The three-dimensional baroclinic quasigeostrophic flow model has been widely used to study basic mechanisms in oceanic flows and climate dynamics. In this paper, we consider this flow model under random wind forcing and time-periodic fluctuations on fluid boundary (the interface between the oceans and the atmosphere). The time-periodic fluctuations are due to periodic rotation of the earth and thus periodic exposure of the earth to the solar radiation. After establishing the well-posedness of the baroclinic quasigeostrophic flow model in the state space, we demonstrate the existence of the random attractors, again in the state space. We also discuss the relevance of our result to climate modeling.
The quasigeostrophic equation is a prototypical geophysical fluid model. In this paper, we consider time-periodic motions of this model under dissipation and time-dependent wind forcing. We show that when the wind forcing is time-periodic and the spatial square-integral of the wind forcing is bounded in time, the full nonlinear quasigeostrophic model has time-periodic motions, under some conditions on $β$ parameter, Ekman number, viscousity and the domain size.
The quasigeostrophic model is a simplified geophysical fluid model at asymptotically high rotation rate or at small Rossby number. We consider the quasigeostrophic equation with dissipation under random forcing in bounded domains. We show that global unique solutions exist for appropriate initial data. Unlike the deterministic quasigeostrophic equation whose well-posedness is well-known, there seems no rigorous result on global existence and uniqueness of the randomly forced quasigeostrophic equation. Our work provides such a rigorous result on global existence and uniqueness, under very mild …
We consider time-periodic patterns of the dissipative three dimensional baroclinic quasigeostrophic model in spherical coordinates, under time-dependent forcing. We show that when the forcing is time-periodic and the spatial square-integral of the forcing is bounded in time, the model has time-periodic solutions.
We investigate the ergodicity of 2D large scale quasigeostrophic flows under random wind forcing. We show that the quasigeostrophic flows are ergodic under suitable conditions on the random forcing and on the fluid domain, and under no restrictions on viscosity, Ekman constant or Coriolis parameter. When these conditions are satisfied, then for any observable of the quasigeostrophic flows, its time average approximates the statistical ensemble average, as long as the time interval is sufficiently long.
he quasigeostrophic model describes large scale and relatively slow fluid motion in geophysical flows. We investigate the quasigeostrophic model under random forcing and random boundary conditions. We first transform the model into a partial differential equation with random coefficients. Then we show that, under suitable conditions on the random forcing, random boundary conditions, viscosity, Ekman constant and Coriolis parameter, all quasigeostrophic motion approach a unique stationary state exponentially fast. This stationary state corresponds to a unique invariant Dirac measure
The two-layer quasigeostrophic flow model is an intermidiate system between the single-layer 2D barotropic flow model and the continuously stratified, 3D baroclinic flow model. This model is widely used to investigate basic mechanisms in geophysical flows, such as baroclinic effects, the Gulf Stream and subtropical gyres. The wind forcing acts only on the top layer. We consider the two-layer quasigeostrophic model under stochastic wind forcing. We first transformed this system into a coupled system of random partial differential equations and then show that the asymptotic probabilistic dynamic…
The baroclinic instability problem is considered in the framework of Laplacian tidal theory. The Hilbert space of the quasigeostrophic vorticity budget is spanned by spheroidal functions. The fluid is linearly stable against quasigeostrophic disturbances. As the essential source of irregular ocean-atmosphere motions, baroclinic instability is ruled out by tidal theory. The midlatitude beta-plane budget of vorticity fluxes is inconsistent with basic laws of motion on the rotating spherical surface. Realistic numerical simulations of global wave dynamics and dynamical circulation instabilities r…
Three-dimensional geophysical fluids support both internal and boundary-trapped waves. To obtain the normal modes in such fluids we must solve a differential eigenvalue problem for the vertical structure (for simplicity, we only consider horizontally periodic domains). If the boundaries are dynamically inert (e.g., rigid boundaries in the Boussinesq internal wave problem, flat boundaries in the quasigeostrophic Rossby wave problem) the resulting eigenvalue problem typically has a Sturm-Liouville form and the properties of such problems are well-known. However, when restoring forces are also pr…
We consider the Navier-Stokes system describing the motion of a compressible barotropic fluid driven by stochastic external forces. Our approach is semi-deterministic, based on solving the system for each fixed representative of the random variable and applying an abstract result on measurability of multi-valued maps. The class of admissible driving forces includes the (temporal) white noise and the random kicks, considered recently in the context of incompressible fluid models.
