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This article is devoted to the energy critical hyperbolic Yang-Mills equation in the (4 + 1)-dimensional Minkowski space, which is considered by the authors in a sequence of four papers. The final outcome of these papers is twofold: (i) the Threshold Theorem, which asserts that global well-posedness and scattering hold for all topologically trivial initial data with energy below twice the ground state energy; and (ii) the Dichotomy Theorem, which for larger data in arbitrary topological classes provides a choice of two outcomes, either a global scattering solution or a soliton bubbling off. In the last case, the bubbling-off phenomena can happen in one of two ways: (a) in finite time, triggering a finite time blowup; or (b) in infinite time. Our goal here is to first describe the equation and the results, and then to provide an overview of the flow of ideas within their proofs in the above-mentioned four papers.

This article is devoted to the energy critical hyperbolic Yang-Mills equation in the (4 + 1)-dimensional Minkowski space, which is considered by the authors in a sequence of four papers. The final outcome of these papers is twofold: (i) the Threshold Theorem, which asserts that global well-posedness and scattering hold for all topologically trivial initial data with energy below twice the ground state energy; and (ii) the Dichotomy Theorem, which for larger data in arbitrary topological classes provides a choice of two outcomes, either a global scattering solution or a soliton bubbling off. In the last case, the bubbling-off phenomena can happen in one of two ways: (a) in finite time, triggering a finite time blowup; or (b) in infinite time. Our goal here is to first describe the equation and the results, and then to provide an overview of the flow of ideas within their proofs in the above-mentioned four papers.

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.

This article constitutes the final and main part of a three-paper sequence (Ann PDE, 2016. doi:10.1007/s40818-016-0006-4; Oh and Tataru, 2015. arXiv:1503.01561), whose goal is to prove global well-posedness and scattering of the energy critical Maxwell-Klein-Gordon equation (MKG) on R1 + 4 for arbitrary finite energy initial data. Using the successively stronger continuation/scattering criteria established in the previous two papers (Ann PDE, 2016. doi:10.1007/s40818-016-0006-4; Oh and Tataru, 2015. arXiv:1503.01561), we carry out a blow-up analysis and deduce that the failure of global well-posedness and scattering implies the existence of a nontrivial stationary or self-similar solution to MKG. Then, by establishing that such solutions do not exist, we complete the proof.

This article constitutes the final and main part of a three-paper sequence (Ann PDE, 2016. doi:10.1007/s40818-016-0006-4; Oh and Tataru, 2015. arXiv:1503.01561), whose goal is to prove global well-posedness and scattering of the energy critical Maxwell-Klein-Gordon equation (MKG) on R1 + 4 for arbitrary finite energy initial data. Using the successively stronger continuation/scattering criteria established in the previous two papers (Ann PDE, 2016. doi:10.1007/s40818-016-0006-4; Oh and Tataru, 2015. arXiv:1503.01561), we carry out a blow-up analysis and deduce that the failure of global well-posedness and scattering implies the existence of a nontrivial stationary or self-similar solution to MKG. Then, by establishing that such solutions do not exist, we complete the proof.

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.

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.

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.

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.

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 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.

In this paper we prove the local well-posedness and global well-posednesswith small initial data of the strong solution to the reduced $3D$ primitivegeostrophic adjustment model with weak dissipation. The term reduced modelstems from the fact that the relevant physical quantities depends only on twospatial variables. The additional weak dissipation helps us overcome theill-posedness of original model. We also prove the global well-posedness of thestrong solution to the Voigt $\alpha$-regularization of this model, andestablish the convergence of the strong solution of the Voigt$\alpha$-regularized model to the corresponding solution of original model.Furthermore, we derive a criterion for finite-time blow-up of reduced $3D$primitive geostrophic adjustment model with weak dissipation based on Voigt$\alpha$-regularization.

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.

In this work we study the global solvability of the primitive equations forthe 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 ascloud water and rain water. This moisture model contains closures for the phasechanges condensation and evaporation, as well as the processes ofautoconversion of cloud water into rainwater and the collection of cloud waterby the falling rain droplets. It has been used by Klein and Majda in \cite{KM}and corresponds to a basic form of the bulk microphysics closure in the spiritof Kessler \cite{Ke} and Grabowski and Smolarkiewicz \cite{GS}. The moisturebalances are strongly coupled to the thermodynamic equation via the latent heatassociated to the phase changes. In \cite{HKLT} we assumed the velocity fieldto be given and proved rigorously the global existence and uniqueness ofuniformly bounded solutions of the moisture balances coupled to thethermodynamic equation. In this paper we present the solvability of a fullmoist atmospheric flow model, where the moisture model is coupled to theprimitive equations of atmospherical dynamics governing the velocity field. Forthe derivation of a priori estimates for the velocity field we thereby use theideas of Cao and Titi \cite{CT}, who succeeded in proving the globalsolvability of the primitive equations.

