On the Well-Posedness of Reduced 3 D Primitive Geostrophic Adjustment Model with Weak Dissipation
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Journal of Mathematical Fluid Mechanics
On the Well-Posedness of Reduced 3D Primitive Geostrophic Adjustment Model with Weak Dissipation Chongsheng Cao, Quyuan Lin and Edriss S. Titi Communicated by M. Hieber
Abstract. In this paper we prove the local well-posedness and global well-posedness with small initial data of the strong solution to the reduced 3D primitive geostrophic adjustment model with weak dissipation. The term reduced model means that the relevant physical quantities depend only on two spatial variables. The weak dissipation helps us overcome the illposedness of the original model. We also prove the global well-posedness of the strong solution to the Voigt α-regularization of this model, and establish the convergence of the strong solution of the Voigt α-regularized model to the corresponding solution of the original model. Furthermore, we derive a criterion for existence of finite-time blow-up of the original model with weak dissipation based on Voigt α-regularization. Mathematics Subject Classification. 35A01, 35B44, 35Q35, 35Q86, 76D03, 86-08, 86A10. Keywords. Primitive geostrophic adjustment model, Regularization, Blow-up criterion.
1. Introduction It is commonly believed that the dynamics of ocean and atmosphere adjusts itself toward a geostrophic balance. The following reduced 3D primitive geostrophic adjustment system is one of the main diagnostic models for studying geostrophic adjustment (cf. e.g., [23,29,46]): ut + u ux + wuz − f0 v + px = 0,
(1)
vt + u vx + wvz + f0 u = 0, pz + T = 0,
(2) (3)
ux + wz = 0, Tt + u Tx + w Tz = 0,
(4) (5)
where the velocity field (u, v, w), the temperature T and the pressure p are the unknown functions of horizontal variable x, vertical variable z, and time t, and f0 is the Coriolis parameter. System (1)–(5) is reduced from the 3D inviscid primitive equations model by assuming that the flow is independent of the third spatial variable. This system has been a standard framework for studying geostrophic adjustment of frontal anomalies in a rotating continuously stratified fluid of strictly rectilinear fronts and jets (cf. e.g., [2,22,23,28,29,31,46,48] and references therein). The first systematically mathematical studies of the viscous primitive equations (PEs) were carried out in the 1990s by Lions–Temam–Wang [40–42]. They considered the PEs with both full viscosities and full diffusivities and established the global existence of weak solutions. The uniqueness of weak solutions in the 3D viscous case is still an open problem, while the weak solutions in 2D turn out to be unique, see Bresch et al. [5]. Concerning the strong solutions for the 2D case, the local existence result was established by Guill´en-Gonz´alez et al. [25], while the global existence for 2D case was proved by Bresch et al. in [6], and by Temam and Ziane in [52]. The global existence of strong solutions for 3D case was established 0123456789().: V,-vol
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by Cao and Titi in [14] and later by Kobelkov in [30], see also the subsequent articles of Kukavica
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