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Stochastic Entropy Solutions for Stochastic Scalar Balance Laws
2019
Entropy
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We are concerned with the initial value problem for a multidimensional balance law with multiplicative stochastic perturbations of Brownian type. ...
Furthermore, as applications, we derive the uniqueness and existence of the stochastic entropy solution for stochastic Buckley-Leverett equations and generalized stochastic Burgers type equations. ...
Acknowledgments: The authors are grateful to the anonymous referees for helpful comments and suggestions that greatly improved the presentation of this paper. ...
doi:10.3390/e21121142
fatcat:fq65rkl75rfo5pbu27p4obcz3m
Existence of stochastic entropy solutions for stochastic scalar balance laws with Lipschitz vector fields
[article]
2016
arXiv
pre-print
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2015 pre-print arXiv:1406.0040v2 fatcat:fffhzt7zgnem5elydq6xyjjzzuOther Versions
In this paper, we consider a scalar stochastic balance law and gain the existence for stochastic entropy solutions. Our proof relies on the BGK approximation and the generalized Itô formula. ...
Moreover, as an application, we derive the existence of stochastic entropy solutions for stochastic Buckley-Leverett type equations. ...
Introduction In this paper, we study the first-order scalar balance law with Stratonovich type perturbations: ∂ t ρ(t, x) + div x (B(ρ)) + ∂ x i ρ(t, x) •Ṁ i (t) = A(t, ρ), in Ω × (0, ∞) × R d , (1.1) ...
arXiv:1406.0040v3
fatcat:fy6as3frkjg57jez57fr4swyqu
On a time-splitting method for a scalar conservation law with a multiplicative stochastic perturbation and numerical experiments
2014
Journal of evolution equations (Printed ed.)
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to show the existence and uniqueness of the stochastic weak entropy solution. ...
The result of convergence of such a sequence is based on the work of Bauzet-Vallet-Wittbold (J Funct Anal, 2013), where the authors used the concept of measure-valued solution and Kruzhkov's entropy formulation ...
Last but not least, many thanks go to the referee for his comments helping to improve the manuscript. ...
doi:10.1007/s00028-013-0215-1
fatcat:bjj2l3xb7fexphjlj7fs3xbrfi
OUP accepted manuscript
2019
IMA Journal of Numerical Analysis
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2017 pre-print arXiv:1709.04351v1 fatcat:ots4giovhba7bfu4ajdfvj7zsiOther Versions
In this article we present an a posteriori error estimator for the spatial-stochastic error of a Galerkin-type discretisation of an initial value problem for a random hyperbolic conservation law. ...
Combined with the relative entropy stability framework of Dafermos dafermos2005hyperbolic, this leads to computable error bounds for the space-stochastic discretisation error. ...
Furthermore, we show how to construct so-called space-time-stochastic reconstructions for the random scalar conservation law. ...
doi:10.1093/imanum/drz004
fatcat:j5esjufobjdmrnnffuxtsxs6gy
Uniqueness of stochastic entropy solutions for stochastic balance laws with Lipschitz fluxes
[article]
2016
arXiv
pre-print
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2014 pre-print arXiv:1405.2614v1 fatcat:f7jdchdx45hpdd2chcw5t4b7fa
In this paper, we consider a stochastic balance law with a Lipschitz flux and gain the uniqueness for stochastic entropy solutions. ...
Furthermore, as an application, we derive the uniqueness of stochastic entropy solutions for stochastic porous media type equations. ...
It is remarked that all above mentioned works are concentrate their attention of stochastic entropy solutions for stochastic balance laws on C 2 -fluxes. ...
arXiv:1405.2614v2
fatcat:wxviqyhlnjhfdoh3mbthqxh5h4
Page 8572 of Mathematical Reviews Vol. , Issue 2002K
[page]
2002
Mathematical Reviews
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The author gives an approximation method for the entropy so-
lution of a 1D scalar conservation law based on signed sticky
particles when the variation of the initial condition is bounded. ...
(English and French summaries) [Signed sticky particles and 1D
scalar conservation laws]
C. R. Math. Acad. Sci. Paris 334 (2002), no. 3, 233-238. ...
Multi-level Monte Carlo Finite Volume Methods for Uncertainty Quantification in Nonlinear Systems of Balance Laws
[chapter]
2013
Lecture Notes in Computational Science and Engineering
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Luc Grosheintz, a student in the ETH Zürich MSc Applied Mathematics curriculum for performing the numerics for the sparse two-point correlation computations reported in Section 6.8. ...
The authors thank the systems support at ETH Zürich parallel Compute Cluster BRUTUS [48] for their support in the production runs for the present paper, and the staff at the Swiss National Supercomputing ...
Random entropy solutions of scalar conservation laws. ...
doi:10.1007/978-3-319-00885-1_6
fatcat:nygpyhj5ujd4pphv5mptscgm4q
Entropy production in mesoscopic stochastic thermodynamics: nonequilibrium kinetic cycles driven by chemical potentials, temperatures, and mechanical forces
2016
Journal of Physics: Condensed Matter
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In the present review, we introduce a mesoscopic stochastic formulation of NET by analyzing entropy production in several simple examples. ...
It is argued that mesoscopic stochastic NET provides a rigorous mathematical basis of fundamental concepts needed for understanding complex processes in chemistry, physics and biology, and which is also ...
