# Entropy formulation for fractal conservation laws.

Abstract: Using an integral formula of Droniou and Imbert (2005) for the fractional Laplacian, we define an entropy formulation for fractal conservation laws with pure fractional diffusion of order λ ∈]0, 1]. This allows to show the existence and the uniqueness of a solution in the L∞ framework. We also establish a result of controled speed of propagation that generalizes the finite propagation speed result of scalar conservation laws. We finally let the non-local term vanish to approximate solutions of scalar conservation laws, with optimal error estimates for BV initial conditions as Kuznecov (1976) for λ = 2 and Droniou (2003) for λ ∈]1, 2].

## 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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### Cites background from "Entropy formulation for fractal con..."

...[4] N. Alibaud, 2007, Entropy formulation for fractal conservation laws, Journal of Evolution Equations, 7, pp. 145-175....

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...entropy formulation is needed to guarantee uniqueness and well-posedness [3, 4]....

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94 citations

### Cites background or methods from "Entropy formulation for fractal con..."

...2) for λ ≤ 1 is that uniqueness of weak solutions is not obvious (precisely because they lack regularity); if the initial data is regular and small enough, some uniqueness results of the weak solution exist in [3], but for general bounded initial data, one has to use the notion of entropy solution developed in [1] in order to ensure existence and uniqueness of the (possibly irregular) solution....

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...3 (see [1]) This definition can be extended to the case λ = 1 and to multidimensional equations, and provides existence and uniqueness of the solution to (2....

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...Splitting method (see [7, 1]): for δ > 0, we construct u : [0, +∞[×R → R the following way: we let u(0, ·) = u0 and, for all even p and all odd q, we define by induction...

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...2) is also a weak solution (see [1]), this means that u satisfies (1....

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...and there is a notion of entropy solution to the fractal Burgers equation (see [1])....

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67 citations

### Cites background from "Entropy formulation for fractal con..."

...When λ ∈ [1,2) it suffices to assume that A ∈ C2, and when λ ∈ (0,1) A ∈ C1 is enough....

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...For a more complete discussion and many more references, we refer the reader to the nice papers [1] and [32]....

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...In [1,32] the fractional diffusion is always linear and non-degenerate....

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...1) is well defined by the dominated convergence theorem since ∣∣gr [φ](x)∣∣ ⎧⎨ ⎩ cλ‖Dφ‖L∞(B(x,r)) ∫ |z|<r |z| |z|d+λ dz when λ ∈ (0,1) cλ 2 ‖D2φ‖L∞(B(x,r)) ∫ |z|<r |z|2 |z|d+λ dz when λ ∈ [1,2) ⎫⎬ ⎭ < ∞....

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...The first part of our proof builds on the ideas developed by Alibaud (and Kružkov!)...

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##### References

6,917 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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442 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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203 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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