Maëlle Nodet
We consider the linear Primitive Equations of the ocean in the three dimensional space, with horizontal periodic and vertical Dirichlet boundary conditions. Thanks to Fourier transforms we are able to calculate explicitly the pressure term. We then state existence, unicity and regularity results for the linear time-depending Primitive Equations, with low-regularity right-hand side.
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In this paper we prove the global existence and uniqueness (regularity) of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere dynamics.
In this article, we show that the continuous data assimilation algorithm is valid for the 3D primitive equations of the ocean. Namely, the $L^2$ norm of the assimilated solution converge to that of the reference solution at an exponential rate in time. We also prove the global existence of strong solution to the assimilated system.
We show that in the limit of small Rossby number $\eps$, the primitive equations of the ocean (OPEs) can be approximated by ``higher-order quasi-geostrophic equations'' up to an exponential accuracy in $\eps$. This approximation assumes well-prepared initial data and is valid for a timescale of order one (independent of $\eps$). Our construction uses Gevrey regularity of the OPEs and a classical method to bound errors in higher-order perturbation theory.
Stochastic interpretations of the oceanic primitive equations with relaxed hydrostatic assumptions
本文研究了在准静力层极限下，基于低通滤波的随机解释的LU原始方程的适定性，并探讨了涡旋（超）粘性模型在描述静力平衡与非静力过程之间现象中的作用。
This paper is concerned with the long-time behavior of solutions for the three dimensional primitive equations of large-scale ocean and atmosphere dynamics in an unbounded domain. Since the Sobolev embedding is no longer compact in an unbounded domain, we cannot obtain the asymptotical compactness of the semigroup generated by problem (2.4)-(2.6) by the Sobolev compactness embedding theorem even if we obtain the existence of an absorbing set in more regular phase space. To overcome this difficulty, we first prove the asymptotical compactness in a weak phase space of the semigroup generated by …
From anisotropic Navier-Stokes equations to primitive equations for the ocean and atmosphere
本文研究了海洋与大气原始方程在两类周期边界区域上的适定性，推导了全局解在合适 Besov 空间中的存在性，并严格论证了从重缩放各向异性 Navier-Stokes 方程到原始方程的极限过渡。
The primitive equations for the ocean and atmosphere in anisotropic spaces
本文研究了海洋与大气原始方程在特定周期边界区域上的适定性，在合适的各向异性Besov空间中证明了小初始数据的全局存在与唯一性，并证明了从各向异性Navier-Stokes方程到该系统的奇异极限。
Global Well-Posedness of the Primitive Equations of Large-Scale Ocean Dynamics with the Gent-McWilliams-Redi Eddy Parametrization Model
本研究针对带Gent-McWilliams-Redi涡旋参数化的海洋原始方程系统，通过正则化密度假设与涡旋算子分析，证明了其全局弱解存在性与全局适定性，结果可推广至小斜率近似场景。
The Primitive Equations are a basic model in the study of large scale Oceanic and Atmospheric dynamics. These systems form the analytical core of the most advanced General Circulation Models. For this reason and due to their challenging nonlinear and anisotropic structure the Primitive Equations have recently received considerable attention from the mathematical community. In view of the complex multi-scale nature of the earth's climate system, many uncertainties appear that should be accounted for in the basic dynamical models of atmospheric and oceanic processes. In the climate community sto…
In an earlier work we have shown the global (for all initial data and all time) well-posedness of strong solutions to the three-dimensional viscous primitive equations of large scale oceanic and atmospheric dynamics. In this paper we show that for certain class of initial data the corresponding smooth solutions of the inviscid (non-viscous) primitive equations blow up in finite time. Specifically, we consider the three-dimensional inviscid primitive equations in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-no…
We establish, for smooth enough initial data, the global well-posedness (existence, uniqueness and continuous dependence on initial data) of solutions, for an inviscid three-dimensional {\it slow limiting ocean dynamics} model. This model was derived as a strong rotation limit of the rotating and stratified Boussinesg equations with periodic boundary conditions. To establish our results we utilize the tools developed for investigating the two-dimensional incompressible Euler equations and linear transport equations. Using a weaker formulation of the model we also show the global existence and …
We establish the continuity of the Markovian semigroup associated with strong solutions of the stochastic 3D Primitive Equations, and prove the existence of an invariant measure. The proof is based on new moment bounds for strong solutions. The invariant measure is supported on strong solutions, but is furthermore shown to have higher regularity properties.
