Nonlinear stability of Ekman boundary layers in rotating stratified fluid (Q2925677)

The author considers the mathematical model of the rotating stratified flow of incompressible viscous fluid in the half-space described by the system of the Navier-Stokes-Boussinesq equations under the Coriolis and stratification effects, particularly in the case when the rotation axis is not perpendicular to the boundary of the half-space NEWLINE\[NEWLINE \begin{cases} \partial _t u -\nu \triangle u + (u,\nabla)u+\Omega d\times u + \bigtriangledown p = \mathcal{G}\theta e_3, \quad x\in \mathbb{R}^3_{+}, \; t>0,\\ \partial _t \theta -\kappa \triangle \theta + (u,\nabla)\theta = -N^2u^3, \quad x\in \mathbb{R}^3_{+}, \; t>0,\\ \nabla\cdot u =0, \quad x\in \mathbb{R}^3_{+}, \; t>0,\\ u|_{x_3=0}=(a_1,b_1,0),\; \theta|_{x_3=0}=c_1, \quad (x_1,x_2)\in \mathbb{R}^2, \; t>0,\\ u|_{t=0}=u_0,\;\theta|_{t=0}=\theta_0,\quad x\in \mathbb{R}^3_{+}. \end{cases} \eqno{(1)} NEWLINE\]NEWLINE with respect to \(u = u(x, t) = (u_1, u_2, u_3), \theta=\theta(x,t)\) and \(p=p(x,t)\) where \( \mathbb{R}^3_{+}= \{x\in \mathbb{R}^3, x_3>0\}, a_1,b_1,c_1\) are real constants, \(\nu >0,\kappa >0\) and \(\mathcal{G}\in \mathbb{R}\backslash \{0\}\) are the viscosity, the thermal diffusivity and the gravity constants; \(\Omega\) and \(N\) are the constant rotation velocity and the Brunt-Väisälä frequency. Here \(e_3\equiv (0,0,1), d=(d_1,d_2,d_3)\in \mathbb{S}^2\equiv\{d\in \mathbb{R}^3_{+}|\;|d|=1\}, d_3\neq 0\) is the unit vector in the direction of the rotation axis.NEWLINENEWLINEThe stationary solution NEWLINE\[NEWLINE u_E=\left(\begin{matrix} u_{E}^{1}\\ u_{E}^2\\ u_{E}^3 \\ \end{matrix}\right)= \left(\begin{matrix} a_2+(a_1-a_2)e^{-(x_3/\delta)}\cos(x_3/{\delta})+(b_1-b_2)e^{-(x_3/\delta)}\sin(x_3/{\delta})\\ b_2+(b_1-b_2)e^{-(x_3/\delta)}\cos(x_3/\delta)+(a_1-a_2)e^{-(x_3/\delta)}\sin(x_3/{\delta})\\ 0 \end{matrix}\right), NEWLINE\]NEWLINE NEWLINE\[NEWLINE\begin{aligned} &\theta _{E}=c_2 x_3+c_1,\\ & p_{E}=\frac{1}{2}\mathcal{G}c_2x_3^2+(\mathcal{G}c_1+\Omega d_2a_2-\Omega d_1b_2)x_3+\\ & +\frac{\delta \Omega}{2}e^{-(x_3/{\delta})}\cos(x_3/{\delta})[(a_2-a_1)(d_1+d_2)+(b_2-b_1)(d_2-d_1)]+\\ & +\frac{\delta \Omega}{2} e^{-(x_3/{\delta})}\sin(x_3/{\delta})[(a_1-a_2)(d_2-d_1)+(b_2-b_1)(d_1+d_2)] -\Omega d_3a_2x_2+\Omega d_3b_2x_1 \end{aligned}NEWLINE\]NEWLINE to (1) for any \(a_2, b_2, c_2 \in \mathbb{R}\) and with the boundary condition is called the Ekman boundary layer.NEWLINENEWLINEThe main results of the article can be briefly presented in the form of two assertionsNEWLINENEWLINE{Theorem 1.