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We consider a non-linear Bayesian data assimilation model for the periodic two-dimensional Navier-Stokes equations with initial condition modelled by a Gaussian process prior. We show that if the system is updated with sufficiently many discrete noisy measurements of the velocity field, then the posterior distribution eventually concentrates near the ground truth solution of the time evolution equation, and in particular that the initial condition is recovered consistently by the posterior mean vector field. We further show that the convergence rate can in general not be faster than inverse logarithmic in sample size, but describe specific conditions on the initial conditions when faster rates are possible. In the proofs we provide an explicit quantitative estimate for backward uniqueness of solutions of the two-dimensional Navier-Stokes equations., 22 pages

We consider the transport equation of a passive scalar $f(x,t)\in\mathbb{R}$ along a divergence-free vector field $u(x,t)\in\mathbb{R}^2$, given by $\frac{\partial f}{\partial t} + \nabla\cdot (u f) = 0$; and the associated advection-diffusion equation of $f$ along $u$, with positive viscosity/diffusivity parameter $\nu>0$, given by $\frac{\partial f}{\partial t} + \nabla\cdot (u f) -\nu\Delta f = 0$. We demonstrate failure of the vanishing viscosity limit of advection-diffusion to select unique solutions, or to select entropy-admissible solutions, to transport along $u$. First, we construct a bounded divergence-free vector field $u$ which has, for each (non-constant) initial datum, two weak solutions to the transport equation. Moreover, we show that both these solutions are renormalised weak solutions, and are obtained as strong limits of a subsequence of the vanishing viscosity limit of the corresponding advection-diffusion equation. Second, we construct a second bounded divergence-free vector field $u$ admitting, for any initial datum, a weak solution to the transport equation which is perfectly mixed to its spatial average, and after a delay, unmixes to its initial state. Moreover, we show that this entropy-inadmissible unmixing is the unique weak vanishing viscosity limit of the corresponding advection-diffusion equation.

We develop a convex integration scheme for constructing nonunique weak solutions to the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics) in both two and three dimensions. We also develop such a scheme for the construction of nonunique weak solutions to the three-dimensional viscous primitive equations, as well as the two-dimensional Prandtl equations. While in [D.W. Boutros, S. Markfelder and E.S. Titi, arXiv:2208.08334 (2022)] the classical notion of weak solution to the hydrostatic Euler equations was generalised, we introduce here a further generalisation. For such generalised weak solutions we show the existence and nonuniqueness for a large class of initial data. Moreover, we construct infinitely many examples of generalised weak solutions which do not conserve energy. The barotropic and baroclinic modes of solutions to the hydrostatic Euler equations (which are the average and the fluctuation of the horizontal velocity in the $z$-coordinate, respectively) that are constructed have different regularities., 73 pages

This paper considers the asymptotic limit of small aspect ratio between vertical and horizontal spatial scales for viscous isothermal compressible flows. In particular, it is observed that fast vertical acoustic waves arise and induce an averaging mechanism of the density in the vertical variable, which at the limit leads to the hydrostatic approximation of compressible flows, i.e., the compressible primitive equations of atmospheric dynamics. We justify the hydrostatic approximation for general as well as ``well-prepared'' initial data. The initial data is called well-prepared when it is close to the hydrostatic balance in a strong topology. Moreover, the convergence rate is calculated in the well-prepared initial data case in terms of the aspect ratio, as the latter goes to zero.