The primitive equations with rough transport noise: Global well-posedness and regularity
本文利用 $q>2$ 的 $L^q$-技术处理粗糙传输噪声，为原始方程识别出新的临界各向异性 Besov 空间，从而建立了其整体适定性与瞬时正则化结果。
In this paper, we show mathematical justification of the data assimilation of nudging type in $L^p$-$L^q$ maximal regularity settings. We prove that the approximate solution of the primitive equations by data assimilation converges to the true solution with exponential order on the Besov space $B^{2/q}_{q,p}(Ω)$ in the periodic layer domain $Ω= \mathbb{T}^2 \times (-h, 0)$.
We consider the 3D viscous primitive equations with periodic boundary conditions. These equations arise in the study of ocean dynamics and generate a dynamical system in a Sobolev H^1 type space. Our main result establishes the so-called squeezing property in the Ladyzhenskaya form for this system. As a consequence of this property we prove (i) the finiteness of the fractal dimension of the corresponding global attractor, (ii) the existence of finite number of determining modes, and (iii) ergodicity of a related random kick model. All these results provide a new information concerning long tim…
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 …
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 de…
A Liouville-type theorem for the 3D primitive equations
本文研究三维原始方程的稳态解，证明无论有无 Coriolis 旋转项或粘性，都不存在紧支集光滑解，并与三维 Euler 方程的相关结果形成对比。
The article is devoted to prove the existence and regularity of the solutions of the $3D$ inviscid Linearized Primitive Equations (LPEs) in a channel with lateral periodicity. This was assumed in a previous work \cite{HJT} which is concerned with the boundary layers generated by the corresponding viscous problem. Although the equations under investigation here are of hyperbolic type, the standard methods do not apply because of the specificity of the hyperbolic system. A set of \textit{non-local} boundary conditions for the inviscid LPEs has to be imposed at the top and bottom of the channel m…
We establish the existence and uniqueness of both local martingale and local pathwise solutions of an abstract nonlinear stochastic evolution system. The primary application of this abstract framework is to infer the local existence of strong, pathwise solutions to the 3D primitive equations of the oceans and atmosphere forced by a nonlinear multiplicative white noise. Instead of developing our results specifically for the 3D primitive equations we choose to develop them in a slightly abstract framework which covers many related forms of these equations (atmosphere, oceans, coupled atmosphere-…
As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and oceans, we study their time discretization by an implicit Euler scheme. From deterministic viewpoint the 3D Primitive Equations are studied with physically realistic boundary conditions. From probabilistic viewpoint we consider a wide class of nonlinear, state dependent, white noise forcings. The proof of convergence of the Euler scheme covers the equations for the oceans, atmosphere, coupled oceanic-atmospheric system and other geophysical equations. We obtain the existence of solutions …
Global well-Posedness and Interior Regularity of 2D Navier-Stokes Equations with Stochastic Boundary Conditions
本文研究带非齐次随机边界条件的二维 Navier-Stokes 方程，证明其整体适定性与内部正则性，并将时白、空间有色噪声解释为海气界面的应力—风力平衡机制。
First, a new sufficient condition for uniqueness of weak solutions is proved for the system of 2D viscous Primitive Equations. Second, global existence and uniqueness are established for several classes of weak solutions with partial initial regularity, including but not limited to those weak solutions with initial horizontal regularity, rather than vertical regularity. Our results and analyses for the problem with phys- ical boundary conditions can be extended to those with other typical boundary conditions. Most of the results were not available before, even for the periodic case.
Global Well-Posedness and Persistence of H2 Regularity for the Three-Dimensional Primitive Equations with Horizontal and Topography-Adapted Thermal Diffusion
该研究针对带全粘性和水平热扩散的三维原始方程，在平坦底圆柱及适配地形的变深度域中，建立了H¹初值的整体适定性，并证明了H²正则性的保持。
Energy equality for the compressible Primitive Equations with vacuum
本文针对含真空的可压缩原始方程弱解，给出各向同性与各向异性空间中保证能量等式成立的正则性充分条件，并利用CPE系统特殊结构得到区别于可压缩Navier-Stokes方程的新正则性结果。
In this paper, we consider the initial-boundary value problem of the viscous 3D primitive equations for oceanic and atmospheric dynamics with only vertical diffusion in the temperature equation. Local and global well-posedness of strong solutions are established for this system with $H^2$ initial data.