We show the relative energy inequality for the compressible Navier-Stokes system driven by a stochastic forcing. As a corollary, we prove the weak-strong uniqueness property (pathwise and in law) and convergence of weak solutions in the inviscid-incompressible limit. In particular, we establish a Yamada-Watanabe type result in the context of the compressible Navier-Stokes system, that is, pathwise weak--strong uniqueness implies weak--strong uniqueness in law.
We consider a model of steady, incompressible non-Newtonian flow with neglected convective term under external forcing. Our structural assumptions allow for certain non-degenerate power-law or Carreau-type fluids. We provide the full-range theory, namely existence, optimal regularity and uniqueness of solutions, not only with respect to forcing belonging to Lebesgue spaces, but also with respect to their refinements, namely the weighted Lebesgue spaces, with weights in a respective Muckenhoupt class. The analytical highlight is derivation of existence and uniqueness theory for forcing with its…
The motion of a thin elastic plate interacting with a viscous fluid is investigated. A periodic force acting on the plate is considered, which in a setting without damping could lead to a resonant response. The interaction with the viscous fluid provides a damping mechanism due to the energy dissipation in the fluid. Moreover, an internal damping mechanism in the plate is introduced. In this setting, we show that the periodic forcing leads to a time-periodic (non-resonant) solution. We employ the Navier-Stokes and the Kirchhoff-Love plate equation in a periodic cell structure to model the moti…
We study the asymptotic behavior of the isentropic Navier-Stokes system driven by a multiplicative stochastic forcing in the compressible regime, where the Mach number approaches zero. Our approach is based on the recently developed concept of weak martingale solution to the primitive system, uniform bounds derived from a stochastic analogue of the modulated energy inequality, and careful analysis of acoustic waves. A stochastic incompressible Navier-Stokes system is identified as the limit problem.
We discuss the regularity of solutions of 2D incompressible Navier-Stokes equations forced by singular forces. The problem is motivated by the study of complex fluids modeled by the Navier-Stokes equations coupled to a nonlinear Fokker-Planck equation describing microscopic corpora embedded in the fluid. This leads naturally to bounded added stress and hence to $W^{-1,\infty}$ forcing of the Navier-Stokes equations.
We are concerned with the long time behavior of the stochastic Navier--Stokes system for compressible fluids in dimension two and three. In this setting, the part of the phase space occupied by the solution depends sensitively on the choice of the initial state. Our main results are threefold. (i) The kinetic energy of a solution is universally and asymptotically bounded, independent of the initial datum. (ii) Time shifts of a solution with initially controlled energy are asymptotically compact and generate an entire solution defined for all $t\in R$. (iii) Every solution with initially contro…
We consider the Burgers equation posed on the outer communication region of a Schwarzschild black hole spacetime. Assuming spherical symmetry for the fluid flow under consideration, we study the propagation and interaction of shock waves under the effect of random forcing. First of all, considering the initial and boundary value problem with boundary data prescribed in the vicinity of the horizon, we establish a generalization of the Hopf--Lax--Oleinik formula, which takes the curved geometry into account and allows us to establish the existence of bounded variation solutions. To this end, we …
We study a singular limit of a scaled compressible Navier--Stokes--Coriolis system driven by both a deterministic and stochastic forcing terms in three dimensions. If the Mach number is comparable to the Froude number with both proportional to say $\varepsilon\ll 1$, whereas the Rossby number scales like $\varepsilon^m$ for $m>1$ large, then we show that any family of weak martingale solution to the $3$-D randomly forced rotating compressible equation (under the influence of a deterministic centrifugal force) converges in probability, as $\varepsilon\rightarrow0$, to the $2$-D incompressible N…
We study a kinetic toy model for a spray of particles immersed in an ambient fluid, subject to some additional random forcing given by a mixing, space-dependent Markov process. Using the perturbed test function method, we derive the hydrodynamic limit of the kinetic system. The law of the limiting density satisfies a stochastic conservation equation in Stratonovich form, whose drift and diffusion coefficients are completely determined by the law of the stationary process associated with the Markovian perturbation.
These expository notes address certain stationary and ergodic properties of the equations of fluid dynamics subject to a spatially degenerate (i.e. frequency localized), white in time gaussian forcing. In order to provide an accessible treatment of some recent progress in this subject we will develop ideas in detail for a class of finite dimensional models.
