We consider the initial value problem for the 2D quasi-geostrophic equation with weak dissipation term $κ(-Δ)^{α/2}θ (0<α\leqslant 1)$ and dispersive forcing term $Au_2$. We establish a unique global solution for a given initial data $θ_0$ which belongs to the scaling subcritical Sobolev space $H^s(\mathbb{R}^2)\ (s>2-α)$ if the size of dispersion parameter is sufficiently large. This phenomenon is so-called the global regularity. We also obtain the relationship between the initial data and the dispersion parameter, which ensures the existence of the global solution. Moreover, we show the global regularity in the scaling critical Sobolev space $H^{2-α}(\mathbb{R}^2)$ and find that the size of dispersion parameter to ensure the global existence is determined by each subset $K\subset H^{2-α}(\mathbb{R}^2)$, which is precompact in some homogeneous Sobolev spaces.
We consider surface quasi-geostrophic equation with dispersive forcing and critical dissipation. We prove global existence of smooth solutions given sufficiently smooth initial data. This is done using a maximum principle for the solutions involving conservation of a certain family of moduli of continuity.
We consider the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing in the whole space. When the dispersive amplitude parameter is large enough, we prove the global well-posedness of strong solution to the equation with large initial data. We also show the strong convergence result as the amplitude parameter goes to $\infty$. Both results rely on the Strichartz-type estimates for the corresponding linear equation.
In this paper, the averaging principle for quasi-geostrophic motions with rapidly oscillating forcing is proved, both on finite but large time intervals and on the entire time axis. This includes comparison estimate, stability estimate, and convergence result between quasi-geostrophic motions and its averaged motions. Furthermore, the existence of almost periodic quasi-geostrophic motions and attractor convergence are also investigated.
We consider the asymptotic behavior of the surface quasi-geostrophic equation, subject to a small external force. Under suitable assumptions on the forcing, we first construct the steady states and we provide a number of useful a posteriori estimates for them. Importantly, to do so, we only impose minimal cancellation conditions on the forcing function. Our main result is that all $L^1\cap L^\infty$ localized initial data produces global solutions of the forced SQG, which converge to the steady states in $L^p(\mathbf R^2), 1<p\leq 2$ as time goes to infinity. This establishes that the steady s…
In this paper, we consider a family of piecewise constant solutions of the quasi-geostrophic shallow-water (QGSW) equation. We derive the contour dynamics equation of the QGSW front, which is a nonlinear, nonlocal dispersive equation, and prove the global existence of the solutions when the initial data is sufficiently close to a flat front.
Inviscid and fast rotation limit for dispersive generalized SQG equations
Indian Institute of Technology Jodhpur（印度理工学院乔德普尔分校）；Malviya National Institute of Technology Jaipur（马拉维亚国家技术学院斋浦尔）；National Institute of Technology Warangal（瓦朗加尔国家技术学院）
J.-Y. Chemin proved the convergence (as the Rossby number $ε$ goes to zero) of the solutions of the Primitive Equations to the solution of the 3D quasi-geostrophic system when the Froude number F = 1 that is when no dispersive property is available. The result was proved in the particular case where the kinematic viscosity $ν$ and the thermal diffusivity $ν$ ' are close. In this article we generalize this result for any choice of the viscosities, the key idea is to rely on a special feature of the quasi-geostrophic structure.
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…
In this paper we study a higher order viscous quasi-geostrophic type equation. This equation was derived in [11] as the limit dynamics of a singularly perturbed Navier-Stokes-Korteweg system with Coriolis force, when the Mach, Rossby and Weber numbers go to zero at the same rate. The scope of the present paper is twofold. First of all, we investigate well-posedness of such a model on the whole space $\R^2$: we prove that it is well-posed in $H^s$ for any $s\geq3$, globally in time. Interestingly enough, we show that this equation owns two levels of energy estimates, for which one gets existenc…
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…
We consider quasi-geostrophic (Q-G) models in two- and three-layers that are useful in theoretical studies of planetary atmospheres and oceans. In these models, the streamfunctions are given by (1+2) partial differen- tial systems of evolution equations. A two-layer Q-G model, in a simpli- fied version, is dependent exclusively on the Rossby radius of deformation. However, the f-plane Q-G point vortex model contains factors such as the density, thickness of each layer, the Coriolis parameter, and the constant of gravitational acceleration, and this two-layered model admits a lesser number of L…
We prove local well-posedness in the Sobolev spaces $\dot H^s(\mathbb{T})$, with $s>7/2$, for an initial value problem for a nonlocal, cubically nonlinear, dispersive equation that provides an approximate description of the evolution of surface quasi-geostrophic (SQG) fronts with small slopes.
