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Anisotropic estimates and strong solutions of the primitive equations. (English) Zbl 1161.76454

The authors prove the local existence of strong solutions of the primitive equations of the ocean with initial data in the Sobolev space ${H}^{1}$. These equations are obtained from the Navier-Stokes equations by an asymptotic process where the small parameter is the ratio of the vertical scale over the horizontal scale. The domain of the ocean is assumed to be a Lipschitz-continuous domain

${\Omega }=\left\{\left(\stackrel{\to }{x},z\right)\in {ℝ}^{n-1}×ℝ\mid \stackrel{\to }{x}\in \omega ,-D\left(\stackrel{\to }{x}\right)

where $\omega \subset {ℝ}^{n-1},n=2,3$, is an open set and $D:\overline{\omega }\to ℝ$ is the depth function. The boundary of ${\Omega }$ is the union of ${{\Gamma }}_{s}=\left\{\left(\stackrel{\to }{x},0\right)\mid \stackrel{\to }{x}\in \omega \right\}$, ${{\Gamma }}_{b}=\left\{\left(\stackrel{\to }{x},-D\left(\stackrel{\to }{x}\right)\right)\mid \stackrel{\to }{x}\in \omega \right\}$, and ${{\Gamma }}_{l}=\left\{\left(\stackrel{\to }{x},z\right)\mid \stackrel{\to }{x}\in \partial \omega \phantom{\rule{0.166667em}{0ex}}-D\left(\stackrel{\to }{x}\right), where $\partial \omega$ is the boundary of $\omega$. The velocity of the fluid $\left(\stackrel{\to }{u},{u}_{3}\right)$ and the pressure ${p}_{s}$ satisfy

$\frac{\partial \stackrel{\to }{u}}{\partial t}+\left(\stackrel{\to }{u}·{\nabla }_{H}\right)\stackrel{\to }{u}+{u}_{3}\frac{\partial \stackrel{\to }{u}}{\partial z}+\alpha {\stackrel{\to }{u}}^{\perp }-{\nu }_{h}{{\Delta }}_{H}\stackrel{\to }{u}-{\nu }_{v}\frac{{\partial }^{2}\stackrel{\to }{u}}{\partial {z}^{2}}+{\nabla }_{H}{p}_{s}=\stackrel{\to }{F}\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{0.166667em}{0ex}}\left(0,T\right)×{\Omega },$
${\nabla }_{H}·{\int }_{-D\left(\stackrel{\to }{x}\right)}^{0}\stackrel{\to }{u}\left(t;\stackrel{\to }{x},z\right)\phantom{\rule{0.166667em}{0ex}}dz=0\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{0.166667em}{0ex}}\left(0,T\right)×\omega ,$
$\frac{\partial \stackrel{\to }{u}}{\partial z}{|}_{{{\Gamma }}_{s}}=\stackrel{\to }{\tau },\phantom{\rule{1.em}{0ex}}\stackrel{\to }{u}{|}_{{{\Gamma }}_{b}\cup {{\Gamma }}_{l}}=0,$
$\stackrel{\to }{u}{|}_{t=0}={\stackrel{\to }{u}}_{0}$

where ${\nabla }_{H}$ and ${{\Delta }}_{H}$ are the dimensional nabla and Laplacian operators respectively. The novel difficulty in this model, compared with the Navier-Stokes equations, is the lack of a priori estimates on the vertical component ${u}_{3}$: instead of having $|\nabla {u}_{3}{|}_{{L}^{2}}$ in ${L}^{2}\left(0,T\right)$ (in the Navier-Stokes equations), one only has $|\partial {u}_{3}{/\partial z|}_{{L}^{2}}$ in ${L}^{2}\left(0,T\right)$. The authors circumvent this difficulty using anisotropic inequalities and the special structure of the nonlinear term containing ${u}_{3}$ to obtain the necessary estimates on the nonlinear term for obtaining the global existence of small strong solutions. In order to obtain the local existence of strong solutions for large data in ${H}^{1}$ the authors linearize the equations around the solution of the associated linear problem and then use the small data argument.

##### MSC:
 76D03 Existence, uniqueness, and regularity theory 76U05 Rotating fluids 35Q35 PDEs in connection with fluid mechanics 35B45 A priori estimates for solutions of PDE 76D05 Navier-Stokes equations (fluid dynamics)