EXISTENCE AND STABILITY OF PERIODIC AND ALMOST PERIODIC SOLUTIONS TO THE BOUSSINESQ SYSTEM IN UNBOUNDED DOMAINS

doi:10.1007/s10473-022-0510-4

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (5): 1875-1901.doi: 10.1007/s10473-022-0510-4

• 论文 • 上一篇

EXISTENCE AND STABILITY OF PERIODIC AND ALMOST PERIODIC SOLUTIONS TO THE BOUSSINESQ SYSTEM IN UNBOUNDED DOMAINS

Thieu Huy NGUYEN¹, Truong Xuan PHAM², Thi Ngoc Ha VU¹, The Sac LE^3,1

1. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology Vien Toan ung dung va Tin hoc, Dai hoc Bach khoa Hanoi 1 Dai Co Viet, Hanoi, Vietnam;
2. Department of Mathematics, Faculty of Information Technology, Thuyloi university Khoa Cong nghe Thong tin, Bo mon Toan, Dai hoc Thuy loi, 175 Tay Son, Dong Da, Ha Noi, Viet Nam;
3. Thuyloi University, Dai hoc Thuy Loi, 175 Tay Son, Dong Da, Hanoi, Viet Nam

收稿日期:2021-05-12 修回日期:2021-09-16 发布日期:2022-11-02
通讯作者: Thieu Huy Nguyen,E-mail:huy.nguyenthieu@hust.edu.vn E-mail:huy.nguyenthieu@hust.edu.vn
基金资助:
This work was financially supported by the Vietnam National Foundation for Science and Technology Development under grant number 101.02-2021.04. The work of the last author was financially supported by Vietnam Ministry of Education and Training under Project B2022-BKA-06.

EXISTENCE AND STABILITY OF PERIODIC AND ALMOST PERIODIC SOLUTIONS TO THE BOUSSINESQ SYSTEM IN UNBOUNDED DOMAINS

Thieu Huy NGUYEN¹, Truong Xuan PHAM², Thi Ngoc Ha VU¹, The Sac LE^3,1

1. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology Vien Toan ung dung va Tin hoc, Dai hoc Bach khoa Hanoi 1 Dai Co Viet, Hanoi, Vietnam;
2. Department of Mathematics, Faculty of Information Technology, Thuyloi university Khoa Cong nghe Thong tin, Bo mon Toan, Dai hoc Thuy loi, 175 Tay Son, Dong Da, Ha Noi, Viet Nam;
3. Thuyloi University, Dai hoc Thuy Loi, 175 Tay Son, Dong Da, Hanoi, Viet Nam

Received:2021-05-12 Revised:2021-09-16 Published:2022-11-02
Contact: Thieu Huy Nguyen,E-mail:huy.nguyenthieu@hust.edu.vn E-mail:huy.nguyenthieu@hust.edu.vn
Supported by:
This work was financially supported by the Vietnam National Foundation for Science and Technology Development under grant number 101.02-2021.04. The work of the last author was financially supported by Vietnam Ministry of Education and Training under Project B2022-BKA-06.

摘要/Abstract

摘要： In this paper we investigate the existence and stability of periodic solutions (on a half-line $\mathbb{R}_{+}$) and almost periodic solutions on the whole line time-axis $\mathbb{R}$ to the Boussinesq system on several classes of unbounded domains of $\mathbb{R}^n$ in the framework of interpolation spaces. For the linear Boussinesq system we combine the L^p — L^q-smoothing estimates and interpolation functors to prove the existence of bounded mild solutions. Then, we prove the existence of periodic solutions by invoking Massera’s principle. We also prove the existence of almost periodic solutions. Then we use the results of the linear Boussinesq system to establish the existence, uniqueness and stability of the small periodic and almost periodic solutions to the Boussinesq system using fixed point arguments and interpolation spaces. Our results cover and extend the previous ones obtained in [13, 34, 38].

关键词: Boussinesq systems, periodic solutions, almost periodic solutions, unbounded domains, smoothing estimates, dual estimates, interpolation spaces, stability

Abstract: In this paper we investigate the existence and stability of periodic solutions (on a half-line $\mathbb{R}_{+}$) and almost periodic solutions on the whole line time-axis $\mathbb{R}$ to the Boussinesq system on several classes of unbounded domains of $\mathbb{R}^n$ in the framework of interpolation spaces. For the linear Boussinesq system we combine the L^p — L^q-smoothing estimates and interpolation functors to prove the existence of bounded mild solutions. Then, we prove the existence of periodic solutions by invoking Massera’s principle. We also prove the existence of almost periodic solutions. Then we use the results of the linear Boussinesq system to establish the existence, uniqueness and stability of the small periodic and almost periodic solutions to the Boussinesq system using fixed point arguments and interpolation spaces. Our results cover and extend the previous ones obtained in [13, 34, 38].

Key words: Boussinesq systems, periodic solutions, almost periodic solutions, unbounded domains, smoothing estimates, dual estimates, interpolation spaces, stability

中图分类号:

35A01

Thieu Huy NGUYEN, Truong Xuan PHAM, Thi Ngoc Ha VU, The Sac LE. EXISTENCE AND STABILITY OF PERIODIC AND ALMOST PERIODIC SOLUTIONS TO THE BOUSSINESQ SYSTEM IN UNBOUNDED DOMAINS[J]. 数学物理学报(英文版), 2022, 42(5): 1875-1901.

Thieu Huy NGUYEN, Truong Xuan PHAM, Thi Ngoc Ha VU, The Sac LE. EXISTENCE AND STABILITY OF PERIODIC AND ALMOST PERIODIC SOLUTIONS TO THE BOUSSINESQ SYSTEM IN UNBOUNDED DOMAINS[J]. Acta mathematica scientia,Series B, 2022, 42(5): 1875-1901.

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EXISTENCE AND STABILITY OF PERIODIC AND ALMOST PERIODIC SOLUTIONS TO THE BOUSSINESQ SYSTEM IN UNBOUNDED DOMAINS

EXISTENCE AND STABILITY OF PERIODIC AND ALMOST PERIODIC SOLUTIONS TO THE BOUSSINESQ SYSTEM IN UNBOUNDED DOMAINS

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