We consider the three-dimensional incompressible Euler equations on a bounded domain $\Omega$ with $C^3$ boundary. We prove that if the velocity field $u \in C^{0,\alpha} (\Omega)$ with $\alpha > 0$ (where we are omitting the time dependence), it follows that the pressure $p \in C^{0, \alpha} (\Omega)$. In order to prove this result we use a local parametrisation of the boundary and a very weak formulation of the boundary condition for the pressure, as was introduced in [C. Bardos and E.S. Titi, Philos. Trans. Royal Soc. A, 380 (2022), 20210073]. Moreover, we provide an example illustrating the necessity of this new very weak formulation of the boundary condition for the pressure. This result is of importance for the proof of the first half of the Onsager Conjecture, the sufficient conditions for energy conservation of weak solutions to the three-dimensional incompressible Euler equations in bounded domains., Comment: 35 pages

We study a nonlinear-nudging modification of the Azouani-Olson-Titi continuous data assimilation (downscaling) algorithm for the 2D incompressible Navier-Stokes equations. We give a rigorous proof that the nonlinear-nudging system is globally well-posed, and moreover that its solutions converge to the true solution exponentially fast in time. Furthermore, we also prove that, once the error has decreased below a certain order one threshold, the convergence becomes double-exponentially fast in time, up until a precision determined by the sparsity of the observed data. In addition, we demonstrate the applicability of the analytical and sharpness of the results computationally., 33 pages, 7 figures

We prove global well-posedness of the ocean primitive equations coupled to advection-diffusion equations of the oceanic tracers temperature and salinity that are supplemented by the eddy parametrization model due to Gent-McWilliams and Redi. This parametrization forms a milestone in global ocean modelling and constitutes a central part of any general ocean circulation model computation. The eddy parametrization adds a secondary transport velocity to the tracer equation and renders the original Laplacian operators in the advection-diffusion equations nonlinear, with a diffusion matrix that depends via the equation of state in a nonlinear fashion on both tracers simultaneously. The eddy parametrization of Gent-McWilliams-Redi augments the complexity of the mathematical analysis of the whole system which we present here. We show first that weak solutions exist globally in time, provided the parametrization uses a regularized density. Then we prove by a detailed analysis of the eddy operators the global well-posedness. Our results apply also to the ``small-slope approximation'' that is commonly used in global ocean simulations., Comment: 31 pages

We consider the three-dimensional incompressible Euler equations on a bounded domain $Ω$ with $C^3$ boundary. We prove that if the velocity field $u \in C^{0,α} (Ω)$ with $α> 0$ (where we are omitting the time dependence), it follows that the pressure $p \in C^{0, α} (Ω)$. In order to prove this result we use a local parametrisation of the boundary and a very weak formulation of the boundary condition for the pressure, as was introduced in [C. Bardos and E.S. Titi, Philos. Trans. Royal Soc. A, 380 (2022), 20210073]. Moreover, we provide an example illustrating the necessity of this new very weak formulation of the boundary condition for the pressure. This result is of importance for the proof of the first half of the Onsager Conjecture, the sufficient conditions for energy conservation of weak solutions to the three-dimensional incompressible Euler equations in bounded domains., 35 pages

This paper is devoted to investigating the rotating Boussinesq equations of inviscid, incompressible flows with both fast Rossby waves and fast internal gravity waves. The main objective is to establish a rigorous derivation and justification of a new generalized quasi-geostrophic approximation in a channel domain with no normal flow at the upper and lower solid boundaries, taking into account the resonance terms due to the fast and slow waves interactions. Under these circumstances, We are able to obtain uniform estimates and compactness without the requirement of either well-prepared initial data (as in [10]) or domain with no boundary (as in [17]). In particular, the nonlinear resonances and the new limit system, which takes into account the fast waves correction to the slow waves dynamics, are also identified without introducing Fourier series expansion. The key ingredient includes the introduction of (full) generalized potential vorticity.

