Search

We analyze a three-dimensional rapidly rotating convection model of tall columnar structure in the limit of infinite Prandtl number, i.e., when the momentum diffusivity is much more dominant than the thermal diffusivity. Consequently, the dynamics of the velocity field takes place at a much faster time scale than the temperature fluctuation, and at the limit the velocity field formally adjusts instantaneously to the thermal fluctuation. We prove the global well-posedness of weak solutions and strong solutions to this model.

This work is devoted to establishing the local-in-time well-posedness of strong solutions to the three-dimensional compressible primitive equations of atmospheric dynamics. It is shown that strong solutions exist, are unique, and depend continuously on the initial data, for a short time in two cases: with gravity but without vacuum, and with vacuum but without gravity.

Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces. On the other hand, the inviscid PEs without rotation is known to be ill-posed in Sobolev spaces, and its smooth solutions can form singularity in finite time. In this paper, we extend the above results in the presence of rotation. We construct finite-time blowup solutions to the inviscid PEs with rotation, and establish that the inviscid PEs with rotation is ill-posed in Sobolev spaces in the sense that its perturbation around a certain steady state background flow is both linearly and nonlinearly ill-posed in Sobolev spaces. Its linear instability is of the Kelvin-Helmholtz type similar to the one appears in the context of vortex sheets problem. This implies that the inviscid PEs is also linearly ill-posed in Gevrey class of order s>1, and suggests that a suitable space for the well-posedness is Gevrey class of order s=1, which is exactly the space of analytic functions.

In this paper, we provide rigorous justification of the hydrostaticapproximation and the derivation of primitive equations as the small aspectratio limit of the incompressible three-dimensional Navier-Stokes equations inthe anisotropic horizontal viscosity regime. Setting $\varepsilon >0$ to be thesmall aspect ratio of the vertical to the horizontal scales of the domain, weinvestigate the case when the horizontal and vertical viscosities in theincompressible three-dimensional Navier-Stokes equations are of orders $O(1)$and $O(\varepsilon^\alpha)$, respectively, with $\alpha>2$, for which thelimiting system is the primitive equations with only horizontal viscosity as$\varepsilon$ tends to zero. In particular we show that for "well prepared"initial data the solutions of the scaled incompressible three-dimensionalNavier-Stokes equations converge strongly, in any finite interval of time, tothe corresponding solutions of the anisotropic primitive equations with onlyhorizontal viscosities, as $\varepsilon$ tends to zero, and that theconvergence rate is of order $O\left(\varepsilon^\frac\beta2\right)$, where$\beta=\min\{\alpha-2,2\}$. Note that this result is different from the case$\alpha=2$ studied in [Li, J.; Titi, E.S.: \emph{The primitive equations as thesmall aspect ratio limit of the Navier-Stokes equations: Rigorous justificationof the hydrostatic approximation}, J. Math. Pures Appl., \textbf{124}\rm(2019), 30--58], where the limiting system is the primitive equations withfull viscosities and the convergence is globally in time and its rate of order$O\left(\varepsilon\right)$.

Differential buoyancy surface sources in the ocean may induce adensity-driven flow that joins faster flow components to create a multi-scale,3D flow. Potential temperature and salinity are active tracers that determinethe ocean's potential density: their distribution strongly affects thedensity-driven component while the overall flow affects their distribution. Wepresent a robust framework that allows one to study the effects of a general 3Dflow on a density-driven velocity component, by constructing a modularobservation-based 3D model of intermediate complexity. The model contains anincompressible velocity that couples two advection-diffusion equations, fortemperature and salinity. Instead of solving the Navier-Stokes equations forthe velocity, we consider a flow composed of several temporally separated,spatially predetermined modes. One of these modes models the density-drivenflow: its spatial form describes the density-driven flow structure and itsstrength is determined dynamically by average density differences. The othermodes are completely predetermined, consisting of any incompressible, possiblyunsteady, 3D flow, e.g. as determined by kinematic models, observations, orsimulations. The model is a non-linear, weakly coupled system of two non-localPDEs. We prove its well-posedness in the sense of Hadamard, and obtain rigorousbounds regarding analytical solutions. The model's relevance to oceanic systemsis demonstrated by tuning the model to mimic the North Atlantic ocean'sdynamics. In one limit the model recovers a simplified oceanic box model and inanother limit a kinematic model of oceanic chaotic advection, suggesting it canbe utilized to study spatially dependent feedback processes in the ocean.

