Spatial asymptotic profile in geophysical fluid dynamics (Q2902605)

The author considers the Boussinesq equations NEWLINE\[NEWLINEu_t - \Delta u + \left(u \cdot \nabla u\right) u + \nabla p = g \theta\;\text{in}\;Q_T,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\theta_t- \Delta \theta + \left(u\cdot \nabla \right) \theta = 0\; \text{in}\;Q_T,NEWLINE\]NEWLINE NEWLINE\[NEWLINE \nabla \cdot u =0\;\text{in}\;Q_T,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0)=u_0,\;\text{in}\;\mathbb{R}^n,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\theta(0)=\theta_0,\;\text{in}\;\mathbb{R}^n,NEWLINE\]NEWLINE where \(Q_T= \mathbb{R}^n \times (0,T)\) and \(u_0,\theta_0\) are given initial data. It is assumed that \(g=\left(g_1,g_2,\cdots,g_n\right) \in L^{\infty}_{n-1}(\mathbb{R}^n)^n\). The solvability of this system in weighted \(L^{\infty}\) spaces is established. For all \(\mu \in (0,n]\) and \(\nu>\max\{0,\mu-n+1\}\) the existence of unique mild solution NEWLINE\[NEWLINE(u,\theta) \in C_{\omega}([0,T]\;L_{\mu}^{\infty}(\mathbb{R}^{n})^n) \times C_{\omega}([0,T]\;L_{\mu}^{\infty}(\mathbb{R}^{n})^n)NEWLINE\]NEWLINE for the system of integral equations NEWLINE\[NEWLINEu(t)= e^{tA}u_0 +\int_{0}^{t}e^{(t-\tau)A}\mathbb{P}\left(u \cdot \nabla u\right)(\tau)\; d \tau + \int_{0}^{t}e^{(t-\tau)}A\;\mathbb{P}(g \theta)(\tau) d \tau,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\theta(t)=-e^{t\Delta}\theta_0 +\int_{0}^{t}e^{(t-\tau)\Delta}\left(u \cdot \nabla \theta\right)(\tau)\; d \tau,NEWLINE\]NEWLINE is proved under the assumption that \((u_0,\theta_0) \in L_{\mu}^{\infty}(\mathbb{R}^{n}) \times L_{\nu}^{\infty}(\mathbb{R}^{n})\), \(\nabla \cdot u_0=0\) and \(g \in L^{\infty}_{n-1}(\mathbb{R}^n)^n\) for all \(T>0\) sufficiently small (depending on the data). The strong solvability of solutions in weighted \(L^{\infty}\) spaces is obtained under the assumption that the gravity \(g\) is more regular. The regularity of the solution \((u,\theta)\) depends on the regularity of the gravity while the initial data has no contribution on the regularity of the solution. Some results are derived on the spatial asymptotic behavior of the solutions. While the temperature \(\theta\) decays as the initial data \(\theta_0\), the velocity of the fluid has slower decays due to the effect of the buoyancy force \(g \theta\). The asymptotic profile of the pressure is obtained for strong solutions.NEWLINENEWLINEIn the second part of the work, the 3D (incompressible) Navier-Stokes equation in \(\mathbb{R}^n\) is considered in a rotating frame NEWLINE\[NEWLINEu_t - \Delta u + \left(u \cdot \nabla u\right) u +\Omega e_3 \times u + \nabla p = f \theta\;\text{in}\;\mathbb{R}^3 \times (0,T),NEWLINE\]NEWLINE NEWLINE\[NEWLINE \nabla \cdot u =0\;\text{in}\;\mathbb{R}^3 \times (0,T),NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0)=u_0,\;\text{in}\;\mathbb{R}^3,NEWLINE\]NEWLINE where \(\Omega \neq 0\) is the constant Coriolis parameter. The solvability of the corresponding integral equation is established in weighted \(L^{\infty}\)-spaces for any \(T>0\) sufficiently small (depending on the data). In the study of strong solvability, the initial data is considered with a vertical averaging property and the external force is assumed to have some spatial regularity. Again, a smooth force yields a smooth solution, while the initial data \(u_0\) has no contribution to the regularity of the solution. Finally, the leading terms of the asymptotic profile of the solution (when \(f=0\)) far from the axis of rotation are obtained.