A fractional degenerate parabolic-hyperbolic Cauchy problem with noise

Bhauryal, Neeraj; Koley, Ujjwal; Vallet, Guy

doi:10.1016/j.jde.2021.02.061

Cited by 19 publications

(12 citation statements)

References 27 publications

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“…We also refer the work of G. Lv et al [30] for stochastic nonlocal conservation laws in whole space. A number of authors have contributed in the area of stochastic conservation laws and we mention the works of Biswas et al [8], Koley et al [25,24], Bhauryal et al [5,6]. In the stochastic setup, the main idea is to successfully capture the noise-noise interaction of the underlying problem since this plays an important role in the well-posedness theory for stochastic PDEs, for details see [5,6,24,26,27,22,13,14].…”

Section: Earlier Workmentioning

confidence: 99%

Stochastic Degenerate Fractional Conservation Laws

Chaudhary¹

2021

Preprint

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We consider the Cauchy problem for a degenerate fractional conservation laws driven by a noise. In particular, making use of an adapted kinetic formulation, a result of existence and uniqueness of solution is established. Moreover, a unified framework is also established to develop the continuous dependence theory. More precisely, we demonstrate L 1 -continuous dependence estimates on the initial data, the order of fractional Laplacian, the diffusion matrix, the flux function, and the multiplicative noise function present in the equation.

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Section: Earlier Workmentioning

confidence: 99%

Stochastic Degenerate Fractional Conservation Laws

Chaudhary¹

2021

Preprint

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“…Our problems need to invoke ideas from spectral methods for deterministic problems and meaningfully fuse them with available approximation methods for SDEs. In the realm of stochastic conservation laws, noise-noise interaction terms play a fundamental role to establish well-posedness theory, for details see [2,3,4,25,26,27,28]. The main contributions of this article are as follows;…”

Section: Aim and Scope Of This Papermentioning

confidence: 99%

Convergence of a spectral method for the stochastic incompressible Euler equations

Chaudhary¹

2021

Preprint

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We propose a spectral viscosity method (SVM) to approximate the incompressible Euler equations driven by a multiplicative noise. We show that SVM solution converges to a dissipative measure-valued martingale solution. These solutions are weak in the probabilistic sense i.e. the probability space and the driving Wiener process are an integral part of the solution. We also exhibit weak (measure-valued)-strong uniqueness principle. Moreover, we establish strong convergence of approximate solutions to the regular solution of the limit system at least on the lifespan of the latter, thanks to the weak (measure-valued)-strong uniqueness principle for the underlying system.

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“…This is easier said than done as any such attempt has to capture the noise-noise interaction as well. In the realm of stochastic conservation laws, noise-noise interaction terms play a fundamental role to establish well-posedness theory, for details see [3][4][5][29][30][31][32].…”

Section: Scope and Outline Of The Papermentioning

confidence: 99%

A convergent finite volume scheme for the stochastic barotropic compressible Euler equations

Chaudhary,

Koley

2021

Preprint

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In this paper, we analyze a semi-discrete finite volume scheme for the three-dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure-valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong convergence of numerical solutions to the regular solution of the limit systems at least on the lifespan of the latter, thanks to the weak (measure-valued)-strong uniqueness principle for the underlying system. To the best of our knowledge, this is the first attempt to prove the convergence of numerical approximations for the underlying system.

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A fractional degenerate parabolic-hyperbolic Cauchy problem with noise

Cited by 19 publications

References 27 publications

Stochastic Degenerate Fractional Conservation Laws

Stochastic Degenerate Fractional Conservation Laws

Convergence of a spectral method for the stochastic incompressible Euler equations

A convergent finite volume scheme for the stochastic barotropic compressible Euler equations

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