This paper establishes the local-in-time well-posedness of solutions to anapproximating system constructed by mildly regularizing the dynamical sea-icemodel of {\it W.D. Hibler, Journal of Physical Oceanography, 1979}. Our choiceof regularization has been carefully designed, prompted by physicalconsiderations, to retain the original coupled hyperbolic-parabolic characterof Hibler's model. Various regularized versions of this model have been usedwidely for the numerical simulation of the circulation and thickness of theArctic ice cover. However, due to the singularity in the ice rheology, thenotion of solutions to the original model is unclear. Instead, an approximatingsystem, which captures current numerical study, is proposed. The well-posednesstheory of such a system provides a first-step groundwork in both numericalstudy and future analytical study.

We study the three-dimensional Navier–Stokes equations of rotating incompressible viscous fluids with periodic boundary conditions. The asymptotic expansions, as time goes to infinity, are derived in all Gevrey spaces for any Leray–Hopf weak solutions in terms of oscillating, exponentially decaying functions. The results are established for all non-zero rotation speeds, and for both cases with and without the zero spatial average of the solutions. Our method makes use of the Poincaré waves to rewrite the equations, and then implements the Gevrey norm techniques to deal with the resulting time-dependent bi-linear form. Special solutions are also found which form infinite dimensional invariant linear manifolds.

The Rayleigh–Bénard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the L2 norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy–palinstrophy plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations.

In this paper, we consider the 3D primitive equations of oceanic and atmospheric dynamics with only horizontal eddy viscosities in the horizontal momentum equations and only vertical diffusivity in the temperature equation. Global well-posedness of strong solutions is established for any initial data such that the initial horizontal velocity v0∈H2(Ω) and the initial temperature T0∈H1(Ω)∩L∞(Ω) with ∇HT0∈Lq(Ω), for some q∈(2,∞). Moreover, the strong solutions enjoy correspondingly more regularities if the initial temperature belongs to H2(Ω). The main difficulties are the absence of the vertical viscosity and the lack of the horizontal diffusivity, which, interact with each other, thus causing the “mismatching” of regularities between the horizontal momentum and temperature equations. To handle this “mismatching” of regularities, we introduce several auxiliary functions, i.e., η,θ,φ, and ψ in the paper, which are the horizontal curls or some appropriate combinations of the temperature with the horizontal divergences of the horizontal velocity v or its vertical derivative ∂zv. To overcome the difficulties caused by the absence of the horizontal diffusivity, which leads to the requirement of some Lt1(Wx1,∞)-type a priori estimates on v, we decompose the velocity into the “temperature-independent” and temperature-dependent parts and deal with them in different ways, by using the logarithmic Sobolev inequalities of the Brézis–Gallouet–Wainger and Beale–Kato–Majda types, respectively. Specifically, a logarithmic Sobolev inequality of the limiting type, introduced in our previous work (Cao et al., 2016), is used, and a new logarithmic type Gronwall inequality is exploited.

This work concerns the zero Mach number limit of the compressible primitive equations. The primitive equations with the incompressibility condition are identified as the limiting equations. The convergence with well-prepared initial data (i.e., initial data without acoustic oscillations) is rigorously justified, and the convergence rate is shown to be of order O(ε) , as ε→ 0 +, where ε represents the Mach number. As a byproduct, we construct a class of global solutions to the compressible primitive equations, which are close to the incompressible flows.

We study the effect of the rotation on the life-span of solutions to the $3D$hydrostatic Euler equations with rotation and the inviscid Primitive equations(PEs) on the torus. The space of analytic functions appears to be the naturalspace to study the initial value problem for the inviscid PEs with generalinitial data, as they have been recently shown to exhibit Kelvin-Helmholtz typeinstability. First, for a short interval of time that is independent of therate of rotation $|\Omega|$, we establish the local well-posedness of theinviscid PEs in the space of analytic functions. In addition, thanks to a fineanalysis of the barotropic and baroclinic modes decomposition, we establish tworesults about the long time existence of solutions. (i) Independently of$|\Omega|$, we show that the life-span of the solution tends to infinity as theanalytic norm of the initial baroclinic mode goes to zero. Moreover, we show inthis case that the solution of the $3D$ inviscid PEs converges to the solutionof the limit system, which is governed by the $2D$ Euler equations. (ii) Weshow that the life-span of the solution can be prolonged unboundedly with$|\Omega|\rightarrow \infty$, which is the main result of this paper. This isestablished for "well-prepared" initial data, namely, when only the Sobolevnorm (but not the analytic norm) of the baroclinic mode is small enough,depending on $|\Omega|$. Furthermore, for large $|\Omega|$ and "well-prepared"initial data, we show that the solution to the $3D$ inviscid PEs isapproximated by the solution to a simple limit resonant system with the sameinitial data.

