Coupling PDEs and SDEs: The Illustrative Example of the Multiscale Simulation of Viscoelastic Flows

Jourdain, Benjamin; Bris, Claude Le; Lelièvre, Tony

doi:10.1007/3-540-26444-2_7

Cited by 5 publications

(8 citation statements)

References 26 publications

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“…A different technique was used by Gomes [9] to express the diffusive Lagrangian [5] as the expected value minimizer of a suitable functional. Finally we mention similar systems have been considered by Jourdain et al in [10].…”

Section: Introductionmentioning

confidence: 85%

A Stochastic Perturbation of Inviscid Flows

Iyer

2006

Commun. Math. Phys.

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Abstract. We prove existence and regularity of the stochastic flows used in the stochastic Lagrangian formulation of the incompressible Navier-Stokes equations (with periodic boundary conditions), and consequently obtain a C k,α local existence result for the Navier-Stokes equations. Our estimates are independent of viscosity, allowing us to consider the inviscid limit. We show that as ν → 0, solutions of the stochastic Lagrangian formulation (with periodic boundary conditions) converge to solutions of the Euler equations at the rate of O( √ νt).

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Section: Introductionmentioning

confidence: 85%

A Stochastic Perturbation of Inviscid Flows

Iyer

2006

Commun. Math. Phys.

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“…In contrast, references concerned with coupled PDE-SDE systems are rather scarce. In [17] the authors considered a system of a 1D parabolic PDE coupled to an SDE in a half-local way, letting the variable satisfying the SDE intervene in the PDE only by its expectation. The possibility of a coupling via compatibility conditions at the mutual interface between the different domains on which a PDE and an SDE are respectively stated was mentioned as well, however without further dwelling on it.…”

mentioning

confidence: 99%

“…The possibility of a coupling via compatibility conditions at the mutual interface between the different domains on which a PDE and an SDE are respectively stated was mentioned as well, however without further dwelling on it. The emphasis of [17] was on numerics.…”

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confidence: 99%

On a coupled SDE-PDE system modeling acid-mediated tumor invasion

Hiremath

Surulescu

Zhigun³

et al. 2018

Discrete &Amp; Continuous Dynamical Systems - B

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We propose and analyze a multiscale model for acid-mediated tumor invasion accounting for stochastic effects on the subcellular level. The setting involves a PDE of reaction-diffusion-taxis type describing the evolution of the tumor cell density, the movement being directed towards pH gradients in the local microenvironment, which is coupled to a PDE-SDE system characterizing the dynamics of extracellular and intracellular proton concentrations, respectively. The global well-posedness of the model is shown and numerical simulations are performed in order to illustrate the solution behavior.2010 Mathematics Subject Classification. Primary: 34F05, 35R60, 92C17; Secondary: 35Q92, 60H10, 92C50.

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“…An approximation similar to that in equation (2.5) is taken. B. Angular velocity induced by the external field: (a linear response system is assumed) 36) where ζ r is the rotational friction coefficient, and may be deduced by Stokes' formula combined with the Shish-Kebab model or more delicate calculations [15]. A common choice for ζ r is…”

Section: Rod-like Modelmentioning

confidence: 99%

“…A publication list [75,76,29,28,82,54,18,48,37,38,84,85,51,11,83,52,7,8,9,36,88], though incomplete, shows the increased popularity. This field is just at the very beginning, and a lot of problems are still left for mathematicians to explore.…”

Section: Introductionmentioning

confidence: 99%