We study the Lagrangian flow associated to velocity fields arising from various models of fluid mechanics subject to white-in-time, $H^s$-in-space stochastic forcing in a periodic box. We prove that in many circumstances, these flows are chaotic, that is, the top Lyapunov exponent is strictly positive. Our main results are for the Navier-Stokes equations on $\mathbb T^2$ and the hyper-viscous regularized Navier-Stokes equations on $\mathbb T^3$ (at arbitrary Reynolds number and hyper-viscosity parameters), subject to forcing which is non-degenerate at high frequencies. As an application, we st…
We consider a statistical limit of solutions to the compressible Navier--Stokes system in the high Reynolds number regime in a domain exterior to a rigid body. We investigate to what extent this highly turbulent regime can be modeled by an external stochastic perturbation, as suggested in the related physics literature. To this end, we interpret the statistical limit as a stochastic process on the associated trajectory space. We suppose that the limit process is statistically equivalent to a solution of the stochastic compressible Euler system. Then, necessarily, (i) the stochastic forcing is …
We consider the system of partial differential equations governing two-dimensional flows of a robust class of viscoelastic rate-type fluids with stress diffusion, involving a general objective derivative. The studied system generalizes the incompressible Navier--Stokes equations for the fluid velocity $v$ and pressure $p$ by the presence of an additional term in the constitutive equation for the Cauchy stress expressed in terms of a positive definite tensor $B$. The tensor $B$ evolves according to a diffusive variant of an equation that can be viewed as a combination of corresponding counterpa…
We consider the time-dependent 2D Ginzburg-Landau equation in the whole plane with terms modeling impurities and applied currents. The Ginzburg-Landau vortices are then subjected to three forces: their mutual repulsive Coulomb-like interaction, the applied current pushing them in a fixed direction, and the pinning force attracting them towards the impurities. The competition between the three is expected to lead to complicated glassy effects. We rigorously study the limit in which the number of vortices $N_ε$ blows up as the inverse Ginzburg-Landau parameter $ε$ goes to $0$, and we derive via …
In [Commun Math Phys 348(1), 129-143, 2016], Cheskidov et al. proved that physically realizable weak solutions of the incompressible 2D Euler equations on a torus conserve kinetic energy. Physically realizable weak solutions are those that can be obtained as limits of vanishing viscosity. The key hypothesis was boundedness of the initial vorticity in $L^p$, $p>1$. In this work we extend their result, by adding forcing to the flow.
The Blackstock-Crighton equations describe the motion of a viscous, heat-conducting, compressible fluid. They are used as models for acoustic wave propagation in a medium in which both nonlinear and dissipative effects are taken into account. In this article, a mathematical analysis of the Blackstock-Crighton equations with a time-periodic forcing term is carried out. For arbitrary time-periodic data (sufficiently restricted in size) it is shown that a time-periodic solution of the same period always exists. This implies that the dissipative effects are sufficient to avoid resonance within the…
We study generalised Navier--Stokes equations governing the motion of an electro-rheological fluid subject to stochastic perturbation. Stochastic effects are implemented through (i) random initial data, (ii) a forcing term in the momentum equation represented by a multiplicative white noise and (iii) a random character of the variable exponent $p=p(ω,t,x)$ (as a result of a random electric field). We show the existence of a weak martingale solution provided the variable exponent satisfies $p\geq p^->\frac{3n}{n+2}$ ($p^->1$ in two dimensions). Under additional assumptions we obtain also pathwi…
We study the global existence of a unique strong solution and its large-time behavior of a two-phase fluid system consisting of the compressible isothermal Euler equations coupled with compressible isentropic Navier-Stokes equations through a drag forcing term. The coupled system can be derived as the hydrodynamic limit of the Vlasov-Fokker-Planck/isentropic Navier-Stokes equations with strong local alignment forces. When the initial data is sufficiently small and regular, we establish the unique existence of the global $H^s$-solutions in a perturbation framework. We also provide the large-tim…
Multilayered fluid-structure interactions: existence of weak solutions for time-periodic and initial-value problems
BCAM – Basque Center for Applied Mathematics（巴斯克应用数学中心）；Gheorghe Mihoc – Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy（罗马尼亚科学院格奥尔基·米霍克-凯乌斯·雅各布数学统计与应用数学研究所）；IKERBASQUE, Basque Foundation for Science（伊卡巴斯科，巴斯克科学基金会）
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 …
