Volume 35, Issue 1
On the Existence of Time-Periodic Solution to the Compressible Heat-Conducting Navier-Stokes Equations

Ming Cheng

Commun. Math. Res., 35 (2019), pp. 35-56.

Published online: 2019-12

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  • Abstract

We study in this article the compressible heat-conducting Navier-Stokes equations in periodic domain driven by a time-periodic external force. The existence of the strong time-periodic solution is established by a new approach. First, we reformulate the system and consider some decay estimates of the linearized system. Under some smallness and symmetry assumptions on the external force, the existence of the time-periodic solution of the linearized system is then identified as the fixed point of a Poincaré map which is obtained by the Tychonoff fixed point theorem. Although the Tychonoff fixed point theorem cannot directly ensure the uniqueness, but we could construct a set-valued function, the fixed point of which is the time-periodic solution of the original system. At last, the existence of the fixed point is obtained by the Kakutani fixed point theorem. In addition, the uniqueness of time-periodic solution is also studied. 

  • Keywords

non-isentropic compressible fluid, strong solution, time period, fixed point theorem

  • AMS Subject Headings

35Q30, 35B10, 76N10

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

mcheng314@jlu.edu.cn (Ming Cheng)

  • BibTex
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@Article{CMR-35-35, author = {Cheng , Ming}, title = {On the Existence of Time-Periodic Solution to the Compressible Heat-Conducting Navier-Stokes Equations}, journal = {Communications in Mathematical Research }, year = {2019}, volume = {35}, number = {1}, pages = {35--56}, abstract = {

We study in this article the compressible heat-conducting Navier-Stokes equations in periodic domain driven by a time-periodic external force. The existence of the strong time-periodic solution is established by a new approach. First, we reformulate the system and consider some decay estimates of the linearized system. Under some smallness and symmetry assumptions on the external force, the existence of the time-periodic solution of the linearized system is then identified as the fixed point of a Poincaré map which is obtained by the Tychonoff fixed point theorem. Although the Tychonoff fixed point theorem cannot directly ensure the uniqueness, but we could construct a set-valued function, the fixed point of which is the time-periodic solution of the original system. At last, the existence of the fixed point is obtained by the Kakutani fixed point theorem. In addition, the uniqueness of time-periodic solution is also studied. 

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2019.01.05}, url = {http://global-sci.org/intro/article_detail/cmr/13473.html} }
TY - JOUR T1 - On the Existence of Time-Periodic Solution to the Compressible Heat-Conducting Navier-Stokes Equations AU - Cheng , Ming JO - Communications in Mathematical Research VL - 1 SP - 35 EP - 56 PY - 2019 DA - 2019/12 SN - 35 DO - http://doi.org/10.13447/j.1674-5647.2019.01.05 UR - https://global-sci.org/intro/article_detail/cmr/13473.html KW - non-isentropic compressible fluid, strong solution, time period, fixed point theorem AB -

We study in this article the compressible heat-conducting Navier-Stokes equations in periodic domain driven by a time-periodic external force. The existence of the strong time-periodic solution is established by a new approach. First, we reformulate the system and consider some decay estimates of the linearized system. Under some smallness and symmetry assumptions on the external force, the existence of the time-periodic solution of the linearized system is then identified as the fixed point of a Poincaré map which is obtained by the Tychonoff fixed point theorem. Although the Tychonoff fixed point theorem cannot directly ensure the uniqueness, but we could construct a set-valued function, the fixed point of which is the time-periodic solution of the original system. At last, the existence of the fixed point is obtained by the Kakutani fixed point theorem. In addition, the uniqueness of time-periodic solution is also studied. 

Ming Cheng. (2019). On the Existence of Time-Periodic Solution to the Compressible Heat-Conducting Navier-Stokes Equations. Communications in Mathematical Research . 35 (1). 35-56. doi:10.13447/j.1674-5647.2019.01.05
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