TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID∗

doi:10.1007/s10473-023-0413-z

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (4): 1645-1667.doi: 10.1007/s10473-023-0413-z

TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID^∗

Jinxia CEN¹, Stanis law MIGÓRSKI², Emilio VILCHES³, Shengda ZENG^4,5,6,†

1. School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China;
2. Jagiellonian University in Krakow, Chair of Optimization and Control, ul. Lojasiewicza 6, 30348 Krakow, Poland;
3. Instituto de Ciencias de la Ingenieria, Universidad de O'Higgins, Av. Libertador Bernardo OHiggins 611, 2841959 Rancagua, Chile;
4. Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, China;
5. Department of Mathematics Nanjing University, Nanjing 210093, China;
6. Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30348 Krakow, Poland

收稿日期:2021-11-15 修回日期:2022-05-27 发布日期:2023-08-08
通讯作者: †Shengda ZENG, E-mail: zengshengda@163.com
作者简介:Jinxia CEN, E-mail: jinxiacen@163.com; Stanis law MIGÓRSKI, E-mail: stanislaw.migorski@uj.edu.pl; Emilio VILCHES, E-mail: emilio.vilches@uoh.cl
基金资助:
*NSF of Guangxi (2021GXNSFFA196004, GKAD23026237), the NNSF of China (12001478), the China Postdoctoral Science Foundation (2022M721560), the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No. 823731 CONMECH, the National Science Center of Poland under Preludium Project (2017/25/N/ST1/00611), the Startup Project of Doctor Scientific Research of Yulin Normal University (G2020ZK07) and the Ministry of Science and Higher Education of Republic of Poland (4004/GGPJII/H2020/2018/0, 440328/PnH2/2019).

TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID^∗

Jinxia CEN¹, Stanis law MIGÓRSKI², Emilio VILCHES³, Shengda ZENG^4,5,6,†

1. School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China;
2. Jagiellonian University in Krakow, Chair of Optimization and Control, ul. Lojasiewicza 6, 30348 Krakow, Poland;
3. Instituto de Ciencias de la Ingenieria, Universidad de O'Higgins, Av. Libertador Bernardo OHiggins 611, 2841959 Rancagua, Chile;
4. Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, China;
5. Department of Mathematics Nanjing University, Nanjing 210093, China;
6. Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30348 Krakow, Poland

Received:2021-11-15 Revised:2022-05-27 Published:2023-08-08
Contact: †Shengda ZENG, E-mail: zengshengda@163.com
About author:Jinxia CEN, E-mail: jinxiacen@163.com; Stanis law MIGÓRSKI, E-mail: stanislaw.migorski@uj.edu.pl; Emilio VILCHES, E-mail: emilio.vilches@uoh.cl
Supported by:
*NSF of Guangxi (2021GXNSFFA196004, GKAD23026237), the NNSF of China (12001478), the China Postdoctoral Science Foundation (2022M721560), the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No. 823731 CONMECH, the National Science Center of Poland under Preludium Project (2017/25/N/ST1/00611), the Startup Project of Doctor Scientific Research of Yulin Normal University (G2020ZK07) and the Ministry of Science and Higher Education of Republic of Poland (4004/GGPJII/H2020/2018/0, 440328/PnH2/2019).

摘要/Abstract

摘要： In this paper we study a nonstationary Oseen model for a generalized Newtonian incompressible fluid with a time periodic condition and a multivalued, nonmonotone friction law. First, a variational formulation of the model is obtained; that is a nonlinear boundary hemivariational inequality of parabolic type for the velocity field. Then, an abstract first-order evolutionary hemivariational inequality in the framework of an evolution triple of spaces is investigated. Under mild assumptions, the nonemptiness and weak compactness of the set of periodic solutions to the abstract inequality are proven. Furthermore, a uniqueness theorem for the abstract inequality is established by using a monotonicity argument. Finally, we employ the theoretical results to examine the nonstationary Oseen model.

关键词: nonstationary Oseen model, Newtonian incompressible fluid, hemivariational inequality, periodic solution, generalized subgradient

Abstract: In this paper we study a nonstationary Oseen model for a generalized Newtonian incompressible fluid with a time periodic condition and a multivalued, nonmonotone friction law. First, a variational formulation of the model is obtained; that is a nonlinear boundary hemivariational inequality of parabolic type for the velocity field. Then, an abstract first-order evolutionary hemivariational inequality in the framework of an evolution triple of spaces is investigated. Under mild assumptions, the nonemptiness and weak compactness of the set of periodic solutions to the abstract inequality are proven. Furthermore, a uniqueness theorem for the abstract inequality is established by using a monotonicity argument. Finally, we employ the theoretical results to examine the nonstationary Oseen model.

Key words: nonstationary Oseen model, Newtonian incompressible fluid, hemivariational inequality, periodic solution, generalized subgradient

Jinxia CEN, Stanis law MIGÓRSKI, Emilio VILCHES, Shengda ZENG. TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1645-1667.

Jinxia CEN, Stanis law MIGÓRSKI, Emilio VILCHES, Shengda ZENG. TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1645-1667.

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TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID^∗

TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID^∗

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