In this paper we prove the local well-posedness and global well-posedness with small initial data of the strong solution to the reduced 3D primitive geostrophic adjustment model with weak dissipation. The term reduced model means that the relevant physical quantities depend only on two spatial variables. The weak dissipation helps us overcome the ill-posedness of the original model. We also prove the global well-posedness of the strong solution to the Voigt α-regularization of this model, and establish the convergence of the strong solution of the Voigt α-regularized model to the corresponding solution of the original model. Furthermore, we derive a criterion for existence of finite-time blow-up of the original model with weak dissipation based on Voigt α-regularization.

We establish the local and global well-posedness of strong solutions to the two- and three-dimensional anelastic equations of stratified viscous flows. In this model, the interaction of the density profile with the velocity field is taken into account, and the density background profile is permitted to have physical vacuum singularity. The existing time of the solutions is infinite in two dimensions, with general initial data, and in three dimensions with small initial data.

In this work we study the global solvability of the primitive equations for the atmosphere coupled to moisture dynamics with phase changes for warm clouds, where water is present in the form of water vapor and in the liquid state as cloud water and rain water. This moisture model contains closures for the phase changes condensation and evaporation, as well as the processes of auto conversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. It has been used by Klein and Majda in [17] and corresponds to a basic form of the bulk microphysics closure in the spirit of Kessler [16] and Grabowski and Smolarkiewicz [12]. The moisture balances are strongly coupled to the thermodynamic equation via the latent heat associated to the phase changes. In [14] we assumed the velocity field to be given and proved rigorously the global existence and uniqueness of uniformly bounded solutions of the moisture balances coupled to the thermodynamic equation. In this paper we present the solvability of a full moist atmospheric flow model, where the moisture model is coupled to the primitive equations of atmospherical dynamics governing the velocity field. For the derivation of a priori estimates for the velocity field we thereby use the ideas of Cao and Titi [6], who succeeded in proving the global solvability of the primitive equations.

Using the convex integration technique for the three-dimensional Navier–Stokes equations introduced by Buckmaster and Vicol, it is shown the existence of non-unique weak solutions for the 3D Navier–Stokes equations with fractional hyperviscosity (- Δ) θ, whenever the exponent θ is less than Lions’ exponent 5/4, i.e., when θ< 5 / 4.

In this paper we analyze a finite element method applied to a continuous downscaling data assimilation algorithm for the numerical approximation of the two- and three-dimensional Navier-Stokes equations corresponding to given measurements on a coarse spatial scale. For representing the coarse mesh measurements we consider different types of interpolation operators including a Lagrange interpolant. We obtain uniform-in-time estimates for the error between a finite element approximation and the reference solution corresponding to the coarse mesh measurements. We consider both the case of a plain Galerkin method and a Galerkin method with grad-div stabilization. For the stabilized method we prove error bounds in which the constants do not depend on inverse powers of the viscosity. Some numerical experiments illustrate the theoretical results.

AbstractIt is well known, by now, that the three-dimensional non-viscous planetary geostrophic model, with vertical hydrostatic balance and horizontal Rayleigh friction/damping, coupled to the heat diffusion and transport, is mathematically ill-posed. This is because the no-normal flow physical boundary condition implicitly produces an additional boundary condition for the temperature at the lateral boundary. This additional boundary condition is different, because of the Coriolis forcing term, than the no-heat-flux physical boundary condition. Consequently, the second order parabolic heat equation is over-determined with two different boundary conditions. In a previous work we proposed one remedy to this problem by introducing a fourth-order artificial hyper-diffusion to the heat transport equation and proved global regularity for the proposed model. A shortcoming of this higher-oder diffusion is the loss of the maximum/minimum principle for the heat equation. Another remedy for this problem was suggested by R. Salmon by introducing an additional Rayleigh-like friction/damping term for the vertical component of the velocity in the hydrostatic balance equation. In this paper we prove the global, for all time and all initial data, well-posedness of strong solutions to the three-dimensional Salmon’s planetary geostrophic model of ocean dynamics. That is, we show global existence, uniqueness and continuous dependence of the strong solutions on initial data for this model. Unlike the 3D viscous PG model, we are still unable to show the uniqueness of the weak solution. Notably, we also demonstrate in what sense the additional damping term, suggested by Salmon, annihilate the ill-posedness in the original system; consequently, it can be viewed as “regularizing” term that can possibly be used to regularize other related systems.