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 $H^2$ initial data.
In this work we consider a stochastic version of the Primitive Equations (PEs) of the ocean and the atmosphere and establish the existence and uniqueness of pathwise, strong solutions. The analysis employs novel techniques in contrast to previous works in order to handle a general class of nonlinear noise structures and to allow for physically relevant boundary conditions. The proof relies on Cauchy estimates, stopping time arguments and anisotropic estimates.
We discuss the discrete data assimilation problem for the 3D viscous primitive equations arising in the modeling of large scale phenomena in oceanic dynamics. Our main result states possibility of asymptotically reliable prognosis based on a discrete sequence of finite number of scalar observations. Our method is quite general and can be applied to a wide class of dissipative systems.
In this paper, we consider the initial boundary value problem in a cylindrical domain to the three dimensional primitive equations with full eddy viscosity in the momentum equations but with only horizontal eddy diffusivity in the temperature equation. Global well-posedness of $z$-weak solution is established for any such initial datum that itself and its vertical derivative belong to $L^2$. This not only extends the results in \cite{Cao5} from the spatially periodic case to general cylindrical domains but also weakens the regularity assumptions on the initial data which are required to be $H^…
The primitive equations of large-scale oceanic dynamics form a fundamental model in geophysical flows. It is well-known that the primitive equations can be formally derived by the hydrostatic approximation. On the other hand, the mathematically rigorous derivation of the primitive equations without coupling with the temperature is also known. In this paper, we generalize the above result from the mathematical point of view. More precisely, we prove that the scaled Boussinesq equations with rotation converge to the full primitive equations in a strong sense, globally in time, with the convergen…
The hydrostatic primitive equations (HPE) form the basis of most numerical weather, climate and global ocean circulation models. Analytical (not statistical) methods are used to find a scaling proportional to $(Nu\,Ra\,Re)^{1/4}$ for the range of horizontal spatial sizes in HPE solutions, which is much broader than currently achievable computationally. The range of scales for the HPE is determined from an analytical bound on the time-averaged enstrophy of the horizontal circulation. This bound allows the formation of very small spatial scales, whose existence would excite unphysically large li…
Global existence and large time behavior for primitive equations with free boundary
本文研究带自由边界的三维原始方程，证明了强解的全局存在唯一性，且解会以指数或代数速率长时间收敛到平衡态，为海洋大气大尺度运动模拟提供理论支撑。
We show that, under reasonably mild hypotheses, the solution of the forced--dissipative rotating primitive equations of the ocean loses most of its fast, inertia--gravity, component in the small Rossby number limit as $t\to\infty$. At leading order, the solution approaches what is known as "geostrophic balance" even under ageostrophic, slowly time-dependent forcing. Higher-order results can be obtained if one further assumes that the forcing is time-independent and sufficiently smooth. If the forcing lies in some Gevrey space, the solution will be exponentially close to a finite-dimensional "s…
Global well-posedness of the 3D primitive equations with horizontal viscosity and vertical diffusivity II: close to $H^1$ initial data
该研究针对仅含水平涡粘性与垂直扩散率的三维海洋大气原始方程，利用各向异性能量估计与极大正则性估计，证明了一类接近$H^1$的初始数据下强解的全局适定性。
Uniqueness of weak solutions to the primitive equations in some anisotropic spaces
本文研究了等温设定下海洋与大气的3D或2D原始方程，通过采用基于齐次环形Besov空间的各向异性方法，建立了弱解在满足尺度不变函数空间及额外假设条件下的新条件唯一性结果，并据此证明了唯一性类中弱解的能量等式。
The three--dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly called the primitive equations. To overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection (or density stratification) equation. In this paper we prove the global regularity of strong solutions to this model in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-penetration…
The primitive equations on curved surfaces
该研究针对$\mathbb R^3$中光滑闭合超曲面领圈邻域内的原始方程，借助极大$L_q$正则性框架与流体静力学亥姆霍兹投影，证明了无初始数据小性假设下强解的整体存在性、唯一性及正则性，且解关于时间和空间联合$C^\text{\infty}$。