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.

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

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 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 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.

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.

In this paper we show that the global attractor of the 1D damped, driven, nonlinear Schrödinger equation (NLS) is embedded in the long-time dynamics of a determining form. The determining form is an ordinary differential equation in a space of trajectories X=Cb1(R,PmH2) where Pm is the L2-projector onto the span of the first m Fourier modes. There is a one-to-one identification with the trajectories in the global attractor of the NLS and the steady states of the determining form. We also give an improved estimate for the number of the determining modes.

We analyse the performance of a data-assimilation algorithm based on a linear feedback control when used with observational data that contains measurement errors. Our model problem consists of dynamics governed by the two-dimensional incompressible Navier-Stokes equations, observational measurements given by finite volume elements or nodal points of the velocity field and measurement errors which are represented by stochastic noise. Under these assumptions, the data-assimilation algorithm consists of a system of stochastically forced Navier-Stokes equations. The main result of this paper provides explicit conditions on the observation density (resolution) which guarantee explicit asymptotic bounds, as the time tends to infinity, on the error between the approximate solution and the actual solutions which is corresponding to these measurements, in terms of the variance of the noise in the measurements. Specifically, such bounds are given for the limit supremum, as the time tends to infinity, of the expected value of the L2-norm and of the H1 Sobolev norm of the difference between the approximating solution and the actual solution. Moreover, results on the average time error in mean are stated.

We analyse the performance of a data-assimilation algorithm based on a linear feedback control when used with observational data that contains measurement errors. Our model problem consists of dynamics governed by the two-dimensional incompressible Navier-Stokes equations, observational measurements given by finite volume elements or nodal points of the velocity field and measurement errors which are represented by stochastic noise. Under these assumptions, the data-assimilation algorithm consists of a system of stochastically forced Navier-Stokes equations. The main result of this paper provides explicit conditions on the observation density (resolution) which guarantee explicit asymptotic bounds, as the time tends to infinity, on the error between the approximate solution and the actual solutions which is corresponding to these measurements, in terms of the variance of the noise in the measurements. Specifically, such bounds are given for the limit supremum, as the time tends to infinity, of the expected value of the L2-norm and of the H1 Sobolev norm of the difference between the approximating solution and the actual solution. Moreover, results on the average time error in mean are stated.

This paper is concerned with the cubic Szego{double acute} equationi ∂t u = Π (| u |2 u), defined on the L2 Hardy space on the one-dimensional torus T, where Π : L2 (T) → L+2 (T) is the Szego{double acute} projector onto the non-negative frequencies. For analytic initial data, it is shown that the solution remains spatial analytic for all time t ∈ (- ∞, ∞). In addition, we find a lower bound for the radius of analyticity of the solution. Our method involves energy-like estimates of the special Gevrey class of analytic functions based on the ℓ1 norm of Fourier transforms (the Wiener algebra). © 2013 Elsevier Masson SAS. All rights reserved.

In this paper, we consider the initial-boundary value problem of the 3D primitive equations for oceanic and atmospheric dynamics with only horizontal diffusion in the temperature equation. Global well-posedness of strong solutions are established with H2 initial data. © 2014 Elsevier Inc. All rights reserved.

In this paper we show that the global attractor of the 1D damped, driven, nonlinear Schrödinger equation (NLS) is embedded in the long-time dynamics of a determining form. The determining form is an ordinary differential equation in a space of trajectories X=Cb1(R,PmH2) where Pm is the L2-projector onto the span of the first m Fourier modes. There is a one-to-one identification with the trajectories in the global attractor of the NLS and the steady states of the determining form. We also give an improved estimate for the number of the determining modes.

In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study. We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow. We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as "freezing" or "pinning") and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation. © 2014 The Author(s).

In this paper, we consider the initial-boundary value problem of the 3D primitive equations for oceanic and atmospheric dynamics with only horizontal diffusion in the temperature equation. Global well-posedness of strong solutions are established with H2 initial data. © 2014 Elsevier Inc. All rights reserved.

In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study. We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow. We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as "freezing" or "pinning") and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation. © 2014 The Author(s).

This paper is concerned with the cubic Szego{double acute} equationi ∂t u = Π (| u |2 u), defined on the L2 Hardy space on the one-dimensional torus T, where Π : L2 (T) → L+2 (T) is the Szego{double acute} projector onto the non-negative frequencies. For analytic initial data, it is shown that the solution remains spatial analytic for all time t ∈ (- ∞, ∞). In addition, we find a lower bound for the radius of analyticity of the solution. Our method involves energy-like estimates of the special Gevrey class of analytic functions based on the ℓ1 norm of Fourier transforms (the Wiener algebra). © 2013 Elsevier Masson SAS. All rights reserved.