We construct a determining form for the surface quasi-geostrophic (SQG) equation with subcritical dissipation. In particular, we show that the global attractor for this equation can be embedded in the long-time dynamics of an ordinary differential equation (ODE) called a determining form. Indeed, there is a one-to-one correspondence between the trajectories in the global attractor of the SQG equation and the steady state solutions of the determining form. The determining form is a true ODE in the sense that its vector field is Lipschitz. This is shown by combining De Giorgi techniques and elementary harmonic analysis. Finally, we provide elementary proofs of the existence of time-periodic solutions, steady state solutions, as well as the existence of finitely many determining parameters for the SQG equation.

An intrinsic property of almost any physical measuring device is that it makes observations which are slightly blurred in time. The authors consider a nudging-based approach for data assimilation that constructs an approximate solution based on a feedback control mechanism that is designed to account for observations that have been blurred by a moving time average. Analysis of this nudging model in the context of the subcritical surface quasi-geostrophic equation shows, provided the time-averaging window is sufficiently small and the resolution of the observations sufficiently fine, that the approximating solution converges exponentially fast to the observed solution over time. In particular, the authors demonstrate that observational data with a small blur in time possess no significant obstructions to data assimilation provided that the nudging properly takes the time averaging into account. Two key ingredients in our analysis are additional bounded-ness properties for the relevant interpolant observation operators and a non-local Gronwall inequality.

We consider the incompressible Euler equations in a bounded domain in threespace dimensions. Recently, the first two authors proved Onsager's conjecturefor bounded domains, i.e., that the energy of a solution to these equations isconserved provided the solution is H\"older continuous with exponent greaterthan 1/3, uniformly up to the boundary. In this contribution we relax thisassumption, requiring only interior H\"older regularity and continuity of thenormal component of the energy flux near the boundary. The significance of thisimprovement is given by the fact that our new condition is consistent with thepossible formation of a Prandtl-type boundary layer in the vanishing viscositylimit.

We construct a determining form for the 2D Rayleigh-B\'enard (RB) system in astrip with solid horizontal boundaries, in the cases of no-slip and stress-freeboundary conditions. The determining form is an ODE in a Banach space oftrajectories whose steady states comprise the long-time dynamics of the RBsystem. In fact, solutions on the global attractor of the RB system can befurther identified through the zeros of a scalar equation to which the ODEreduces for each initial trajectory. The twist in this work is that thetrajectories are for the velocity field only, which in turn determines thecorresponding trajectories of the temperature.

In this paper we survey the various implementations of a new data assimilation (downscaling) algorithm based on spatial coarse mesh measurements. As a paradigm, we demonstrate the application of this algorithm to the 3D Leray-α subgrid scale turbulence model. Most importantly, we use this paradigm to show that it is not always necessary to collect coarse mesh measurements of all the state variables that are involved in the underlying evolutionary system, in order to recover the corresponding exact reference solution. Specifically, we show that in the case of the 3D Leray-α model of turbulence, the solutions of the algorithm, constructed using only coarse mesh observations of any two components of the three-dimensional velocity field, and without any information on the third component, converge, at an exponential rate in time, to the corresponding exact reference solution of the 3D Leray-α model. This study serves as an addendum to our recent work on abridged continuous data assimilation for the 2D Navier-Stokes equations. Notably, similar results have also been recently established for the 3D viscous Planetary Geostrophic circulation model in which we show that coarse mesh measurements of the temperature alone are sufficient for recovering, through our data assimilation algorithm, the full solution; i.e. the three components of velocity vector field and the temperature. Consequently, this proves the Charney conjecture for the 3D Planetary Geostrophic model; namely, that the history of the large spatial scales of temperature is sufficient for determining all the other quantities (state variables) of the model.

