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

doi:10.1007/s10473-022-0510-4

Abstract

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

CLC Number:

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].Acta mathematica scientia,Series B, 2022, 42(5): 1875-1901.

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