} Let be \(\Omega d_3>0, \mathcal{M}\equiv \sqrt{(a_1-a_2)^2+(b_1-b_2)^2}\). Then at the assumptions \((u_0-u_E,\theta _0-\theta_E)\in \mathcal{L}^2_{\sigma}(\mathbb{R}^3_{+})\times \mathcal{L}^2(\mathbb{R}^3_{+}),\; (c_2+N^2)\mathcal{G}>0\) and \(\frac{2\mathcal{M}}{\sqrt{\nu\Omega d_3}}<1,\) there exists at least one global weak solution \((\widetilde{u},\widetilde{\theta},\widetilde{p})\) to (1) with the initial data \((u_0, \theta_0)\) satisfying the strong energy inequality and NEWLINE\[NEWLINE\lim\limits_{T\to \infty}\int\limits_T^{T+1}[\|u(t)-u_E\|^2_{L^2(\mathbb{R}^3_{+})}+\|\theta(t)-\theta_E\|^2_{L^2(\mathbb{R}^3_{+})}]dt=0. NEWLINE\]NEWLINE If \((\widetilde{u}-u_E,\widetilde{\theta}-\theta_E)\in L^{p_1}(0,T;L^{p_2}(\mathbb{R}^3_{+}))\) with \(2/p_1 +3/p_2 =1\) for some \(p_2 >3\) and \(T>0\), then this solution is unique.NEWLINENEWLINE{Definition. } Let \(w_0 \in {L}^2_{\sigma}=\widetilde{{L}}^{(2)}_{\sigma}(\mathbb{R}^3_{+})={L}^2_{\sigma}\times {L}^2(\mathbb{R}^3_{+})\) and \((w,\widetilde{q})=(w^1,w^2,w^3,w^4,\widetilde{q})\) be a weak solution to (1) near the Ekman-type stationary solution \((u_E,\theta_E,p_E)\) with the initial data \(w_0\). It is said that \((w,\widetilde{q})\) satisfied the strong energy inequality, if NEWLINE\[NEWLINE\|w(t)\|^2_{L^2}+2\nu\int\limits_s^t\|\nabla \overline{w}(t)\|^2_{L^2}d\tau +2\kappa \int\limits_s^t\|\nabla {w}^4(t)\|^2_{L^2}d\tau +2\int\limits_s^t \left<w^3 \partial_3\widetilde{u}_E,w\right>d \tau \leq \|w(s)\|^2_{L^2} NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\begin{aligned} &w\equiv(w^1,w^2,w^3,w^4)\equiv \left( u-u_E, \sqrt{\frac{\mathcal{G}}{c_2+N^2}}(\theta -\theta _E)\right),\; \widetilde{q}\equiv p-p_E \quad \text{if}\quad \mathcal{G}>0\\ & w\equiv(w^1,w^2,w^3,w^4)\equiv \left( u-u_E,- \sqrt{\frac{\mathcal{G}}{c_2+N^2}}(\theta -\theta _E)\right),\; \widetilde{q}\equiv p-p_E \quad \text{if}\quad \mathcal{G}<0.\end{aligned} NEWLINE\]NEWLINENEWLINENEWLINESuch substitution is needed in order to study the solution stability to (1) near the Ekman-type stationary solution.NEWLINENEWLINE{Theorem 2.} Let \((u_0-u_E,\theta _0-\theta_E)\in H^1_{0,\sigma}(\mathbb{R}^3_{+})\times H^1_0(\mathbb{R}^3_{+})\). Then in the Theorem 1 conditions there is a positive constant \(\delta_0\) independent of \(u_0, \theta _0\), such that if \((\|u_0-u_E\|_{H^1_(\mathbb{R}^3_{+})})+\|\theta _0-\theta_E\|_{H^1_(\mathbb{R}^3_{+})})<\delta\), then there exists a unique global-in-time strong solution \((u,\theta, p)\) to (1) with the unique data \((u_0,\theta _0)\) and \(p\) is a associated with the \((u,\theta)\) pressure. Moreover, the solution \((u, \theta, p)\) satisfies the strong energy equality and \(\lim_{t\to \infty}(\|u(t)-u_E\|_{L^2_(\mathbb{R}^3_{+})}+\|\theta(t)-\theta_E\|_{L^2_(\mathbb{R}^3_{+})})=0\).NEWLINENEWLINEVarious Corollaries are derived for the cases when the rotating axis is perpendicular to the half-plane boundary or the absence of stratification effects, stability results when \(\Omega=0\).NEWLINENEWLINESeparate Chapters are devoted to the weak nonlinear stability of the solution to (1), smoothness of weak solutions and some extensions of the theory for the Navier-Stokes system with Coriolis force. Three appendices contain the auxiliary results.