The aim of this work is to extend and prove the Onsager conjecture for a class of conservation laws that possess generalized entropy. One of the main findings of this work is the “universality” of the Onsager exponent, α> 1 / 3 , concerning the regularity of the solutions, say in C ,α , that guarantees the conservation of the generalized entropy, regardless of the structure of the genuine nonlinearity in the underlying system.

An important feature of the planetary oceanic dynamics is that the aspect ratio (the ratio of the depth to horizontal width) is very small. As a result, the hydrostatic approximation (balance), derived by performing the formal small aspect ratio limit to the Navier–Stokes equations, is considered as a fundamental component in the primitive equations of the large-scale ocean. In this paper, we justify rigorously the small aspect ratio limit of the Navier–Stokes equations to the primitive equations. Specifically, we prove that the Navier–Stokes equations, after being scaled appropriately by the small aspect ratio parameter of the physical domain, converge strongly to the primitive equations, globally and uniformly in time, and the convergence rate is of the same order as the aspect ratio parameter. This result validates the hydrostatic approximation for the large-scale oceanic dynamics. Notably, only the weak convergence of this small aspect ratio limit was rigorously justified before.

Based on a previously introduced downscaling data assimilation algorithm, which employs a nudging term to synchronize the coarse mesh spatial scales, we construct a determining map for recovering the full trajectories from their corresponding coarse mesh spatial trajectories, and investigate its properties. This map is then used to develop a downscaling data assimilation scheme for statistical solutions of the two-dimensional Navier–Stokes equations, where the coarse mesh spatial statistics of the system is obtained from discrete spatial measurements. As a corollary, we deduce that statistical solutions for the Navier–Stokes equations are determined by their coarse mesh spatial distributions. Notably, we present our results in the context of the Navier–Stokes equations; however, the tools are general enough to be implemented for other dissipative evolution equations.

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.

We describe a spectrally filtered discrete-in-time downscaling data assimilation algorithm and prove, in the context of the two-dimensional Navier-Stokes equations, that this algorithm works for a general class of interpolants, such as those based on local spatial averages as well as point measurements of the velocity. Our algorithm is based on the classical technique of inserting new observational data directly into the dynamical model as it is being evolved over time, rather than nudging, and extends previous results in which the observations were defined directly in terms of an orthogonal projection onto the large-scale (lower) Fourier modes. In particular, our analysis does not require the interpolant to be represented by an orthogonal projection, but requires only the interpolant to satisfy a natural approximation of the identity.

We introduce a continuous (downscaling) data assimilation algorithm for the 2D Bénard convection problem using vorticity or local circulation measurements only. In this algorithm, a nudging term is added to the vorticity equation to constrain the model. Our numerical results indicate that the approximate solution of the algorithm is converging to the unknown reference solution (vorticity and temperature) corresponding to the measurements of the 2D Bénard convection problem when only spatial coarse-grain measurements of vorticity are assimilated. Moreover, this convergence is realized using data which is much more coarse than the resolution needed to satisfy rigorous analytical estimates.

We study a three-dimensional fluid model describing rapidly rotatingconvection that takes place in tall columnar structures. The purpose of thismodel is to investigate the cyclonic and anticyclonic coherent structures.Global existence, uniqueness, continuous dependence on initial data, andlarge-time behavior of strong solutions are shown provided the model isregularized by a weak dissipation term.

We study the three-dimensional Hasegawa-Mima model of turbulent magnetizedplasma with horizontal viscous terms and a weak vertical dissipative term. Inparticular, we establish the global existence and uniqueness of strongsolutions for this model.

The goal of this note is to show that, in a bounded domain Ω ⊂ Rn, with ∂Ω ∈ C2, any weak solution (u(x, t) , p(x, t)) , of the Euler equations of ideal incompressible fluid in Ω × (0 , T) ⊂ Rn× Rt, with the impermeability boundary condition u· n→ = 0 on ∂Ω × (0 , T) , is of constant energy on the interval (0,T), provided the velocity field u∈ L3((0 , T) ; C0,α(Ω ¯)) , with α>13.

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler–Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler–Voigt equations also require less resolution than simulations of the 3D Euler equations for fixed values of the regularization parameter α> 0. Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly, namely by simulating the better-behaved 3D Euler–Voigt equations. The new criteria are only known to be sufficient criterion for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well known to occur.

