This article reports several important mathematical and physical results related to the 3D primitive equations modeling 3D stratified flows. The primitive equations considered here assume the following form: \[ \partial_t U+U\cdot\nabla U+f_0 e_3\times U= -\nabla p+\rho_1 e_3 +\nu_1\Delta U +F,\quad\nabla\cdot U=0,\tag{1} \]
\[ \partial_t\rho_1+U\cdot\nabla \rho_1=-N_0^2 U_3+\nu_2\Delta\rho_1+F_4, \] where \(U=(U_1,U_2,U_3)\) is the velocity field, \(\rho_1\) is the buoyancy variable, \(N_0\) is the Brunt-Väisälä wave frequency, \(f_0\) is twice the frequency of background rotation, and \(F=(F_1,F_2,F_3)\). The dimensionless parameters associated with \(f_0\) and \(N_0\) are \(f=f_0 L/U_h\) and \(N=N_0L/U_h\), respectively, where \(L\) is the typical horizontal length scale and \(U_h\) the characteristic horizontal velocity scale.
To obtain their major results, the authors fix the ratio \(\eta=f/N\) and the domain parameters \(a_1\), \(a_2\) and \(a_3\), and assume that \(\tilde{F}\equiv(F_1,F_2,F_3, F_4)\) satisfies \[ \sup_{T}\int_T^{T+1}\|\tilde{F}(t,\cdot)\|_{H^{\alpha-1}}\,dt\leq M_{\alpha F}^2 \] for \(\alpha\) to be specified below. The article contains three major theorems. Assuming \(\nu_1>0\), \(\nu_2>0\) and the initial datum \((U_0,\rho_{10})\) is in \(H^\alpha\) with \(\alpha>3/4\), namely \(\|(U_0,\rho_{10})\|_{H^\alpha}\leq M_\alpha\), the first theorem establishes the global regularity of strong solutions to the 3D primitive equations (1) with \(N>N_1(M_\alpha, M_{\alpha F}, \nu_1,\nu_2,a_1,a_2,a_3)\). The condition \(N>N_1\) physically means strong stratification. In the second theorem, assume \(\nu_1>0\), \(\nu_2>0\) and \(\|(U_0,\rho_{10})\|_{H^0}\leq M_0\) and let \(T=T(M_0, M_{\alpha F}, \nu_1,\nu_2)\). Then the authors show that any Leray weak solution of (1) defined on \([0,T]\) can be extended to \(t\in (0,\infty)\). In addition, if \(F\) is independent of \(t\), (1) has a global attractor that attracts every Leray weak solution. The attractor is bounded in \(H^\alpha\) and has a finite fractal dimension. The third theorem is for the inviscid primitive equations, namely (1) with \(\nu_1=0\) and \(\nu_2=0\). It establishes the global regularity for those parameters \(\eta\), \(a_1\) and \(a_2\) that do not belong to a resonant set.
The proofs of these theorems and other details are given in nine sections. The first section is an introduction. The second section rewrites (1) in Craya cyclic basis and introduces the limit resonant equations that govern the large-time dynamics. The third section details the algebraic structure of resonant and quasi-resonant sets and provides a uniform convergence estimate for the difference between \((U,\rho_1)\) of (1) and \(W_{QG}+ W_{AG}\), where \(W_{QG}\) and \(W_{AG}\) are the solutions of the corresponding 3D quasi-geostrophic (QG) equations and the ageostrophic (AG) equations, respectively. Section 4 investigates the global regularity of solutions to the 3D QG equations. Section 5 studies the regularity of solutions to the AG equations. Section 6 establishes the global regularity of strong solutions to the “\(2\frac12\)-dimensional” infinite-time limit equations. Section 7 proves the first two major theorems mentioned previously. Section 8 provides the global regularity result for the inviscid primitive equations. Section 9 describes the dynamics of the anisotropic AG baroclinic waves and explains the genesis of fronts in the regime of strong stratification and weak rotation.
For the entire collection see [Zbl 0994.00019].