Stochastic Boussinesq Research Articles

Global Existence for the Stochastic Boussinesq Equations with Transport Noise and Small Rough Data

Global Existence for the Stochastic Boussinesq Equations with Transport Noise and Small Rough Data

Study on the well-posedness of stochastic Boussinesq equations with Navier boundary conditions

Study on the well-posedness of stochastic Boussinesq equations with Navier boundary conditions

Invariant measure of stochastic Boussinesq equation with zero viscosity in Banach space

In this paper, we investigate the stochastic Boussinesq equations driven by Gaussian noise on a two-dimensional domain . By the regularity estimation on the high-order Sobolev space, we prove the existence of weak solutions in non-separable Banach space. We also show that the Markov semigroup generated by the solution of stochastic Boussinesq equations is also weak Feller. Endowed with the weak-★ topology on , we prove the existence of the invariant measure by Krylov–Bogoliubov theorem.

The Global Existence and Averaging Theorem for the Strong Solution of the Stochastic Boussinesq Equations with the Low Froude Number

The Global Existence and Averaging Theorem for the Strong Solution of the Stochastic Boussinesq Equations with the Low Froude Number

Existence and Uniqueness Results for Two-Dimensional Stochastic Linearised Boussinesq Equation

The water flow in saturated zones of the soil is described by two-dimensional Boussinesq equation. This paper is devoted to investigating the linearised stochastic Boussinesq problem in the presence of randomness in hydraulic conductivity, drainable porosity, recharge, evapotranspiration, initial condition and boundary condition. We use the Sabolev spaces and Galerkin method. Under some suitable assumptions, we prove the existence and uniqueness results, as well as, the continuous dependence on the data for the solution of linearised stochastic Boussinesq problem. Keywords: linearised stochastic Boussinesq equation, Galerkin method, existence and uniqueness results, and continuous dependence on the data.

Large Prandtl number asymptotics in randomly forced turbulent convection

We establish the convergence of statistically invariant states for the stochastic Boussinesq equations in the infinite Prandtl number limit and in particular demonstrate the convergence of the Nusselt number (a measure of heat transport in the fluid). This is a singular parameter limit significant in mantle convection and for gasses under high pressure. The equations are subject to a both temperature gradient on the boundary and internal heating in the bulk driven by a stochastic, white in time, gaussian forcing. Here, the stochastic source terms have a strong physical motivation for example as a model of radiogenic heating. Our approach uses mixing properties of the formal limit system to reduce the convergence of invariant states to an analysis of the finite time asymptotics of solutions and parameter-uniform moment bounds. Here, it is notable that there is a phase space mismatch between the finite Prandtl system and the limit equation, and we implement methods to lift both finite and infinite time convergence results to an extended phase space which includes velocity fields. For the infinite Prandtl stochastic Boussinesq equations, we show that the associated invariant measure is unique and that the dual Markovian dynamics are contractive in an appropriate Kantorovich–Wasserstein metric. We then address the convergence of solutions on finite time intervals, which is still a singular perturbation. In the process we derive well-posed equations which accurately approximate the dynamics up to the initial time when the Prandtl number is large.

On the Well-Posedness of Stochastic Boussinesq Equations with Transport Noise

The Boussinesq equations are fundamental in meteorology. Among other aspects, they aim to model the process of front formation. We use the approach presented in [Hol15] to introduce stochasticity into the incompressible Boussinesq equations. This is, we introduce cylindrical transport noise in a way that the geometric properties in the Euler-Poincar\'{e} formulation are preserved. One of our main results establishes the local well-posedness of regular solutions for these new stochastic Boussinesq equations. We also construct a blow-up criterion and derive some general estimates, which are crucial for showing well-posedness of a wide range of similar SPDEs.

Local and global existence of pathwise solution for the stochastic Boussinesq equations with multiplicative noises

Considering the stochastic Boussinesq equations in Td with the nonlinear multiplicative noises, we establish the local existence of pathwise solutions. Furthermore, we establish the global existence of pathwise solution when the noises are non-degenerate, which show that the non-degenerate multiplicative noises would provide a regularizing effect: the global existence of solution occurs with high probability if the initial data are sufficiently small, or if the noise coefficients are sufficiently large.

The local existence of strong solution for the stochastic 3D Boussinesq equations

The stochastic 3D Boussinesq equations with additive noise are considered. We prove the local existence of the strong solution in H^{s} (frac{1}{2}< sleq 1). We also obtain a new stopping time, which shows that the H^{frac{1}{2}^{+}} norm of u controls the breakdown of the strong solution. Furthermore, we give the probability estimate of the lifespan larger than δ (0<delta <1).