Onsager's conjecture, which relates the conservation of energy to the regularity of weak solutions of the Euler equations, was completely resolved in recent years. In this work, we pursue an analogue of Onsager's conjecture in the context of the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics). In this case the relevant conserved quantity is the horizontal kinetic energy. We first consider the standard notion of weak solution which is commonly used in the literature. We show that if the horizontal velocity $(u,v)$ is sufficiently regular then the horizontal kinetic energy is conserved. Interestingly, the spatial H\"older regularity exponent which is sufficient for energy conservation in the context of the hydrostatic Euler equations is $\frac{1}{2}$ and hence larger than the corresponding regularity exponent for the Euler equations (which is $\frac{1}{3}$). This is due to the anisotropic regularity of the velocity field: Unlike the Euler equations, in the case of the hydrostatic Euler equations the vertical velocity $w$ is one degree spatially less regular with respect to the horizontal variables, compared to the horizontal velocity $(u,v)$. Since the standard notion of weak solution is not able to deal with this anisotropy properly, we introduce two new notions of weak solutions for which the vertical part of the nonlinearity is interpreted as a paraproduct. We finally prove several sufficient conditions for such weak solutions to conserve energy., Comment: 43 pages

Funder: Deutsche Forschungsgemeinschaft, Funder: Simons Foundation, We study the effect of the fast rotation and vertical viscosity on the lifespan of solutions to the three-dimensional primitive equations (also known as the hydrostatic Navier-Stokes equations) with impermeable and stress-free boundary conditions. Firstly, for a short time interval, independent of the rate of rotation |Ω| , we establish the local well-posedness of solutions with initial data that is analytic in the horizontal variables and only L2 in the vertical variable. Moreover, it is shown that the solutions immediately become analytic in all the variables with increasing-in-time (at least linearly) radius of analyticity in the vertical variable for as long as the solutions exist. On the other hand, the radius of analyticity in the horizontal variables might decrease with time, but as long as it remains positive the solution exists. Secondly, with fast rotation, i.e., large |Ω| , we show that the existence time of the solution can be prolonged, with "well-prepared" initial data. Finally, in the case of two spatial dimensions with Ω=0 , we establish the global well-posedness provided that the initial data is small enough. The smallness condition on the initial data depends on the vertical viscosity and the initial radius of analyticity in the horizontal variables.

In this work we study the global solvability of moisture dynamics with phase changes for warm clouds. We thereby in comparison to previous studies [Hittmeir-Klein-Li-Titi (2017)] take into account the different gas constants for dry air and water vapor as well as the different heat capacities for dry air, water vapor and liquid water, which leads to a much stronger coupling of the moisture balances and the thermodynamic equation. This refined thermodynamic setting has been demonstrated to be essential e.g. in the case of deep convective cloud columns in [Hittmeir-Klein (2017)]. The more complicated structure requires careful derivations of sufficient a priori estimates for proving global existence and uniqueness of solutions., arXiv admin note: text overlap with arXiv:1907.11199

A compressible, viscous and heat conducting fluid is confined between two parallel plates maintained at a constant temperature and subject to a strong stratification due to the gravitational force. We consider the asymptotic limit, where the Mach number and the Froude number are of the same order proportional to a small parameter. We show the limit problem can be identified with Majda's model of layered ``stack-of-pancake'' flow., Comment: arXiv admin note: text overlap with arXiv:2206.14041

The purpose of this note is to give a complete proof of a C^{0,α} regularity result for the pressure for weak solutions of the two-dimensional “incompressible Euler equations” when the fluid velocity enjoys the same type of regularity in a compact simply connected domain with C^2 boundary. To accomplish our result we realize that it is compulsory to introduce a new weak formulation for the boundary condition of the pressure which is consistent with, and equivalent to, that of classical solutions.

