Continuous dependence estimate for conservation laws with Lévy noise

Advances in Nonlinear Analysis

Vallet

2017

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Abstract. In this article we deal with stochastic perturbation of degenerate parabolic partial differential equations (PDEs). The particular emphasise is on analysing the effect of multiplicative Lévy noise to such problems and establishing wellposedness by developing a suitable weak entropy solution framework. The proof of existence is based on the vanishing viscosity technique. The uniqueness is settled by interpreting Kruzkov's doubling technique in the presence noise.

Section: 2mentioning

confidence: 99%

On the Cauchy problem of a degenerate parabolic-hyperbolic PDE with Lévy noise

Biswas

Advances in Nonlinear Analysis

Vallet

2017

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“…As we mentioned earlier, the aim of this paper is to fill the gap left by the previous authors by introducing a rate of convergence result for a nonlinear scalar conservation laws forced by a Lévy noise. In this paper, drawing preliminary motivation from [8,9], we intend to prove that the expected value of the L 1 difference between the approximate solution and the unique entropy solution converges at a rate O( √ ∆x), ∆x being the spatial discretization parameter. Moreover, we also prove the convergence of approximate numerical solutions, generated by the semi-discrete finite difference scheme, to the unique stochastic BV-entropy solution of the underlying equation (1.1).…”

Section: Numerical Schemesmentioning

confidence: 99%

“…Therefore, by using A.5, we have from (4.47) Our aim is to estimate each of the above terms suitably. To do this, we follow the ideas from [8,9]. At this point we first let R → ∞ in (4.51).…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

A finite difference scheme for conservation laws driven by Lévy noise

Koley

IMA Journal of Numerical Analysis

Vallet³

2017

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In this paper, we analyze a semi-discrete finite difference scheme for a conservation laws driven by a homogeneous multiplicative Lévy noise. Thanks to BV estimates, we show a compact sequence of approximate solutions, generated by the finite difference scheme, converges to the unique entropy solution of the underlying problem, as the spatial mesh size ∆x → 0. Moreover, we show that the expected value of the L 1 -difference between the approximate solution and the unique entropy solution converges at a rate O( √ ∆x).the given initial function, and f : R → R is a given (sufficiently smooth) scalar valued flux function (see Section 2 for the complete list of assumptions). Moreover, W (t) is a real valued Brownian noise and N (dz, dt) = N (dz, dt) − m(dz) dt, where N is a Poisson random measure on R × (0, ∞) with intensity measure m(dz), a Radon measure on R \ {0} with Definition 1.2 (Entropy-Entropy Flux Pair). An ordered pair (β, ζ) is called an entropy-entropy flux pair if β ∈ C 2 (R) with β ≥ 0, and ζ : R → R such that ζ ′ (r) = β ′ (r)f ′ (r), for all r.Moreover, an entropy-entropy flux pair (β, ζ) is called convex if β ′′ (·) ≥ 0.With the help of a convex entropy-entropy flux pair (β, ζ), the notion of stochastic entropy solution is defined as follows:1 Here we denote x ∧ y := min {x, y}

“…The study of stochastic balance laws driven by noise is comparatively new area of pursuit. Only recently balance laws with stochastic forcing have attracted the attention of many authors [2,4,5,8,9,10,11,12,14,15,16,18,22,23] and resulted a significant momentum in the theoretical development of such problems. Due to nonlinear nature of the underlying problem, explicit solution formula is hard to obtain and hence robust numerical schemes for approximating such equation are very important.…”

Section: Introductionmentioning

confidence: 99%

Convergence of a flux-splitting finite volume scheme for conservation laws driven by Lévy noise

Applied Mathematics and Computation

2018

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We explore numerical approximation of multidimensional stochastic balance laws driven by multiplicative Lévy noise via flux-splitting finite volume method. The convergence of the approximations is proved towards the unique entropy solution of the underlying problem.2000 Mathematics Subject Classification. 45K05, 46S50, 49L20, 49L25, 91A23, 93E20.However, since there are infinitely many weak solutions, one needs to define an extra admissibility criteria to select physically relevant solution in a unique way, and one such condition is called entropy condition. Let us begin with the notion of entropy flux pair. Definition 2.2 (Entropy flux pair). (β, φ) is called an entropy flux pair if β ∈ C 2 (R) and ζ : R → R is such that ζ ′ (r) = β ′ (r)f ′ (r).