Time-periodic solutions of the compressible Navier–Stokes equations in $${\mathbb{R}^{4}}$$ R 4

Abstract

This paper deals with the existence of time-periodic solutions to the compressible Navier-Stokes equations effected by general form external force in R N with N = 4. Using a fixed point method, we establish the existence and uniqueness of time-periodic solutions. This paper extends Ma, UKai, Yang's result [5], in which, the existence is obtained when the space dimension N ≥ 5. Mathematics Subject Classification. 35Q30 · 35B10.

Time‐periodic solutions for the full quantum Euler equation

“…Notice that the assumption holds in general due to the conservation of mass. Different from the previous works, 16,17,21,24,26,29 the symmetry of the external force is unnecessary since we can obtain the L 2 estimates for the velocity and the temperature by the structure of (1.3) directly. Moreover, inspired by the idea in Tan et al, 24 we linearize the term u · ∇ u in the case of the quantum Euler system in order to guarantee the closed estimate in Lemma 2.2.…”

“…Kagei and Tsuda 22 also considered the existence of time‐periodic solutions via the spectral properties on the case n ≥ 3. Moreover, the existence of time‐periodic solutions of Navier–Stokes equations in the whole space was present in Jin and Yang 26 by uniform estimates and the topological degree theory. Then the similar problem in the periodic domain was discussed in Jin and Yang 21 .…”

“…There exists an extensive literature in past decades about the time-periodic solutions for the classical compressible flow. [16][17][18][19][20][21][22][23][24] In the study of the Navier-Stokes equation, one can refer to Jin and Yang 21 and Tan et al 24 for the case of periodic domain and other studies 22,23,25,26 for the whole space. See also Cai et al 17 and Hong et al 27 for the Navier-Stokes-Korteweg system and other studies 16,19,28,29 for the magnetohydrodynamic equations.…”

“…Even though we have employed the similar argument from Tan et al 24 and Jin and Yang, 26 the so‐called full quantum hydrodynamic model (1.1) carries some new feature such as the strong nonlinearity and coupling, which makes the study more mathematically challenging. We also remark that the case of full quantum Navier–Stokes equations is easier than the system under consideration since the high‐order viscous term μ Δ u + λ ∇ div u can be used to control the quantum effects terms as Pu and Guo 13 and Pu and Li 31 .…”

On the existence and uniqueness of time periodic solutions for a semilinear heat equation in the whole space

“…By using the fixed point method, we proved the existence and uniqueness of solutions. In the process of making the estimates, the main difficulty lies in the norm estimate of solutions; to overcome this difficulty, we employ the method used in , that is, we use the periodicity of solutions to obtain the a priori estimate of the first derivative of solutions, which is used as a bound of the ‘initial value’ and further obtain the estimate of solutions in L ∞ ((0, T ); L p ).…”