Well-posedness of the stochastic Boussinesq equation driven by Levy processes

In this paper, we develop a new progressive stopping time technique to prove the existence and uniqueness of a special type of global solutions for the stochastic Boussinesq equations driven by Levy processes. Then we prove the existence of invariant measure.

Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise

We present the time-spatial regularity of the nonlocal stochastic convolution for Caputo-type time fractional nonlocal Ornstein-Ulenbeck equations, and establish the existence and uniqueness of mild solutions for time fractional and space nonlocal stochastic Boussinesq equations driven by Gaussian white noise.

Global martingale solution for the stochastic Boussinesq system with zero dissipation

ABSTRACTWe study the two-dimensional stochastic Boussinesq system with zero dissipation and multiplicative noise. We show the existence of a martingale solution by a priori estimates using stochastic calculus, and applications of Prokhorov's, Skorokhod's, and martingale representation theorems. Due to the lack of dissipation, the proof requires higher regularity estimates, taking advantage of the structure of the nonlinear term. Moreover, we obtain the existence of the pressure term via an application of de Rham's theorem for processes.

Large deviation principle for stochastic Boussinesq equations driven by Lévy noise

The current paper is devoted to dynamics of stochastic Boussinesq equation driven by Lévy pure jump noise. The large deviation principle of solution is established, and the effect of the highly nonlinear and unbounded drift is stated. Moreover, the large deviation principle for stochastic two dimensional hydrodynamical systems with Lévy noise is also obtained.

Hausdorff Dimension of a Random Attractor for Stochastic Boussinesq Equations with Double Multiplicative White Noises

This paper investigates the existence of random attractor for stochastic Boussinesq equations driven by multiplicative white noises in both the velocity and temperature equations and estimates the Hausdorff dimension of the random attractor.

The Stampacchia maximum principle for stochastic partial differential equations and applications

Stochastic partial differential equations (SPDEs) are considered, linear and nonlinear, for which we establish comparison theorems for the solutions, or positivity results a.e., and a.s., for suitable data. Comparison theorems for SPDEs are available in the literature. The originality of our approach is that it is based on the use of truncations, following the Stampacchia approach to maximum principle. We believe that our method, which does not rely too much on probability considerations, is simpler than the existing approaches and to a certain extent, more directly applicable to concrete situations. Among the applications, boundedness results and positivity results are respectively proved for the solutions of a stochastic Boussinesq temperature equation, and of reaction–diffusion equations perturbed by a non-Lipschitz nonlinear noise. Stabilization results to a Chafee–Infante equation perturbed by a nonlinear noise are also derived.

Well-Posedness of the Stochastic Fractional Boussinesq Equation With Lévy Noise

This article is devoted to the well-posedness of the stochastic fractional Boussinesq equation with Lévy noise. The commutator estimates are applied to overcome the difficulty in the convergence since the nonlocal fractional diffusion has lower regularity. Based on stopping time technique, weak convergence method, and monotonicity arguments, the global existence and uniqueness of the weak solution are obtained in a fixed probability space.

Random attractor of non-autonomous stochastic Boussinesq lattice system

In this paper, we first consider the existence of tempered random attractor for second-order non-autonomous stochastic lattice dynamical system of nonlinear Boussinesq equations effected by time-dependent coupled coefficients and deterministic forces and multiplicative white noise. Then, we establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

The existence of martingale solutions to the stochastic Boussinesq equations

The existence of martingale solutions to the stochastic Boussinesq equations

Ergodicity of stochastic Boussinesq equations driven by Lévy processes

We consider a class of stochastic Boussinesq equations driven by Levy processes and establish the uniqueness of its invariant measure. The proof is based on the progressive stopping time technique.

H1-Random Attractors and Asymptotic Smoothing Effect of Solutions for Stochastic Boussinesq Equations with Fluctuating Dynamical Boundary Conditions

This work is concerned with the random dynamics of two-dimensional stochastic Boussinesq system with dynamical boundary condition. The white noises affect the system through a dynamical boundary condition. Using a method based on the theory of omega-limit compactness of a random dynamical system, we prove that theL2-random attractor for the generated random dynamical system is exactly theH1-random attractor. This improves a recent conclusion derived by Brune et al. on the existence of theL2-random attractor for the same system.