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).
Did you find this useful? Give us your feedback
Citations
12 citations
Cites background from "Entropy formulation for fractal con..."
...5) holds for some β ∈ [12 , 1], then there is at most one energy solution u of (1....
[...]
...4), additional entropy conditions are needed to have uniqueness [1, 20, 21]; a counterexample for uniqueness of distributional solutions is given in [2]....
[...]
12 citations
Cites background from "Entropy formulation for fractal con..."
...In [2], Alibaud shows that existence and uniqueness hold for entropy solutions of (0....
[...]
...2 Vanishing viscosity (σN → 0) We consider the Burgers equation ∂tv = ∂x(u (2)/2) with initial condition u0(x) = 1[−3,−2] − 1[2,3], which is the cumulative distribution function of the measure δ−3 − δ−2 + δ2 − δ3....
[...]
...1), defined in [2] as functions v in L∞((0,∞)× R) satisfying the relation...
[...]
...1) has a non-increasing total variation (see [2]), which can be interpreted probabilistically as a compensation of merging sample paths having opposite signs....
[...]
10 citations
Cites background from "Entropy formulation for fractal con..."
...The notion of entropy solution of [2] is therefore required to obtain a well-posed formulation....
[...]
10 citations
10 citations
Cites background from "Entropy formulation for fractal con..."
...Let us recall the notion of entropy solutions to (1) from [20], see also [22, 30, 1, 32]....
[...]
...It was adapted later in [1] for fractional diffusions in space with a focus on the equation (11) ∂tu+∇ · F (u) + (−△) α 2 u = 0....
[...]
...Now the well-posedness is well-understood: If α ≥ 1, there is a unique smooth solution [10, 28, 16, 48, 23]; if α < 1, shocks can occur [6, 35] and weak solutions can be nonunique [2]; for any α ∈ (0, 2), there exists a unique entropy solution corresponding to the classical one when it exists as well [1]....
[...]
...also [22, 30, 1]; A natural question is whether it is possible to reformulate (13) in the spirit of (12)....
[...]
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]....
[...]
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)))....
[...]
...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])....
[...]
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....
[...]
...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])....
[...]
...[1] which deals also with asymptotic behaviour of solutions....
[...]
...Moreover, the general case λ ∈]0, 1] has many other applications to hydrodynamics, molecular biology, etc [1, 2]....
[...]