TY - JOUR

T1 - Boussinesq System with Partial Viscous Diffusion or Partial Thermal Diffusion Forced by a Random Noise

AU - Yamazaki, Kazuo

N1 - Funding Information:
The author expresses deep gratitude to the Editor and the reviewers for valuable comments and suggestions which improved this manuscript significantly.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.

PY - 2021/12

Y1 - 2021/12

N2 - We study the Boussinesq system in a two-dimensional domain. In case only the velocity equation is forced by a white-in-time multiplicative noise, we prove the global existence of a martingale solution in the following two cases: positive horizontal viscous diffusion but zero vertical viscous diffusion and zero thermal diffusion; positive horizontal thermal diffusion but zero viscous diffusion and zero vertical thermal diffusion. Finally, in case both velocity and temperature equations are forced by a white-in-time multiplicative noise, and the velocity equation has zero viscous diffusion while the temperature equation has full thermal diffusion, the global existence of a martingale solution was shown in Yamazaki (Stoch Anal Appl, 34:404–426, 2016); in this manuscript we additionally prove significantly higher regularity.

AB - We study the Boussinesq system in a two-dimensional domain. In case only the velocity equation is forced by a white-in-time multiplicative noise, we prove the global existence of a martingale solution in the following two cases: positive horizontal viscous diffusion but zero vertical viscous diffusion and zero thermal diffusion; positive horizontal thermal diffusion but zero viscous diffusion and zero vertical thermal diffusion. Finally, in case both velocity and temperature equations are forced by a white-in-time multiplicative noise, and the velocity equation has zero viscous diffusion while the temperature equation has full thermal diffusion, the global existence of a martingale solution was shown in Yamazaki (Stoch Anal Appl, 34:404–426, 2016); in this manuscript we additionally prove significantly higher regularity.

KW - Boussinesq system

KW - Inviscid

KW - Martingale solution

KW - Navier-Stokes equations

KW - Regularity

UR - http://www.scopus.com/inward/record.url?scp=85101128318&partnerID=8YFLogxK

U2 - 10.1007/s00245-021-09756-w

DO - 10.1007/s00245-021-09756-w

M3 - Article

AN - SCOPUS:85101128318

VL - 84

JO - Applied Mathematics and Optimization

JF - Applied Mathematics and Optimization

SN - 0095-4616

ER -