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Volume 38, Issue 4
Zero Viscosity-Diffusivity Limit for the Incompressible Boussinesq Equations in Gevrey Class

Feng Cheng

DOI: 10.4208/cmr.2021-0075

Commun. Math. Res., 38 (2022), pp. 579-604.

Published online: 2022-10

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  • Abstract

In this paper, we study the zero viscosity-diffusivity limit for the incompressible Boussinesq equations in a periodic domain in the framework of Gevrey class. We first prove that there exists an interval of time, independent of the viscosity coefficient and the diffusivity coefficient, for the solutions to the viscous incompressible Boussinesq equations. Then, based on these uniform estimates, we show that the solutions of the viscous incompressible Boussinesq equations converge to that of the ideal incompressible Boussinesq equations as the viscosity and diffusivity coefficients go to zero. Moreover, the convergence rate is also given.

  • Keywords

Gevrey class, incompressible Boussinesq equation, analyticity, zero viscosity-diffusivity limit, convergence rate.

  • AMS Subject Headings

35Q35, 76D03, 76D09

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-38-579, author = {Cheng , Feng}, title = {Zero Viscosity-Diffusivity Limit for the Incompressible Boussinesq Equations in Gevrey Class}, journal = {Communications in Mathematical Research }, year = {2022}, volume = {38}, number = {4}, pages = {579--604}, abstract = {

In this paper, we study the zero viscosity-diffusivity limit for the incompressible Boussinesq equations in a periodic domain in the framework of Gevrey class. We first prove that there exists an interval of time, independent of the viscosity coefficient and the diffusivity coefficient, for the solutions to the viscous incompressible Boussinesq equations. Then, based on these uniform estimates, we show that the solutions of the viscous incompressible Boussinesq equations converge to that of the ideal incompressible Boussinesq equations as the viscosity and diffusivity coefficients go to zero. Moreover, the convergence rate is also given.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2021-0075}, url = {http://global-sci.org/intro/article_detail/cmr/21073.html} }
TY - JOUR T1 - Zero Viscosity-Diffusivity Limit for the Incompressible Boussinesq Equations in Gevrey Class AU - Cheng , Feng JO - Communications in Mathematical Research VL - 4 SP - 579 EP - 604 PY - 2022 DA - 2022/10 SN - 38 DO - http://doi.org/10.4208/cmr.2021-0075 UR - https://global-sci.org/intro/article_detail/cmr/21073.html KW - Gevrey class, incompressible Boussinesq equation, analyticity, zero viscosity-diffusivity limit, convergence rate. AB -

In this paper, we study the zero viscosity-diffusivity limit for the incompressible Boussinesq equations in a periodic domain in the framework of Gevrey class. We first prove that there exists an interval of time, independent of the viscosity coefficient and the diffusivity coefficient, for the solutions to the viscous incompressible Boussinesq equations. Then, based on these uniform estimates, we show that the solutions of the viscous incompressible Boussinesq equations converge to that of the ideal incompressible Boussinesq equations as the viscosity and diffusivity coefficients go to zero. Moreover, the convergence rate is also given.

Cheng , Feng. (2022). Zero Viscosity-Diffusivity Limit for the Incompressible Boussinesq Equations in Gevrey Class. Communications in Mathematical Research . 38 (4). 579-604. doi:10.4208/cmr.2021-0075
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