We study a moisture model for warm clouds that has been used by Klein and Majda (2006 Theor. Comput. Fluid Dyn. 20 525-551) as a basis for multiscale asymptotic expansions for deep convective phenomena. These moisture balance equations correspond to a bulk microphysics closure in the spirit of Kessler (1969 Meteorol. Monogr. 10 1-84) and Grabowski and Smolarkiewicz (1996 Mon. Weather Rev. 124 487-97), in which water is present in the gaseous state as water vapor and in the liquid phase as cloud water and rain water. It thereby contains closures for the phase changes condensation and evaporation, as well as the processes of autoconversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. Phase changes are associated with enormous amounts of latent heat and therefore provide a strong coupling to the thermodynamic equation. In this work we assume the velocity field to be given and prove rigorously the global existence and uniqueness of uniformly bounded solutions of the moisture model with viscosity, diffusion and heat conduction. To guarantee local well-posedness we first need to establish local existence results for linear parabolic equations, subject to the Robin boundary conditions on the cylindric type of domains under consideration. We then derive a priori estimates, for proving the maximum principle, using the Stampacchia method, as well as the iterative method by Alikakos (1979 J. Differ. Equ. 33 201-25) to obtain uniform boundedness. The evaporation term is of power law type, with an exponent in general less or equal to one and therefore making the proof of uniqueness more challenging. However, these difficulties can be circumvented by introducing new unknowns, which satisfy the required cancellation and monotonicity properties in the source terms.

We show that the global attractor of a weakly damped and driven Korteweg–de Vries equation (KdV) is embedded in the long-time dynamics of an ordinary differential equation called a determining form. In particular, there is a one-to-one identification of the trajectories in the global attractor of the damped and driven KdV and the steady state solutions of the determining form. Moreover, we analyze a data assimilation algorithm (down-scaling) for the weakly damped and driven KdV. We show that given a certain number of low Fourier modes of a reference solution of the KdV equation, the algorithm recovers the full reference solution at an exponential rate in time.

In this paper we propose a continuous data assimilation (downscaling) algorithm for a two-dimensional Bénard convection problem. Specifically we consider the two-dimensional Boussinesq system of a layer of incompressible fluid between two solid horizontal walls, with no-normal flow and stress-free boundary conditions on the walls, and the fluid is heated from the bottom and cooled from the top. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the horizontal velocity. We show that under an appropriate choice of the nudging parameter and the size of the spatial coarse mesh observables, and under the assumption that the observed data are error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed data on the horizontal component of the velocity.

In this paper we introduce a finite-parameters feedback control algorithm forstabilizing solutions of the Navier-Stokes-Voigt equations, the strongly dampednonlinear wave equations and the nonlinear wave equation with nonlinear dampingterm, the Benjamin-Bona-Mahony-Burgers equation and the KdV-Burgers equation.This algorithm capitalizes on the fact that such infinite-dimensionaldissipative dynamical systems posses finite-dimensional long-time behaviorwhich is represented by, for instance, the finitely many determining parametersof their long-time dynamics, such as determining Fourier modes, determiningvolume elements, determining nodes , etc..The algorithm utilizes these finiteparameters in the form of feedback control to stabilize the relevant solutions.For the sake of clarity, and in order to fix ideas, we focus in this work onthe case of low Fourier modes feedback controller, however, our results andtools are equally valid for using other feedback controllers employing otherspatial coarse mesh interpolants.

In this article, we prove that data assimilation by feedback nudging can be achieved for the three-dimensional quasi-geostrophic equation in a simplified scenario using only large spatial scale observables on the dynamical boundary. On this boundary, a scalar unknown (buoyancy or surface temperature of the fluid) satisfies the surface quasi-geostrophic equation. The feedback nudging is done on this two-dimensional model, yet ultimately synchronizes the streamfunction of the three-dimensional flow. The main analytical difficulties are due to the presence of a nonlocal dissipative operator in the surface quasi-geostrophic equation. This is overcome by exploiting a suitable partition of unity, the modulus of continuity characterization of Sobolev space norms, and the Littlewood-Paley decomposition to ultimately establish various boundedness and approximation-of-identity properties for the observation operators.

