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
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Cites background from "Entropy formulation for fractal con..."
...[6, 1] and [7, 29, 25], fractional porous medium equations [16] (A = |u|m−1u for m ≥ 1 and α-stable μ), and strongly degenerate equations where A vanishes on a set of positive measure....
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...[18, 1], but note that our problem is different and much more difficult....
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...3 is not optimal for Levy operators L with order in the interval [1, 2)....
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...The Kruzkov entropy solution theory of scalar conservation laws [27] was extended to cover fractional conservation laws in [1], to more general Lévy conservation laws in [25], and then finally to setting of this paper, equations with non-linear fractional diffusion and general Lévy measures in [11]....
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...A thorough description of the mathematical background for such equations, relevant bibliography, and applications to several disciplines of interest can be found in [1, 2, 7, 11, 16, 25]....
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16 citations
16 citations
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...At least it includes all the Lévy measures found in finance, see Remark 8.3, and also many singular measures like e.g. delta-measures....
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...Here and in the following we use the shorthand dw = dxdt dy ds. Note that η′(u(x, t), v(y, s)) L̃µ,r [u(·, t)− v(·, s)](x, y) ≤ L̃µ,r[η(u(·, t), v(·, s))](x, y)....
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...Remember that αj ≥ 0, |α| = α1 + · · · + αd, and that xα = xα11 · · ·xαdd for any x ∈ Rd....
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...Here Λ-periodic means 2π-periodic in each coordinate direction....
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...1 Here and in the rest of the paper, a ∧ b = min(a, b), ∂t = ∂ ∂t , ∂j = ∂ ∂xj and ∂x = (∂1, ∂2, . . . , ∂d)....
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15 citations
References
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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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