In this paper, we consider the initial-boundary value problem of the 3D 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 strong solution is established for any $H^2$ initial data. An $N$-dimensional logarithmic Sobolev embedding inequality, which bounds the $L^\infty$ norm in terms of the $L^q$ norms up to a logarithm of the $L^p$-norm, for $p>N$, of the first order derivatives, and a system version of the classic Gronwal…
In this paper we consider the stochastic primitive equation for geophysical flows subject to transport noise and turbulent pressure. Admitting very rough noise terms, the global existence and uniqueness of solutions to this stochastic partial differential equation are proven using stochastic maximal $L^2$-regularity, the theory of critical spaces for stochastic evolution equations, and global a priori bounds. Compared to other results in this direction, we do not need any smallness assumption on the transport noise which acts directly on the velocity field and we also allow rougher noise terms…
This article presents the maximal regularity approach to the primitive equations. It is proved that the $3D$ primitive equations on cylindrical domains admit a unique, global strong solution for initial data lying in the critical solonoidal Besov space $B^{2/p}_{pq}$ for $p,q\in (1,\infty)$ with $1/p+1/q \leq 1$. This solution regularize instantaneously and becomes even real analytic for $t>0$.
Two models based on the hydrostatic primitive equa- tions are proposed. The first model is the primitive equations with partial viscosity only, and is oriented towards large-scale wave structures in the ocean and atmosphere. The second model is the viscous primitive equations with spectral eddy viscosity, and is oriented towards turbulent geophysical flows. For both models, the existence and uniqueness of global strong solutions is estab- lished. For the second model, the convergence of the solutions to the solutions of the classical primitive equations as eddy viscosity parameters tend to zer…
In this paper, the three-dimensional primitive equations with magnetic field (PEM) are considered on a thin domain. We showed the global existence and uniqueness (regularity) of strong solutions to the three-dimensional incompressible PEM without any small assumption on the initial data. More precisely, there exists a unique strong solution globally in time for any given $H^2$-smooth initial data.
We study well-posedness and ill-posedness for Cauchy problem of the three-dimensional viscous primitive equations describing the large scale ocean and atmosphere dynamics. By using the Littlewood-Paley analysis technique, in particular Chemin-Lerner's localization method, we prove that the Cauchy problem with Prandtl number $P=1$ is locally well-posed in the Fourier-Besov spaces $[\dot{FB}^{2-\frac{3}{p}}_{p,r}(\mathbb{R}^3)]^4$ for $1<p\leq\infty,1\leq r<\infty$ and $[\dot{FB}^{-1}_{1,r}(\mathbb{R}^3)]^4$ for $1\leq r\leq 2$, and globally well-posed in these spaces when the initial data $(u_0…
We study the problems of controllability and ergodicity of the system of 3D primitive equations modeling large-scale oceanic and atmospheric motions. The system is driven by an additive force acting only on a finite number of Fourier modes in the temperature equation. We first show that the velocity and temperature components of the equations can be simultaneously approximately controlled to arbitrary position in the phase space. The proof is based on Agrachev-Sarychev type geometric control approach. Next, we study the controllability of the linearisation of primitive equations around a non-s…
The primitive equations (PEs) model large scale dynamics of the oceans and the atmosphere. While it is by now well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces, and that there are solutions to the inviscid PEs (also called the hydrostatic Euler equations) that develop singularities in finite time, the qualitative description of the blowup still remains undiscovered. In this paper, we provide a full description of two blowup mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and …
Global well-posedness of the primitive equations with stochastic wind-driven boundary conditions
针对随机风驱动边界的三维原始方程，在不假设初始数据垂直导数的情况下，证明了全局路径存在唯一性，扩展了局部理论，方法结合局部最大正则性与加权估计。