The purpose of this article is to give a complete proof of a [Formula: see text] regularity result for the pressure for weak solutions of the two-dimensional 'incompressible Euler equations' when the fluid velocity enjoys the same type of regularity in a compact simply connected domain with [Formula: see text] boundary. To accomplish our result, we realize that it is compulsory to introduce a new weak formulation for the boundary condition of the pressure, which is consistent with, and equivalent to, that of classical solutions. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

In this paper, we provide rigorous justification of the hydrostatic approximation and the derivation of primitive equations as the small aspect ratio limit of the incompressible three-dimensional Navier-Stokes equations in the anisotropic horizontal viscosity regime. Setting $\varepsilon >0$ to be the small aspect ratio of the vertical to the horizontal scales of the domain, we investigate the case when the horizontal and vertical viscosities in the incompressible three-dimensional Navier-Stokes equations are of orders $O(1)$ and $O(\varepsilon^\alpha)$, respectively, with $\alpha>2$, for which the limiting 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-dimensional Navier-Stokes equations converge strongly, in any finite interval of time, to the corresponding solutions of the anisotropic primitive equations with only horizontal viscosities, as $\varepsilon$ tends to zero, and that the convergence 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 the small aspect ratio limit of the Navier-Stokes equations: Rigorous justification of the hydrostatic approximation}, J. Math. Pures Appl., \textbf{124} \rm(2019), 30--58], where the limiting system is the primitive equations with full 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 a density-driven flow that joins faster flow components to create a multi-scale, 3D flow. Potential temperature and salinity are active tracers that determine the ocean's potential density: their distribution strongly affects the density-driven component while the overall flow affects their distribution. We present a robust framework that allows one to study the effects of a general 3D flow on a density-driven velocity component, by constructing a modular observation-based 3D model of intermediate complexity. The model contains an incompressible velocity that couples two advection-diffusion equations, for temperature and salinity. Instead of solving the Navier-Stokes equations for the velocity, we consider a flow composed of several temporally separated, spatially predetermined modes. One of these modes models the density-driven flow: its spatial form describes the density-driven flow structure and its strength is determined dynamically by average density differences. The other modes are completely predetermined, consisting of any incompressible, possibly unsteady, 3D flow, e.g. as determined by kinematic models, observations, or simulations. The model is a non-linear, weakly coupled system of two non-local PDEs. We prove its well-posedness in the sense of Hadamard, and obtain rigorous bounds regarding analytical solutions. The model's relevance to oceanic systems is demonstrated by tuning the model to mimic the North Atlantic ocean's dynamics. In one limit the model recovers a simplified oceanic box model and in another limit a kinematic model of oceanic chaotic advection, suggesting it can be utilized to study spatially dependent feedback processes in the ocean., Comment: 15 pages text, 20 pages appendix, 5 figures

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

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

An intrinsic property of almost any physical measuring device is that it makes observations which are slightly blurred in time. We 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, we 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 boundedness properties for the relevant interpolant observation operators and a non-local Gronwall inequality., 44 pages

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

The Rayleigh-B\'enard 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 $L^2$ 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.

We construct a determining form for the 2D Rayleigh-B��nard (RB) system in a strip with solid horizontal boundaries, in the cases of no-slip and stress-free boundary conditions. The determining form is an ODE in a Banach space of trajectories whose steady states comprise the long-time dynamics of the RB system. In fact, solutions on the global attractor of the RB system can be further identified through the zeros of a scalar equation to which the ODE reduces for each initial trajectory. The twist in this work is that the trajectories are for the velocity field only, which in turn determines the corresponding trajectories of the temperature.

The Rayleigh-B��nard 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 $L^2$ 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.

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$$(-\Delta )^{\theta }$$(-Δ)θ, whenever the exponent$$\theta $$θis less than Lions’ exponent 5/4, i.e., when$$\theta < 5/4$$θ<5/4.

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 $\alpha>0$. 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 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., Comment: 7 pages, 8 figures

We consider fully discrete numerical schemes for a downscaling data assimilation algorithm aimed at approximating the velocity field of the 2D Navier-Stokes equations corresponding to given coarse mesh observational measurements. The time discretization is done by considering semi- and fully-implicit Euler schemes, and the spatial discretization is based on a spectral Galerkin method. The two fully discrete algorithms are shown to be unconditionally stable, with respect to the size of the time step, number of time steps and the number of Galerkin modes. Moreover, explicit, uniform in time error estimates between the fully discrete solution and the reference solution corresponding to the observational coarse mesh measurements are obtained, in both the $L^2$ and $H^1$ norms. Notably, the two-dimensional Navier-Stokes equations, subject to the no-slip Dirichlet or periodic boundary conditions, are used in this work as a paradigm. The complete analysis that is presented here can be extended to other two- and three-dimensional dissipative systems under the assumption of global existence and uniqueness.

