) seems to be unappropriate to use the doubling variable technique.
- Thus, the authors should put the operator g on η(u), but this need some regularity on η(u).
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Citations
Cites background or methods from "Entropy formulation for fractal con..."
...Then Alibaud [11] proved the same for α ∈ (0, 2)....
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...There is non-uniqueness of weak solutions, as proved by Alibaud and Andreianov [15]....
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...In [14], for α ∈ (1, 2), and [11] for 0 < α < 1, the authors prove that the entropy solutions in the sense of Definition 3....
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...Using this representation, we introduce, according to [11], the following definition of the entropy solution for system (1....
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...First, by construction in [11, 14], u is an entropy solution of Problem (2....
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Cites background from "Entropy formulation for fractal con..."
...1 in [1], because we cannot prove the solutions of (1....
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...1 in [1], we get that the above Definition 2....
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Cites background from "Entropy formulation for fractal con..."
...7) (this is an extension of Alibaud’s proof, stated in [5], of a similar result)....
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...[4, 1, 5]), and the proof of iii) - which follows - is in some extent based upon these references....
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References
7,118 citations
527 citations
"Entropy formulation for fractal con..." refers background in this paper
...Such a result is well-known since the work of Lévy [14]....
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485 citations
"Entropy formulation for fractal con..." refers background or methods in this paper
...Let us mention that Carrillo [5] has also used an entropy formulation to study a scalar conservation law perturbed by a local degenerate diffusion operator (of the form −△(b(u)))....
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...In the case of a local degenerate diffusion of the form −△(b(u)), this problem can be resolved by putting a gradient operator on test-functions, thanks to an integration by parts (see [5])....
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215 citations
"Entropy formulation for fractal con..." refers background in this paper
...[1] and these results have recently been strengthened in [7], where the existence and the uniqueness of a smooth solution is proved....
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...Most of the studies (well-posedness, asymptotic behaviour, etc) are concerned with the range of exponent λ ∈]1, 2] (see [1, 2, 7, 6, 8, 10])....
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...[1] which deals also with asymptotic behaviour of solutions....
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...Moreover, the general case λ ∈]0, 1] has many other applications to hydrodynamics, molecular biology, etc [1, 2]....
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