Acknowledgments We thank the Lorentz Center of Leiden University for hosting the workshop on 'Nanothermodynamics: For Equilibrium and Nonequilibrium' (1-5 December 2014) where the authors started this ...
doi:10.1088/0953-8984/28/15/153004
pmid:26986039
fatcat:mv76rbtyvrf47efnvyb2ebz6dm
Numerical Solution of Scalar Conservation Laws with Random Flux Functions
2016
SIAM/ASA Journal on Uncertainty Quantification
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The first aim of this paper is to develop an appropriate mathematical framework of random entropy solutions for scalar hyperbolic conservation laws with random flux functions with correlated random perturbations ...
Efficient MLMC discretization of balance laws with random source terms was investigated in [22] . ...
We consider the Cauchy problem for scalar conservation laws (SCL) such as (1.1). ...
doi:10.1137/120896967
fatcat:biuk5jfl4ffg5nzd53ururulva
Kinematic Basis of Emergent Energetics of Complex Dynamics
[article]
2020
arXiv
pre-print
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Stochastic kinematic description of a complex dynamics is shown to dictate an energetic and thermodynamic structure. ...
The present theory provides a mathematical basis for P. W. Anderson's emergent behavior in the hierarchical structure of complexity science. ...
We thank Jin Feng, Hao Ge, Yi-An Ma, Mark Peletier, and Jin Wang for helpful discussions, and an anonymous reviewer for feedback. ...
arXiv:1704.01828v3
fatcat:kej3ngylere4nk3zebzihqdq3u
Convergence of flux-splitting finite volume schemes for hyperbolic scalar conservation laws with a multiplicative stochastic perturbation
2016
Mathematics of Computation
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Under a stability condition on the time step, we prove the convergence of the finite volume approximation towards the unique stochastic entropy solution of the equation. ...
We study here explicit flux-splitting finite volume discretizations of multi-dimensional nonlinear scalar conservation laws perturbed by a multiplicative noise with a given initial data in L 2 (R d ). ...
Definition 1 (Stochastic entropy solution) A function u of N 2 w 0, T, L 2 (R d ) ∩ L ∞ 0, T ; L 2 Ω × R d is an entropy solution of the stochastic scalar conservation law (1) with the initial condition ...
doi:10.1090/mcom/3084
fatcat:z5acvop5nne3bmpqq2j32j22pe
The zeroth law of thermodynamics and volume-preserving conservative system in equilibrium with stochastic damping
2014
Physics Letters A
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2013 pre-print arXiv:1206.7079v2 fatcat:tcivmuxpbzarhigoo3susiw5l4Other Versions
We propose a mathematical formulation of the zeroth law of thermodynamics and develop a stochastic dynamical theory, with a consistent irreversible thermodynamics, for systems possessing sustained conservative ...
Stochastic thermodynamics based on time reversal (t,ϕ,g)→(-t,ϕ,-g) is formulated: entropy production e_p^#(t)=-dF(t)/dt; generalized "heat" h_d^#(t)=-dU(t)/dt, U(t)=∫_R^nϕ(x)u(x,t)dx being "internal energy ...
Detailed balance in systems with even and odd variables. There have been extensive discussions on detailed balance in stochastic differential equation with even and odd variables. ...
doi:10.1016/j.physleta.2013.12.028
fatcat:a5p7wb2ypbfbdfwzmg2bk7kt7q
An a posteriori error analysis based on non-intrusive spectral projections for systems of random conservation laws
[article]
2019
arXiv
pre-print
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We derive an a posteriori error estimator using smooth reconstructions of the numerical solution, which combined with the relative entropy stability framework yields computable error bounds for the space-stochastic ...
We present an a posteriori error analysis for one-dimensional random hyperbolic systems of conservation laws. ...
Following the definition in [9] for scalar problems, we call u P L 1 ξ pΞ; L 1 pp0, T qˆR; Uqq a random entropy solution of (RIVP), if up¨,¨, yq is a classical entropy solution, cf. [3, Def. 4.5.1],P-a.s ...
arXiv:1908.09612v1
fatcat:xezvebvcc5fzddmtm6fvwntm5a
Stochastic Dynamics, Large Deviations Principle, and Non-equilibrium Thermodynamics
[article]
2021
arXiv
pre-print
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cases, the intrinsic connections among mesoscopic stochastic dynamics, deterministic ODEs or PDEs, large deviations rate function, and macroscopic thermodynamic potential are established. ...
By examining the deterministic limit of a general ϵ-dependent generator for Markovian dynamics, which includes the continuous Fokker-Planck equations and discrete chemical master equations as two special ...
functional We now show that the stationary solution to Eq. (11) is an entropy functional for the nonlinear differential equation (4) : d dt ϕ ss z(t) = F (z) · ∇ z ϕ ss (z) = A(z) + m =−m ν R (z) · ∇ ...
arXiv:2002.11311v2
fatcat:pv2phw66sbeyxhnrylugkt2wpq
A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws
2020
BIT Numerical Mathematics
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Compared to scalar equations, little is known about existence and uniqueness of entropy solutions for systems of hyperbolic conservation laws, especially in multiple space dimensions. ...
Based on Kružkov's work [23], a firm theory for random scalar conservation laws in several space dimensions has been developed [25, 26, 31] . ...
We therefore use the residuals for the momentum and the energy balance as indicators for our space-stochastic mesh refinements. ...
doi:10.1007/s10543-019-00794-z
fatcat:akldwwq5tnftjferznyzirc43a
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