We introduce a continuous (downscaling) data assimilation algorithm for the 2D B\'enard 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 B\'enard 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.

The evolution of a determining form for the 2D Navier–Stokes equations (NSE) which is an ODE on a space of trajectories is completely described. It is proved that at every stage of its evolution, the solution is a convex combination of the initial trajectory and a chosen, fixed steady state, with a dynamical convexity parameter θ, which will be called the characteristic determining parameter. That is, we show a separation of variables formula for the solution of the determining form. Moreover, for a given initial trajectory, the dynamics of the infinite-dimensional determining form are equivalent to those of the characteristic determining parameter θ which is governed by a one-dimensional ODE. This one-dimensional ODE is used to show that if the solution to the determining form converges to the fixed state it does so no faster than O(τ- 1 / 2) , otherwise it converges to a projection of some other trajectory in the global attractor of the NSE, but no faster than O(τ- 1) , as τ→ ∞, where τ is the evolutionary variable in determining form. The one-dimensional ODE is also exploited in computations which suggest that the one-sided convergence rate estimates are in fact achieved. The ODE is then modified to accelerate the convergence to an exponential rate. It is shown that the zeros of the scalar function that governs the dynamics of θ, which are called characteristic determining values, identify in a unique fashion the trajectories in the global attractor of the 2D NSE.

We study the persistence of the Gevrey class regularity of solutions to nonlinear wave equations with real analytic nonlinearity. Specifically, it is proven that the solution remains in a Gevrey class, with respect to some of its spatial variables, during its whole life-span, provided the initial data is from the same Gevrey class with respect to these spatial variables. In addition, for the special Gevrey class of analytic functions, we find a lower bound for the radius of the spatial analyticity of the solution that might shrink either algebraically or exponentially, in time, depending on the structure of the nonlinearity. The standard $L^2$ theory for the Gevrey class regularity is employed; we also employ energy-like methods for a generalized version of Gevrey classes based on the $\ell^1$ norm of Fourier transforms (Wiener algebra). After careful comparisons, we observe an indication that the $\ell^1$ approach provides a better lower bound for the radius of analyticity of the solutions than the $L^2$ approach. We present our results in the case of period boundary conditions, however, by employing exactly the same tools and proofs one can obtain similar results for the nonlinear wave equations and the nonlinear Schr\"odinger equation, with real analytic nonlinearity, in certain domains and manifolds without physical boundaries, such as the whole space $\mathbb{R}^n$, or on the sphere $\mathbb{S}^{n-1}$.

It is well known, by now, that the three-dimensional non-viscous planetary geostrophic model, with vertical hydrostatic balance and horizontal Rayleigh friction, 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 literal 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. Another remedy for this problem was suggested by R. Salmon by introducing an additional Rayleigh-like friction 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.

The global existence of weak solutions for the three-dimensional axisymmetric Euler-$\alpha$ (also known as Lagrangian-averaged Euler-$\alpha$) equations, without swirl, is established, whenever the initial unfiltered velocity $v_0$ satisfies $\frac{\nabla \times v_0}{r}$ is a finite Randon measure with compact support. Furthermore, the global existence and uniqueness, is also established in this case provided $\frac{\nabla \times v_0}{r} \in L^p_c(\mathbb{R}^3)$ with $p>{3/2}$. It is worth mention that no such results are known to be available, so far, for the three-dimensional Euler equations of ideal incompressible flows.

