Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing

Pang, Peter H. C.

doi:10.1007/s11401-019-0169-x

Cited by 7 publications

(4 citation statements)

References 103 publications

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“…The existence of invariant measure is also established for the stochastic anisotropic parabolic-hyperbolic equation in [15]. It is clear from the above discussion that our assumption (2.14) on the flux F and the diffusive function A is in line with the assumptions in recently developed works.…”

Section: 3supporting

confidence: 74%

“…We mention some results under similar type of conditions on flux functions: hyperbolic conservation laws [21,18], and diffusion matrix in anisotropic degenerate parabolic-hyperbolic equations [15,14]. These results can summarized as follows: Consider…”

Section: 3mentioning

confidence: 99%

“…Moreover, Debussche and Vovelle [18] established the existence of an invariant measure for a scalar first-order conservation laws with stochastic forcing under a hypothesis of non-degeneracy on the flux. Furthermore, Chen and Pang [15] also have proved existence of an invariant measure for the stochastic anisotropic parabolic-hyperbolic equation. A number of authors have contributed since then, and we mention the works of [34,35].…”

Section: Introductionmentioning

confidence: 99%

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Stochastic fractional conservation laws

Chaudhary¹

2022

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In this paper, we consider the Cauchy problem for the nonlinear fractional conservation laws driven by a multiplicative noise. In particular, we are concerned with the well-posedness theory and the study of the long-time behavior of solutions for such equations. We show the existence of desired kinetic solution by using the vanishing viscosity method. In fact, we establish strong convergence of the approximate viscous solutions to a kinetic solution. Moreover, under a nonlinearity-diffusivity condition, we prove the existence of an invariant measure using the well-known Krylov-Bogoliubov theorem. Finally, we show the uniqueness and ergodicity of the invariant measure.

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Section: 3supporting

confidence: 74%

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stochastic fractional conservation laws

Chaudhary¹

2022

Preprint

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“…Debussche and Vovelle [11] studied scalar conservation laws with additive stochastic forcing on toruses of any dimension and proved the existence and uniqueness of an invariant measure for sub-cubic fluxes and sub-quadratic fluxes, respectively. Later, Chen and Pang [4] extend the result of [11] to degenerate second-order parabolic-hyperbolic conservation laws driven by additive noise. We want to stress that in the above papers, only additive noise was considered and no convergence rate to the invariant measure was obtained.…”

mentioning

confidence: 87%