The quasi-geostrophic equation or the Euler equation with dissipation studied in the present paper is a simplified form of the atmospheric circulation model introduced by Charney and DeVore [J. Atmos. Sci. 36(1979), 1205-1216] on the existence of multiple steady states to the understanding of the persistence of atmospheric blocking. The fluid motion defined by the equation is driven by a zonal thermal forcing and an Ekman friction forcing measured by $κ>0$. It is proved that the steady-state solution is unique for $κ>1$ while multiple steady-state solutions exist for $κ<κ_{crit}$ with respect …
In this paper we study the time evolution of a temperature patch in $R^2$ according to the modified Surface Quasi-Geostrophic Equation (SQG) patch equation. In particular we give a temporal estimate on the growth of the support, providing a rigorous proof of the confinement of a hot patch of temperature in absence of external forcing, under the quasi-geostrophic approximation.
In a recent article \textit{"Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane", to appear on Pacific Journal of Mathematics [15]}, the authors gave a starting point of the study on a series of problems concerning the initial boundary value problem and control theory of Biharmonic NLS in some non-standard domains. In this direction, this article deals to present answers for some questions left in [15] concerning the study of the cubic fourth order Schrödinger equation in a star graph structure $\mathcal{G}$. Precisely, consider $\mathcal{G}$ composed by $N$ e…
We prove the global existence of solutions with small and smooth initial data of a nonlinear dispersive equation for the motion of generalized surface quasi-geostrophic (GSQG) fronts in a parameter regime $1<α<2$, where $α=1$ corresponds to the SQG equation and $α=2$ corresponds to the incompressible Euler equations. This result completes previous global well-posedness results for $0<α\le 1$. We also use contour dynamics to derive the GSQG front equations for $1<α<2$.
We study the stochastic nonlinear Schrödinger equations with additive stochastic forcing. By using the dispersive estimate, we present a simple argument, constructing a unique local-in-time solution with rougher stochastic forcing than those considered in the literature.
We investigate here the interactions of waves governed by a Boussinesq system with a partially immersed body allowed to move freely in the vertical direction. We show that the whole system of equations can be reduced to a transmission problem for the Boussinesq equations with transmission conditions given in terms of the vertical displacement of the object and of the average horizontal discharge beneath it; these two quantities are in turn determined by two nonlinear ODEs with forcing terms coming from the exterior wave-field. Understanding the dispersive contribution to the added mass phenome…
In the present paper, we study the fast rotation and inviscid limits for the 2-D dissipative surface quasi-geostrophic equation with a dispersive forcing term $A \mathcal{R}_{1} \vartheta$, in the domain $Ω=\mathbb{T}^{1} \times \mathbb{R}$. In the case when we perform the fast rotation limit (keeping the viscosity fixed), in the context of general ill-prepared initial data, we prove that the limit dynamics is described by a linear equation. On the other hand, performing the combined fast rotation and inviscid limits, we show that the initial data $\overline{\vartheta}_{0}$ is transported alon…
We construct an invariant measure $μ$ for the Surface Quasi-Geostrophic (SQG) equation and show that almost all functions in the support of $μ$ are initial conditions of global, unique solutions of SQG, that depend continuously on the initial data. In addition, we show that the support of $μ$ is infinite dimensional, meaning that it is not locally a subset of any compact set with finite Hausdorff dimension. Also, there are global solutions that have arbitrarily large initial condition. The measures $μ$ is obtained via fluctuation-dissipation method, that is, as a limit of invariant measures fo…
In this paper, we present a globalization argument for stochastic nonlinear dispersive PDEs with additive noises by adapting the $I$-method (= the method of almost conservation laws) to the stochastic setting. As a model example, we consider the defocusing stochastic cubic nonlinear Schrödinger equation (SNLS) on $\mathbb R^3$ with additive stochastic forcing, white in time and correlated in space, such that the noise lies below the energy space. By combining the $I$-method with Ito's lemma and a stopping time argument, we construct global-in-time dynamics for SNLS below the energy space.
The initial-boundary value problem (ibvp) for the $m$-th order dispersion Korteweg-de Vries (KdV) equation on the half-line with rough data and solution in restricted Bourgain spaces is studied using the Fokas Unified Transform Method (UTM). Thus, this work advances the implementation of the Fokas method, used earlier for the KdV on the half-line with smooth data and solution in the classical Hadamard space, consisting of function that are continuous in time and Sobolev in the spatial variable, to the more general Bourgain spaces framework of dispersive equations with rough data on the half-li…
We consider a two-dimensional convection model augmented with the rotational Coriolis forcing, $U_t + U\cdot\nabla_x U = 2k U^\perp$, with a fixed $2k$ being the inverse Rossby number. We ask whether the action of dispersive rotational forcing alone, $U^\perp$, prevents the generic finite time breakdown of the free nonlinear convection. The answer provided in this work is a conditional yes. Namely, we show that the rotating Euler equations admit global smooth solutions for a subset of generic initial configurations. With other configurations, however, finite time breakdown of solutions may and…
