In this paper we consider the stochastic primitive equation for geophysical flows subject to transport noise and turbulent pressure. Admitting very rough noise terms, the global existence and uniqueness of solutions to this stochastic partial differential equation are proven using stochastic maximal L 2regularity, the theory of critical spaces for stochastic evolution equations, and global a priori bounds. Compared to other results in this direction, we do not need any smallness assumption on the transport noise which acts directly on the velocity field and we also allow rougher noise terms. The adaptation to Stratonovich type noise and, more generally, to variable viscosity and/or conductivity are discussed as well. Contents 1. Introduction 1 2. Preliminaries 6 3. Local and global existence in the strong-weak setting 8 4. L 2 -estimates for the linearized problem 14 5. Proof of the main results in the strong-weak setting 21 6. Local and global existence in the strong-strong setting 39 7. Inhomogeneous viscosity and conductivity 49 8. Transport noise of Stratonovich type 55 Appendix A. Kadlec's formulas 59 References 62

It is shown that the hydrostatic Stokes operator on L σ ¯ p ( Ω ) L^p_{\overline {\sigma }}(\Omega ) , where Ω ⊂ R 3 \Omega \subset \mathbb {R}^3 is a cylindrical domain subject to mixed periodic, Dirichlet and Neumann boundary conditions, admits a bounded H ∞ H^\infty -calculus on L σ ¯ p ( Ω ) L^p_{\overline {\sigma }}(\Omega ) for p ∈ ( 1 , ∞ ) p\in (1,\infty ) of H ∞ H^\infty -angle 0 0 . In particular, maximal L q − L p L^q-L^p -regularity estimates for the linearized primitive equations are obtained.

On finite metric graphs we consider Laplace operators, subject to various classes of non-self-adjoint boundary conditions imposed at graph vertices. We investigate spectral properties, existence of a Riesz basis of projectors and similarity transforms to self-adjoint Laplacians. Among other things, we describe a simple way how to relate the similarity transforms between Laplacians on certain graphs with elementary similarity transforms between matrices defining the boundary conditions. * On a leave from b.

Consider the full primitive equations, i.e. the three dimensional primitive equations coupled to the equation for temperature and salinity, and subject to outer forces. It is shown that this set of equations is globally strongly well-posed for arbitrary large initial data lying in certain interpolation spaces, which are explicitly characterized as subspaces of H 2/p,p , 1 < p < ∞, satisfying certain boundary conditions. In particular, global well-posedeness of the full primitive equations is obtained for initial data having less differentiability properties than H 1 , hereby generalizing by result by Cao and Titi [6] to the case of non-smooth data. In addition, it is shown that the solutions are exponentially decaying provided the outer forces possess this property.2010 Mathematics Subject Classification. Primary: 35Q35; Secondary: 76D03, 47D06, 86A05.The system is complemented by the boundary conditionsand α > 0. The rigorous analysis of the primitive equations started with the pioneering work of Lions, Temam and Wang [25][26][27], who proved the existence of a global weak solution for this set of equations for initial data a ∈ L 2 and b τ ∈ L 2 , b σ ∈ L 2 . For recent results on the uniqueness problem for weak solutions, we refer to the work of Li and Titi [28] and Kukavica, Pei, Rusin and Ziane [21]. The existence of a local, strong solution for the decoupled velocity equation with data a ∈ H 1 was proved by Guillén-González, Masmoudi and Rodiguez-Bellido in [15].In 2007, Cao and Titi [6] proved a breakthrough result for this set of equation which says, roughly speaking, that there exists a unique, global strong solution to the primitive equations for arbitrary initial data a ∈ H 1 and b τ ∈ H 1 neglecting salinity. Their proof is based on a priori H 1 -bounds for the solution, which in turn are obtained by L ∞ (L 6 ) energy estimates. Note that the boundary conditions on Γ b ∪ Γ l considered there are different from the ones we are imposing in (1.2). Kukavica and Ziane considered in [23,24] the primitive equations subject to the boundary conditions on Γ u ∪ Γ b as in (1.2) and they proved global strong well-posedness of the primitive equations with respect to arbitrary H 1 -data. For a different approach see also Kobelkov [20].Modifications of the primitive equations dealing with either only horizontal viscosity and diffusion or with horizontal or vertical eddy diffusivity were recently investigated by Cao and Titi in [7], by Cao, Li and Titi in [8][9][10]. Here, global well-posedness results are established for initial data in H 2 . For recent results concerning the presence of vapor, we refer to the work of Coti-Zelati, Huang, Kukavica, Teman and Ziane [37].The existence of a global attractor for the primitive equations was proved by Ju [19] and its properties were investigated by Chueshov [11].For local well-posedness results concerning the inviscid primitive equations, we refer to Brenier [4], Masmoudi and Wong [30], Kukavica, Temam, Vicol and Ziane [22] as well as Hamouda, Jung and Temam [16].Recentl...

Consider the anisotropic Navier-Stokes equations as well as the primitive equations. It is shown that the horizontal velocity of the solution to the anisotropic Navier-Stokes equations in a cylindrical domain of height ε with initial data u 0 = (v 0 , w 0) ∈ B 2−2/p q,p , 1/q + 1/p ≤ 1 if q ≥ 2 and 4/3q +2/3p ≤ 1 if q ≤ 2, converges as ε → 0 with convergence rate O(ε) to the horizontal velocity of the solution to the primitive equations with initial data v 0 with respect to the maximal-L p-L q-regularity norm. Since the difference of the corresponding vertical velocities remains bounded with respect to that norm, the convergence result yields a rigorous justification of the hydrostatic approximation in the primitive equations in this setting. It generalizes in particular a result by Li and Titi for the L 2-L 2-setting. The approach presented here does not rely on second order energy estimates but on maximal L p-L q-estimates for the heat equation.