Global strong L well-posedness of the 3D primitive equations with heat and salinity diffusion

Abstract

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...

Analyticity of solutions to the primitive equations

“…The additional assumption on the time regularity of in Theorem assures that this solution is also an L 2 solution, and by Proposition for . (c)Concerning analyticity of solutions, for solutions u to the Navier–Stokes equations in with initial values in estimates on , , have been established in using heat kernel estimates. However, the method used there is not applicable in the presence of boundaries. (d)The surface pressure can be reconstructed from v , compare [, Equation (6.2)]. This gives , where .…”

“…The regularizing effect plays an important role when extending local solutions to global ones by means of certain a priori bounds. So far, in order to control the existence time in ‐spaces, H 2 ‐a priori bounds have been used in . In the following, we show that a priori bounds in the maximal regularity space are already sufficient to prove the global existence of a solution in ‐‐spaces.…”

“…Since odd and even function have vanishing traces and traces of the derivatives, respectively, the traces can be found in the interpolation spaces as long as they are well‐defined. Alternatively, the retractions and co‐retractions given in [, Section 4] can be used.…”

“…Choosing in particular and and noting that , we rediscover in particular the celebrated result by Cao and Titi . Furthermore, choosing allows us to enlarge the space of admissible initial values as constructed in to the above more general Besov space setting since . This class of initial values is also used in our works on rough initial data lying in for the case of mixed Dirichlet and Neumann boundary conditions.…”

Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity

“…Remarkably, different from the three-dimensional Navier-Stokes equations, global existence and uniqueness of strong solutions to the three-dimensional primitive equations has already been known since the breakthrough work by Cao-Titi [16]. This global existence of strong solutions to the primitive equations were also proved later by Kobelkov [23] and Kukavica-Ziane [25], by using some different approaches, see also Hieber-Kashiwabara [22] and Hieber-Hussien-Kashiwabara [21] for some generalizations in the L p settings.…”