We obtain a logarithmicaly sharp estimate for the space-analyticity radius of the solutions of the damped-driven 2D Navier-Stokes equations with periodic boundary conditions and relate this to the small scales in this system. This system is inspired by the Stommel-Charney barotropic ocean circulation model.

In this paper we introduce a finite-parameters feedback control algorithm for stabilizing solutions of various classes of damped nonlinear wave equations. Specifically, stabilization the zero steady state solution of initial boundary value problems for nonlinear weakly and strongly damped wave equations, nonlinear wave equation with nonlinear damping term and some related nonlinear wave equations, introducing a feedback control terms that employ parameters, such as, finitely many Fourier modes, finitely many volume elements and finitely many nodal observables and controllers. In addition, we also establish the stabilization of the zero steady state solution to initial boundary value problem for the damped nonlinear wave equation with a controller acting in a proper subdomain. Notably, the feedback controllers proposed here can be equally applied for stabilizing other solutions of the underlying equations.

In this paper we prove that an operator which projects weak solutions of the two- or three-dimensional Navier-Stokes equations onto a finite-dimensional space is determining if it annihilates the difference of two "nearby" weak solutions asymptotically, and if it satisfies a single appoximation inequality. We then apply this result to show that the long-time behavior of weak solutions to the Navier-Stokes equations, in both two- and three-dimensions, is determined by the long-time behavior of a finite set of bounded linear functionals. These functionals are constructed by local surface averages of solutions over certain simplex volume elements, and are therefore well-defined for weak solutions. Moreover, these functionals define a projection operator which satisfies the necessary approximation inequality for our theory. We use the general theory to establish lower bounds on the simplex diameters in both two- and three-dimensions. Furthermore, in the three dimensional case we make a connection between their diameters and the Kolmogoroff dissipation small scale in turbulent flows.

In this paper, we consider the Cauchy problem to the tropical climate model derived by Frierson-Majda-Pauluis in [15], which is a coupled system of the barotropic and baroclinic modes of the velocity and the typical midtropospheric temperature. The system considered in this paper has viscosities in the momentum equations, but no diffusivity in the temperature equation. We establish here the global well-posedness of strong solutions to this model. In proving the global existence of strong solutions, to overcome the difficulty caused by the absence of the diffusivity in the temperature equation, we introduce a new velocity w (called the pseudo baroclinic velocity), which has more regularities than the original baroclinic mode of the velocity. An auxiliary function φ, which looks like the effective viscous flux for the compressible Navier-Stokes equations, is also introduced to obtain the L∞ bound of the temperature. Regarding the uniqueness, we use the idea of performing suitable energy estimates at level one order lower than the natural basic energy estimates for the system.

In this paper we show the global well-posedness of solutions to a three-dimensional magnetohydrodynamical (MHD) model in porous media. Compared to the classical MHD equations, our system involves a nonlinear damping term in the momentum equations due to the "Brinkman-Forcheimerextended-Darcy" law of ow in porous media.

We show the existence of global weak solutions to the three-dimensional compressible primitive equations of atmospheric dynamics with degenerate viscosities. In analogy with the case of the compressible Navier-Stokes equations, the weak solutions satisfy the basic energy inequality, the Bresh-Desjardins entropy inequality, and the Mellet-Vasseur estimate. These estimates play an important role in establishing the compactness of the vertical velocity of the approximating solutions, and therefore are essential to recover the vertical velocity in the weak solutions.

In light of the question of finite-time blow-up vs. global well-posedness of solutions to problems involving nonlinear partial differential equations, we provide several cautionary examples which indicate that modifications to the boundary conditions or to the nonlinearity of the equations can effect whether the equations develop finite-time singularities. In particular, we aim to underscore the idea that in analytical and computational investigations of the blow-up of three-dimensional Euler and Navier-Stokes equations, the boundary conditions may need to be taken into greater account. We also examine a perturbation of the nonlinearity by dropping the advection term in the evolution of the derivative of the solutions to the viscous Burgers equation, which leads to the development of singularities not present in the original equation, and indicates that there is a regularizing mechanism in part of the nonlinearity. This simple analytical example corroborates recent computational observations in the singularity formation of fluid equations.

This paper is devoted to reviewing several recent developments concerning certain class of geophysical models, including the primitive equations (PEs) of atmospheric and oceanic dynamics and a tropical atmosphere model. The PEs for large-scale oceanic and atmospheric dynamics are derived from the Navier-Stokes equations coupled to the heat convection by adopting the Boussinesq and hydrostatic approximations, while the tropical atmosphere model considered here is a nonlinear interaction system between the barotropic mode and the first baroclinic mode of the tropical atmosphere with moisture. We are mainly concerned with the global well-posedness of strong solutions to these systems, with full or partial viscosity, as well as certain singular perturbation small parameter limits related to these systems, including the small aspect ratio limit from the Navier-Stokes equations to the PEs, and a small relaxation-parameter in the tropical atmosphere model. These limits provide a rigorous justification to the hydrostatic balance in the PEs, and to the relaxation limit of the tropical atmosphere model, respectively. Some conditional uniqueness of weak solutions, and the global well-posedness of weak solutions with certain class of discontinuous initial data, to the PEs are also presented.

We establish some conditional uniqueness of weak solutions to the viscous primitive equations, and as an application, we prove the global existence and uniqueness of weak solutions, with the initial data taken as small L∞ perturbations of functions in the space X = {v ∈ (L6(Ω))2|∂z v ∈ (L2(Ω))2}; in particular, the initial data are allowed to be discontinuous. Our result generalizes in a uniform way the result on the uniqueness of weak solutions with continuous initial data and that of the so-called z-weak solutions.

We present a computational study of a simple finite-dimensional feedback control algorithm for stabilizing solutions of infinite-dimensional dissipative evolution equations such as reaction-diffusion systems, the Navier-Stokes equations and the Kuramoto-Sivashinsky equation. This feedback control scheme takes advantage of the fact that such systems possess finite number of determining parameters or degrees of freedom, namely, finite number of determining Fourier modes, determining nodes, and determining interpolants and projections. In particular, the feedback control scheme uses finitely many of such observables and controllers that are acting on the coarse spatial scales. We demonstrate our numerical results for the stabilization of the unstable zero solution of the 1D Chafee-Infante equation and 1D Kuramoto-Sivashinksky equation. We give rigorous stability analysis for the feedback control algorithm and derive sufficient conditions relating the control parameters and model parameter values to attune to the control objective.