GLOBAL H-REGULARITY RESULTS OF THE 3D PRIMITIVE EQUATIONS OF THE OCEAN

In this article, we consider the 3D viscous primitive equations (PEs for brevity) of the ocean under two physically relevant boundary conditions for the H 1 and H 2 smooth initial data, respectively. The H 2 regularity result of the solution for the viscous PEs of the ocean has been unknown since the work by Cao and Titi (3), and Kobelkov (26). In this article we provide the global H 2 -regularity results of the solution and its time derivatives for the 3D viscous primitive equations of the ocean by using the L 6 estimates developed in (3) and some new energy estimate techniques.

[1]  Weiwei Sun,et al.  Stability and Convergence of the Crank-Nicolson/Adams-Bashforth scheme for the Time-Dependent Navier-Stokes Equations , 2007, SIAM J. Numer. Anal..

[2]  M. Petcu.  On the three-dimensional primitive equations , 2006, Advances in Differential Equations.

[3]  E. Titi,et al.  Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics , 2005, math/0503028.

[4]  Li Shan,et al.  A fully discrete stabilized finite element method for the time-dependent Navier-Stokes equations , 2009, Appl. Math. Comput..

[5]  Changbing Hu.  Finite Dimensional Behaviors of the Primitive Equations Under Small Depth Assumption , 2007 .

[6]  J. Pedlosky.  Geophysical Fluid Dynamics , 1979 .

[7]  J. Peixoto,et al.  Physics of climate , 1992 .

[8]  Roger Temam,et al.  On the equations of the large-scale ocean , 1992 .

[9]  W. Washington,et al.  An Introduction to Three-Dimensional Climate Modeling , 1986 .

[10]  Roger Temam,et al.  Some Mathematical Problems in Geophysical Fluid Dynamics , 2009 .

[11]  R. Rannacher,et al.  Finite element approximation of the nonstationary Navier-Stokes problem. I : Regularity of solutions and second-order error estimates for spatial discretization , 1982 .

[12]  E. Titi,et al.  Global well‐posedness and finite‐dimensional global attractor for a 3‐D planetary geostrophic viscous model , 2003 .

[13]  O. H. Lowry.  Academic press. , 1972, Analytical chemistry.

[14]  R. Temam.  Navier-Stokes Equations , 1977 .

[15]  J. Lions,et al.  New formulations of the primitive equations of atmosphere and applications , 1992 .

[16]  Giovanni P. Galdi,et al.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems , 2011 .

[17]  Roger Temam,et al.  Some Mathematical Aspects of Geophysical Fluid Dynamic Equations , 2003 .

[18]  Yinnian He,et al.  First order decoupled method of the primitive equations of the ocean I: Time discretization , 2014 .

[19]  R. Rannacher,et al.  Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: error estimates for second-order time discretization , 1990 .

[20]  G. J. Haltiner.  Numerical Prediction and Dynamic Meteorology , 1980 .

[21]  Yinnian He,et al.  Two-Level Method Based on Finite Element and Crank-Nicolson Extrapolation for the Time-Dependent Navier-Stokes Equations , 2003, SIAM J. Numer. Anal..

[22]  I. Kukavica,et al.  On the regularity of the primitive equations of the ocean , 2007 .

[23]  Rolf Rannacher,et al.  Finite element approximation of the nonstationary Navier-Stokes problem, part II: Stability of solutions and error estimates uniform in time , 1986 .

[24]  G. M. Kobel’kov,et al.  Existence of a Solution “in the Large” for Ocean Dynamics Equations , 2007 .

[25]  Changbing Hu.  Asymptotic analysis of the primitive equations under the small depth assumption , 2005 .

[26]  G. Kobelkov,et al.  Existence of a solution ‘in the large’ for the 3D large-scale ocean dynamics equations , 2006 .

[27]  Francisco Guillén-González,et al.  Anisotropic estimates and strong solutions of the primitive equations , 2001, Differential and Integral Equations.

[28]  M. Ziane.  Regularity results for the stationary primitive equations of the atmosphere and the ocean , 1997 .

[29]  N. Ju.  The global attractor for the solutions to the 3D viscous primitive equations , 2006 .

[30]  R. Temam,et al.  REGULARITY RESULTS FOR LINEAR ELLIPTIC PROBLEMS RELATED TO THE PRIMITIVE EQUATIONS , 2002 .

[31]  Yinnian He,et al.  Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes equations , 2005, Math. Comput..

[32]  Yinnian He,et al.  The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data , 2008, Math. Comput..

[33]  Gerd Baumann,et al.  Navier–Stokes Equations on R3 × [0, T] , 2016 .

[34]  Regularity results for stokes type systems related to climatology , 1995 .

[35]  R. Temam,et al.  The primitive equations on the large scale ocean under the small depth hypothesis , 2002 .

[36]  Roger Temam,et al.  A two-grid finite difference method for the primitive equations of the ocean , 2008 .

[37]  Roger Temam,et al.  Mathematical theory for the coupled atmosphere-ocean models (CAO III) , 1995 .