We adapt a previously introduced continuous in time data assimilation (downscaling) algorithm for the 2D Navier-Stokes equations to the more realistic case when the measurements are obtained discretely in time and may be contaminated by systematic errors. Our algorithm is designed to work with a general class of observables, such as low Fourier modes and local spatial averages over finite volume elements. Under suitable conditions on the relaxation (nudging) parameter, the spatial mesh resolution and the time step between successive measurements, we obtain an asymptotic in time estimate of the difference between the approximating solution and the unknown reference solution corresponding to the measurements, in an appropriate norm, which shows exponential convergence up to a term which depends on the size of the errors. A stationary statistical analysis of our discrete data assimilation algorithm is also provided.

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.

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 $v_0\in H^2(\Omega)$ and the initial temperature $T_0\in H^1(\Omega)\cap L^\infty(\Omega)$ with $\nabla_HT_0\in L^q(\Omega)$, for some $q\in(2,\infty)$. Moreover, the strong solutions enjoy correspondingly more regularities if the initial temperature belongs to $H^2(\Omega)$. 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., $\eta, \theta, \varphi,$ and $\psi$ 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 $\partial_zv$. To overcome the difficulties caused by the absence of the horizontal diffusivity, which leads to the requirement of some $L^1_t(W^{1,\infty}_\textbf{x})$-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 Br\'ezis-Gallouet-Wainger and Beale-Kato-Majda types, respectively. Specifically, a logarithmic Sobolev inequality of the limiting type, introduced in our previous work [12], is used, and a new logarithmic type Gronwall inequality is exploited.

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., Comment: 34 pages

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., arXiv admin note: text overlap with arXiv:1507.05231

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.

Motivated by the presence of a finite number of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages for dissipative dynamical systems, we present here a continuous data assimilation algorithm for three-dimensional viscous hydrodynamic models. However, to validate the convergence of this algorithm our proofs require the existence of uniform global bounds on the gradients of the solutions of the underlying system in terms of certain combinations of the physical parameters (such as kinematic viscosity, the size of the domain and the forcing term). Therefore our proofs cannot be applied to the three-dimensional Navier-Stokes equations; instead we demonstrate the implementation of this algorithm, for instance, in the context of the three-dimensional Navier-Stokes-α equations. This algorithm consists of introducing a nudging process through a general type of approximation interpolation operator (which is constructed from observational measurements) that synchronizes the large spatial scales of the approximate solutions with those of unknown solutions of the Navier-Stokes-α equations corresponding to these measurements. Our main result provides conditions on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, which is obtained from this collected data, converges to the unknown reference solution over time. These conditions are given in terms of the physical parameters.

The paper studies the issue of stability of solutions to the Navier-Stokes and damped Euler systems in periodic boxes. We show that under action of fast oscillating-in- time external forces all two dimensional regular solutions converge to a time periodic flow. Unexpectedly, effects of stabilization can be also obtained for systems with stationary forces with large total momentum (average of the velocity). Thanks to the Galilean transformation and space boundary conditions, the stationary force changes into one with time oscillations. In the three dimensional case we show an analogical result for weak solutions to the Navier- Stokes equations.

© 2016 World Scientific Publishing Company 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.

We apply the Postprocessing Galerkin method to a recently introduced continuous data assimilation (downscaling) algorithm for obtaining a numerical approximation of the solution of the two-dimensional Navier-Stokes equations corresponding to given measurements from a coarse spatial mesh. Under suitable conditions on the relaxation (nudging) parameter, the resolution of the coarse spatial mesh and the resolution of the numerical scheme, we obtain uniform in time estimates for the error between the numerical approximation given by the Postprocessing Galerkin method and the reference solution corresponding to the measurements. 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 our results apply equally to other evolution equations, such as the Boussinesq system of Benard convection and other oceanic and atmospheric circulation models.

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.