In this paper, we consider the initial-boundary value problem of the three-dimensional primitive equations for oceanic and atmospheric dynamics with only 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 also establish the global well-posedness of strong solutions for this system, with any initial data $(v_0, T_0)\in H^1\cap L^\infty$, such that $\partial_zv_0\in L^m$, for some $m\in(2,\infty)$, by using the logarithmic type anisot…
Estimation of Anisotropic Viscosities for the Stochastic Primitive Equations
针对随机原始方程（SPE）中各向异性粘性系数的估计问题，本文利用单样本路径的有限傅里叶模态开发估计器，分析其一致性与渐近正态性，为SPE相关应用提供参数估计方法，且该方法或可推广至其他模型。
Nonuniqueness of generalised weak solutions to the primitive and Prandtl equations
本文通过凸积分方案构造了二维和三维流体静力欧拉方程、三维粘性原始方程及二维Prandtl方程的非唯一广义弱解，证明了大类初值下解的存在性与非唯一性，并构造了无穷多个不守恒能量的解。
The Navier-Stokes equations in the primitive formulation for incompressible flow describe the evolution of velocity and pressure, without recourse to vorticity. We show that, beyond the finite Leray-Hopf regularity interval, every postulated strong solution is accompanied by infinitely many diffusion-dominated percolations of arbitrary size, while the momentum deficit caused by the non-linearity is compensated by the pressure gradient. In the upper half space, we demonstrate how sequences of these collective companions can be re-scaled into an absurd singularity. Owning to the passive nature o…
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. First, 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 th…
The Poincaré problem is a model of two-dimensional internal waves in stable-stratified fluid. The chess billiard flow, a variation of a typical billiard flow, drives the formation behind and describes the evolution of these internal waves, and its trajectories can be represented as rotations around the boundary of a given domain. We find that for sufficiently irrational rotation in the square, or when the rotation number $r(λ)$ is Diophantine, the regularity of the solution $u(t)$ of the evolution problem correlates directly to the regularity of the forcing function $f(x)$. Additionally, we sh…
We consider here the $1D $ Green-Naghdi equations that are commonly used in coastal oceanography to describe the propagation of large amplitude surface waves. We show that the solution of the Green-Naghdi equations can be constructed by a standard Picard iterative scheme so that there is no loss of regularity of the solution with respect to the initial condition.
Well-posedness and ill-posedness of the primitive equations with fractional horizontal dissipation
本文研究具有分数阶水平耗散的二维原始方程，确定了在临界指数α=1处适定性与不适定性的尖锐转变，并建立了α≥6/5时的全局适定性理论。
We study the stochastic effect on the three-dimensional inviscid primitive equations (PEs, also called the hydrostatic Euler equations). Specifically, we consider a larger class of noises than multiplicative noises, and work in the analytic function space due to the ill-posedness in Sobolev spaces of PEs without horizontal viscosity. Under proper conditions, we prove the local existence of martingale solutions and pathwise uniqueness. By adding vertical viscosity, i.e., considering the hydrostatic Navier-Stokes equations, we can relax the restriction on initial conditions to be only analytic i…
The three limits of the hydrostatic approximation
本文研究静水近似的三个极限，依据垂直黏度假设分别得到全黏度原始方程、二维Navier-Stokes方程及仅水平黏度原始方程，并提出最大正则性与二次不等式方法证明收敛性。
Consider the anisotropic Navier-Stokes equations as well as the primitive equations. It is shown that the horizontal velocity of the solution to the anisotropic Navier-Stokes equations in a cylindrical domain of height $\varepsilon $ with initial data $u_0=(v_0,w_0)\in B^{2-2/p}_{q,p}$, $1/q+1/p\le 1$ if $q\ge 2$ and $4/3q+2/3p\le 1$ if $q\le 2$, converges as $\varepsilon \to 0$ with convergence rate $\mathcal{O} (\varepsilon )$ to the horizontal velocity of the solution to the primitive equations with initial data $v_0$ with respect to the maximal-$L^p$-$L^q$-regularity norm. Since the differ…
Existence, regularity, and symmetry of periodic traveling waves for Gardner-Ostrovsky type equations
本文利用局部分岔、正则性分析和移动平面法，系统研究Gardner–Ostrovsky型方程周期行波的存在性、光滑或尖峰奇异性及对称性。
Local well-posedness of strong solutions to the non-isentropic compressible primitive equations with vertical diffusion
本文研究了非等熵可压缩原始方程组在仅含垂直扩散且无重力情况下的局部适定性，证明了强解的局部存在性、唯一性及对初始数据的连续依赖性。
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