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 apply the Postprocessing Galerkin method to a recently introducedcontinuous data assimilation (downscaling) algorithm for obtaining a numericalapproximation of the solution of the two-dimensional Navier-Stokes equationscorresponding to given measurements from a coarse spatial mesh. Under suitableconditions on the relaxation (nudging) parameter, the resolution of the coarsespatial mesh and the resolution of the numerical scheme, we obtain uniform intime estimates for the error between the numerical approximation given by thePostprocessing Galerkin method and the reference solution corresponding to themeasurements. Our results are valid for a large class of interpolant operators,including low Fourier modes and local averages over finite volume elements.Notably, we use here the 2D Navier-Stokes equations as a paradigm, but ourresults apply equally to other evolution equations, such as the Boussinesqsystem of Benard convection and other oceanic and atmospheric circulationmodels.

In this paper, we consider a nonlinear interaction system between the barotropic mode and the first baroclinic mode of the tropical atmosphere with moisture, which was derived in Frierson et al (2004 Commum. Math. Sci. 2 591-626). We establish the global existence and uniqueness of strong solutions to this system, with initial data in H1, for each fixed convective adjustment relaxation time parameter ϵ > 1. Moreover, if the initial data possess slightly more regularity than H 1, then the unique strong solution depends continuously on the initial data. Furthermore, by establishing several appropriate ϵ-independent estimates, we prove that the system converges to a limiting system as the relaxation time parameter ϵ tends to zero, with a convergence rate of the order O(√ϵ). Moreover, the limiting system has a unique global strong solution for any initial data in H1 and such a unique strong solution depends continuously on the initial data if the initial data posses slightly more regularity than H1. Notably, this solves the viscous version of an open problem proposed in the above mentioned paper of Frierson, Majda and Pauluis.

Analyzing the validity and success of a data assimilation algorithm when somestate variable observations are not available is an important problem inmeteorology and engineering. We present an improved data assimilation algorithmfor recovering the exact full reference solution (i.e. the velocity andtemperature) of the 3D Planetary Geostrophic model, at an exponential rate intime, by employing coarse spatial mesh observations of the temperature alone.This provides, in the case of this paradigm, a rigorous justification to anearlier conjecture of Charney which states that temperature history of theatmosphere, for certain simple atmospheric models, determines all other statevariables.

In this paper, we consider the initial boundary value problem of the three-dimensional primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well-posedness of the strong solution is established for any H2 initial data. An N-dimensional logarithmic Sobolev embedding inequality, which bounds the L∞-norm in terms of the Lq-norms up to a logarithm of the Lp-norm for p > N of the first-order derivatives, and a system version of the classic Grönwall inequality are exploited to establish the required a~priori H2 estimates for global regularity.© 2016 Wiley Periodicals, Inc.

In the present study, we are interested in the Davey-Stewartson equations (DSE) that model packets of surface and capillary-gravity waves. We focus on the elliptic-elliptic case, for which it is known that DSE may develop a finite-time singularity. We propose three systems of non-viscous regularization to the DSE in a variety of parameter regimes under which the finite-time blow-up of solutions to the DSE occurs. We establish the global well-posedness of the regularized systems for all initial data. The regularized systems, which are inspired by the α-models of turbulence and therefore are called the α-regularized DSE, are also viewed as unbounded, singularly perturbed DSE. Therefore, we also derive reduced systems of ordinary differential equations for the α-regularized DSE by using the modulation theory to investigate the mechanism with which the proposed non-viscous regularization prevents the formation of the singularities in the regularized DSE. This is a follow-up of the work [Cao et al., Nonlinearity 21, 879-898 (2008); Cao et al., Numer. Funct. Anal. Optim. 30, 46-69 (2009)] on the non-viscous α-regularization of the nonlinear Schrödinger equation.

In this paper, we consider the initial-boundary value problem of thethree-dimensional primitive equations for oceanic and atmospheric dynamics withonly horizontal viscosity and horizontal diffusivity. We establish the local,in time, well-posedness of strong solutions, for any initial data $(v_0,T_0)\in H^1$, by using the local, in space, type energy estimate. We alsoestablish the global well-posedness of strong solutions for this system, withany initial data $(v_0, T_0)\in H^1\cap L^\infty$, such that $\partial_zv_0\inL^m$, for some $m\in(2,\infty)$, by using the logarithmic type anisotropicSobolev inequality and a logarithmic type Gronwall inequality. This paperimproves the previous results obtained in [Cao, C.; Li, J.; Titi, E.S.: Globalwell-posedness of the 3D primitive equations with only horizontal viscosity anddiffusivity, Comm. Pure Appl.Math., Vol. 69 (2016), 1492-1531.], where theinitial data $(v_0, T_0)$ was assumed to have $H^2$ regularity.