We propose a new blow-up criterion for the 3D Euler equations of incompressible fluid flows, based on the 3D Euler-Voigt inviscid regularization. This criterion is similar in character to a criterion proposed in a previous work by the authors, but it is stronger, and better adapted for computational tests. The 3D Euler-Voigt equations enjoy global well-posedness, and moreover are more tractable to simulate than the 3D Euler equations. A major advantage of these new criteria is that one only needs to simulate the 3D Euler-Voigt, and not the 3D Euler equations, to test the blow-up criteria, for the 3D Euler equations, computationally.

We prove that every solution of a KdV-Burgers-Sivashinsky type equation blows up in the energy space, backward in time, provided the solution does not belong to the global attractor. This is a phenomenon contrast to the backward behavior of the periodic 2D Navier-Stokes equations studied by Constantin-Foias-Kukavica-Majda [18], but analogous to the backward behavior of the Kuramoto-Sivashinsky equation discovered by Kukavica-Malcok [50]. Also we study the backward behavior of solutions to the damped driven nonlinear Schrodinger equation, the complex Ginzburg-Landau equation, and the hyperviscous Navier-Stokes equations. In addition, we provide some physical interpretation of various backward behaviors of several perturbations of the KdV equation by studying explicit cnoidal wave solutions. Furthermore, we discuss the connection between the backward behavior and the energy spectra of the solutions. The study of backward behavior of dissipative evolution equations is motivated by the investigation of the Bardos-Tartar conjecture stated in [5]., 34 pages

In this paper we propose a continuous data assimilation (downscaling) algorithm for the B\'enard convection in porous media using only coarse 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., Comment: arXiv admin note: text overlap with arXiv:1504.05978

The second-grade fluid equations are a model for viscoelastic fluids, with two parameters: α >0, corresponding to the elastic response, and $${\nu > 0}$$ν>0, corresponding to viscosity. Formally setting these parameters to 0 reduces the equations to the incompressible Euler equations of ideal fluid flow. In this article we study the limits $${\alpha, \nu \to 0}$$α,ν→0 of solutions of the second-grade fluid system, in a smooth, bounded, two-dimensional domain with no-slip boundary conditions. This class of problems interpolates between the Euler-α model ($${\nu = 0}$$ν=0), for which the authors recently proved convergence to the solution of the incompressible Euler equations, and the Navier-Stokes case (α =0), for which the vanishing viscosity limit is an important open problem. We prove three results. First, we establish convergence of the solutions of the second-grade model to those of the Euler equations provided $${\nu = \mathcal{O}(\alpha^2)}$$ν=O(α2), as α → 0, extending the main result in (Lopes Filho etal., Physica D 292(293):51–61, 2015). Second, we prove equivalence between convergence (of the second-grade fluid equations to the Euler equations) and vanishing of the energy dissipation in a suitably thin region near the boundary, in the asymptotic regime $${\nu = \mathcal{O}(\alpha^{6/5})}$$ν=O(α6/5), $${\nu/\alpha^{2} \to \infty}$$ν/α2→∞ as α → 0. This amounts to a convergence criterion similar to the well-known Kato criterion for the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations. Finally, we obtain an extension of Kato’s classical criterion to the second-grade fluid model, valid if $${\alpha = \mathcal{O}(\nu^{3/2})}$$α=O(ν3/2), as $${\nu \to 0}$$ν→0. The proof of all these results relies on energy estimates and boundary correctors, following the original idea by Kato.

© 2015 Elsevier B.V. An algorithm for continuous data assimilation for the two-dimensional Bénard convection problem is introduced and analyzed. It is inspired by the data assimilation algorithm developed for the Navier-Stokes equations, which allows for the implementation of variety of observables: low Fourier modes, nodal values, finite volume averages, and finite elements. The novelty here is that the observed data is obtained for the velocity field alone; i.e. no temperature measurements are needed for this algorithm. 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, the unique exact unknown solution of the Bénard convection problem associated with the observed (finite dimensional projection of) velocity.

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.