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# A Multi-scale Limit of a Randomly Forced Rotating 3-D Compressible Fluid

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# A Multi-scale Limit of a Randomly Forced Rotating 3-D Compressible Fluid

We study a singular limit of a scaled compressible Navier--Stokes--Coriolis system driven by both a deterministic and stochastic forcing terms in three dimensions. If the Mach number is comparable to the Froude number with both proportional to say $\varepsilon\ll 1$, whereas the Rossby number scales like $\varepsilon^m$ for $m>1$ large, then we show that any family of weak martingale solution to the $3$-D randomly forced rotating compressible equation (under the influence of a deterministic centrifugal force) converges in probability, as $\varepsilon\rightarrow0$, to the $2$-D incompressible Navier--Stokes system with a corresponding random forcing term.

40 pages

- Heriot-Watt University Malaysia
- Heriot Watt University
- Heriot-Watt University United Kingdom
- Heriot-Watt University United Arab Emirates

arXiv: Mathematics::Analysis of PDEs Physics::Fluid Dynamics

Microsoft Academic Graph classification: Random forcing Compressible flow Rossby number symbols.namesake Froude number Mathematics Mathematical analysis Mach number symbols Compressibility Stochastic forcing Martingale (probability theory)

FOS: Physical sciences, Mathematics - Analysis of PDEs, FOS: Mathematics, Mathematical Physics, Applied Mathematics, Probability (math.PR), Fluid Dynamics (physics.flu-dyn), Physics - Fluid Dynamics, Condensed Matter Physics, Computational Mathematics, Mathematics - Probability, Analysis of PDEs (math.AP)

FOS: Physical sciences, Mathematics - Analysis of PDEs, FOS: Mathematics, Mathematical Physics, Applied Mathematics, Probability (math.PR), Fluid Dynamics (physics.flu-dyn), Physics - Fluid Dynamics, Condensed Matter Physics, Computational Mathematics, Mathematics - Probability, Analysis of PDEs (math.AP)

arXiv: Mathematics::Analysis of PDEs Physics::Fluid Dynamics

Microsoft Academic Graph classification: Random forcing Compressible flow Rossby number symbols.namesake Froude number Mathematics Mathematical analysis Mach number symbols Compressibility Stochastic forcing Martingale (probability theory)

###### 20 references, page 1 of 2

[1] A. Babin, A. Mahalov, B. Nicolaenko : Global regularity of 3D rotating NavierStokes equations for resonant domains. Indiana Univ. Math. J., 48(3):1133- 1176, (1999).

[2] A. Babin, A. Mahalov, B. Nicolaenko : 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity. Indiana Univ. Math. J., 50(Special Issue):1-35, (2001). [OpenAIRE]

[3] D. Breit, E. Feireisl, M. Hofmanova´ : Incompressible limit for compressible fluids with stochastic forcing. Arch. Ration. Mech. Anal., 222(2):895-926, (2016).

[4] D. Breit, E. Feireisl, M. Hofmanova´ : Compressible fluids driven by stochastic forcing: The relative energy inequality and applications. Comm. Math. Phys., 350(2):443-473, (2017).

[5] D. Breit, E. Feireisl, M. Hofmanova´ : Stochastically Forced Compressible Fluid Flows. Berlin, Boston: De Gruyter, (2018).

[6] D. Breit, M. Hofmanova´ : Stochastic Navier-Stokes equations for compressible fluids. Indiana Univ. Math. J., 65(4):1183-1250, (2016).

[7] J.-Y. Chemin, B. Desjardins, I. Gallagher, E. Grenier : Mathematical geophysics, volume 32 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, Oxford, (2006).

[8] G. Da Prato, J. Zabczyk : Stochastic equations in infinite dimensions, volume 152 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, second edition, (2014).

[9] D. G. Ebin : Viscous fluids in a domain with frictionless boundary. In Global analysis-analysis on manifolds, volume 57 of Teubner-Texte Math., pages 93- 110. Teubner, Leipzig, (1983).

[10] E. Feireisl, I. Gallagher, D. Gerard-Varet, A. n. Novotny´ : Multi-scale analysis of compressible viscous and rotating fluids. Comm. Math. Phys., 314(3):641- 670, (2012).

###### 3 Research products, page 1 of 1

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- Heriot-Watt University Malaysia
- Heriot Watt University
- Heriot-Watt University United Kingdom
- Heriot-Watt University United Arab Emirates

We study a singular limit of a scaled compressible Navier--Stokes--Coriolis system driven by both a deterministic and stochastic forcing terms in three dimensions. If the Mach number is comparable to the Froude number with both proportional to say $\varepsilon\ll 1$, whereas the Rossby number scales like $\varepsilon^m$ for $m>1$ large, then we show that any family of weak martingale solution to the $3$-D randomly forced rotating compressible equation (under the influence of a deterministic centrifugal force) converges in probability, as $\varepsilon\rightarrow0$, to the $2$-D incompressible Navier--Stokes system with a corresponding random forcing term.

40 pages

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