In this paper we study the finite dimensional behavior of solutions of Primitive Equation (PEs for brevity) for large time. More precisely, we will prove that the long time behaviors of solutions of PEs are determined by the sets of finite fourier modes, line and volume elements respectively. Our results show that the long time behavior of PEs are dominated by the barotropic flow. This present article builds upon the above article concerning the existence and uniqueness of strong solutions of PEs in thin domains.
We consider the question of stability for planar wave solutions that arise in multidimensional conservation laws with fourth order regularization only. Such equations arise, for example, in the study of thin films, for which planar waves correspond with fluid coating a pre-wetted surface. An interesting feature of these equations is that both compressive and undercompressive planar waves arise as solutions (compressive or undercompressive with respect to asymptotic behavior relative to the un-regularized hyperbolic system), and numerical investigation by Bertozzi, M \ddot{\textrm{u}} nch, and Shearer indicates that undercompressive waves can be nonlinearly stable. Proceeding with pointwise estimates on the Green's function for the linear fourth order convection--regularization equation that arises upon linearization of the conservation law about the planar wave solution, we establish that under general spectral conditions consistent with shock fronts arising in our motivating thin films equations, compressive waves are stable for all dimensions d\ge2 and undercompressive waves are stable for dimensions d\ge3 . (In the special case d=1 , compressive waves are stable under a very general spectral condition.) We also consider an alternative spectral criterion, valid for example in the case of constant-coefficient regularization, for which we can establish nonlinear stability for compressive waves in dimensions d\ge3 and undercompressive waves in dimensions d\ge5 . The case of stability for undercompressive waves in the thin films equations for the critical dimensions d=1 and d=2 remains an interesting open problem.
For the case of multidimensional viscous conservation laws with fourth order smoothing only, we develop detailed pointwise estimates on the Green's function for the linear fourth order convection equation that arises upon linearization of the conservation law about a viscous planar wave solution. As in previous analyses in the case of second order smoothing, our estimates are sufficient to establish that spectral stability implies nonlinear stability, though the full development of this result will be considered in a companion paper.