In this paper we propose a continuous data assimilation (downscaling) algorithm for the Bénard convection in porous media using only discrete spatial-mesh measurements of the temperature. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the temperature. We show that under an appropriate choice of the nudging parameter and the size of the mesh, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed (finite dimensional projection of) temperature data. Moreover, we note that in the case where the observational measurements are not error free, one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements.

We prove the global well-posedness of the two-dimensional Boussinesq equations with only vertical dissipation. The initial data (Formula presented.) are required to be only in the space (Formula presented.) , and thus our result generalizes that of Cao and Wu (Arch Rational Mech Anal, 208:985–1004, 2013), where the initial data are assumed to be in (Formula presented.). The assumption on the initial data is at the minimal level that is required to guarantee the uniqueness of the solutions. A logarithmic type limiting Sobolev embedding inequality for the (Formula presented.) norm, in terms of anisotropic Sobolev norms, and a logarithmic type Gronwall inequality are established to obtain the global in time a priori estimates, which guarantee the local solution to be a global one.

This paper is devoted to reviewing several recent developments concerningcertain class of geophysical models, including the primitive equations (PEs) ofatmospheric and oceanic dynamics and a tropical atmosphere model. The PEs forlarge-scale oceanic and atmospheric dynamics are derived from the Navier-Stokesequations coupled to the heat convection by adopting the Boussinesq andhydrostatic approximations, while the tropical atmosphere model considered hereis a nonlinear interaction system between the barotropic mode and the firstbaroclinic mode of the tropical atmosphere with moisture. We are mainly concerned with the global well-posedness of strong solutions tothese systems, with full or partial viscosity, as well as certain singularperturbation small parameter limits related to these systems, including thesmall aspect ratio limit from the Navier-Stokes equations to the PEs, and asmall relaxation-parameter in the tropical atmosphere model. These limitsprovide a rigorous justification to the hydrostatic balance in the PEs, and tothe relaxation limit of the tropical atmosphere model, respectively. Someconditional uniqueness of weak solutions, and the global well-posedness of weaksolutions with certain class of discontinuous initial data, to the PEs are alsopresented.

This paper concerns the uniqueness of weak solutions to the Cauchy problem to the Ericksen-Leslie system of liquid crystal models in ℝ2, with both general Leslie stress tensors and general Oseen-Frank density. It is shown here that such a system admits a unique weak solution provided that the Frank coefficients are close to some positive constant. One of the main ideas of our proof is to perform suitable energy estimates at the level one order lower than the natural basic energy estimates for the Ericksen-Leslie system.

In this paper we introduce and analyze an algorithm for continuous data assimilation for a three-dimensional Brinkman-Forchheimer-extended Darcy (3D BFeD) model of porous media. This model is believed to be accurate when the flow velocity is too large for Darcy's law to be valid, and additionally the porosity is not too small. The algorithm is inspired by ideas developed for designing finite-parameters feedback control for dissipative systems. It aims to obtain improved estimates of the state of the physical system by incorporating deterministic or noisy measurements and observations. Specifically, the algorithm involves a feedback control that nudges the large scales of the approximate solution toward those of the reference solution associated with the spatial measurements. In the first part of the paper, we present a few results of existence and uniqueness of weak and strong solutions of the 3D BFeD system. The second part is devoted to the convergence analysis of the data assimilation algorithm.

We introduce a continuous data assimilation (downscaling) algorithm for the two-dimensional Navier–Stokes equations employing coarse mesh measurements of only one component of the velocity field. This algorithm can be implemented with a variety of finitely many observables: low Fourier modes, nodal values, finite volume averages, or finite elements. We provide conditions on the spatial resolution of the observed data, under the assumption that the observed data is free of noise, which are sufficient to show that the solution of the algorithm approaches, at an exponential rate asymptotically in time, to the unique exact unknown reference solution, of the 2D Navier–Stokes equations, associated with the observed (finite dimensional projection of) velocity.