2D Boussinesq Research Articles

Correction to: Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $$C^{1,\alpha }$$ Velocity and Boundary

Correction to: Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $$C^{1,\alpha }$$ Velocity and Boundary

Speed of Convergence of Time Euler Schemes for a Stochastic 2D Boussinesq Model

We prove that an implicit time Euler scheme for the 2D Boussinesq model on the torus D converges. The various moments of the W1,2-norms of the velocity and temperature, as well as their discretizations, were computed. We obtained the optimal speed of convergence in probability, and a logarithmic speed of convergence in L2(Ω). These results were deduced from a time regularity of the solution both in L2(D) and W1,2(D), and from an L2(Ω) convergence restricted to a subset where the W1,2-norms of the solutions are bounded.

Stability and Large Time Behavior of the 2D Boussinesq Equations with Mixed Partial Dissipation Near Hydrostatic Equilibrium

Stability and Large Time Behavior of the 2D Boussinesq Equations with Mixed Partial Dissipation Near Hydrostatic Equilibrium

Asymptotic stability of the 2D Boussinesq equations without thermal conduction

The paper is concerned with asymptotic stability of certain stationary solution to Boussinesq equations without thermal conduction in the infinite flat strip Ω=R×(0,1). It is shown that the solution starting from initial data close to the stationary solution will converge to it with explicit algebraic rates as time tends to infinity. This is a kind of results on nonlinear Rayleigh-Taylor stability.

Stabilization and exponential decay for 2D Boussinesq equations with partial dissipation

Stabilization and exponential decay for 2D Boussinesq equations with partial dissipation

The optimal start control problem for 2D Boussinesq equations

Abstract We consider the problem of the optimal start control for two-dimensional Boussinesq equations describing non-isothermal flows of a viscous fluid in a bounded domain. Using the study of the properties of admissible tuples and of the evolution operator, we prove the solubility of the optimization problem under natural assumptions about the model data. We derive a variational inequality which is satisfied by the optimal control provided that the objective functional is determined by the final observation. We also obtain sufficient conditions for the uniqueness of an optimal control.

Global Well-Posedness of the 2D Boussinesq Equations with Partial Dissipation

Global Well-Posedness of the 2D Boussinesq Equations with Partial Dissipation

Low regularity global well-posedness for 2D Boussinesq equations with variable viscosity

As a continuation of our previous work, we consider lower regularity global well-posedness for a model of the two-dimensional zero diffusivity Boussinesq equations with variable viscosity. More precisely, based on De Giorgi–Nash–Moser estimates and the refined logarithmic Gronwall-type inequality, we prove that it is globally well-posed, provided that the initial data belong to Hs with s > 1. Finally, we show that it is also valid for the two-dimensional zero diffusivity Boussinesq equations with variable viscosity in the non-divergence form.

Well-posedness and attractors for a 2D Boussinesq system with partial dissipation

This article is devoted to the global well-posedness and the long-time behavior of solutions of a 2D Boussinesq equations with partial dissipation. We prove that this system is global well-posed under some weaker assumptions on the initial data and has a weak sigma-attractor which retains some of the common properties of global attractors for the dissipative dynamical system, moreover, the local attractor which is the composition of the weak sigma-attractor is upper semicontinuous under small viscosity perturbations.

Smoothing effect and well-posedness for 2D Boussinesq equations in critical Sobolev space

<p style='text-indent:20px;'>In this paper, we investigate the fractional dissipation 2D Boussinesq equations with initial data in the critical space <inline-formula><tex-math id="M1">\begin{document}$ u_0\in H^{2-2\alpha}(\mathbb{R}^2) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \theta_0\in H^{2-2\beta}(\mathbb{R}^2) $\end{document}</tex-math></inline-formula>. The local well-posedness for the equations is firstly established by using some <i>a priori</i> estimates for the solution in <inline-formula><tex-math id="M3">\begin{document}$ L^{p}(0, T;{H}^{2-\frac{p-1}{p} 2\alpha}(\mathbb{R}^2))\times L^{p}(0, T;{H}^{2-\frac{p-1}{p} 2\beta}(\mathbb{R}^2)) $\end{document}</tex-math></inline-formula> with some suitable <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula>. And then the generalized blow-up criterion and smoothing effect are obtained in turn, which improves some of the previous results for (critical, subcritcial or supcritical) Boussnesq equations. The results of the present paper are based on the Littlewood-Paley theory and the nonlinear lower bounds estimates for the fractional Laplacian, and can be treated as a generalization of results for 2D quasi-geostrophic equation.</p>

Global well-posedness and large time behavior to 2D Boussinesq equations for MHD convection

Global well-posedness and large time behavior to 2D Boussinesq equations for MHD convection

On the stability of hydrostatic equilibrium in a uniform magnetic field with partial dissipation

In this paper, we establish the nonlinear stability and large time behavior of hydrostatic equilibrium in a uniform magnetic field for the 2D Boussinesq system with magnetohydrodynamics convection on spatial domain with partial dissipation. This is the important part of further work about the stability/instability of shear flow which is the combination of hydrostatic equilibrium and Couette/shear flow in a uniform magnetic field with partial dissipation.

Convergence of stochastic 2D inviscid Boussinesq equations with transport noise to a deterministic viscous system

The inviscid 2D Boussinesq system with thermal diffusivity and multiplicative noise of transport type is studied in the L 2-setting. It is shown that, under a suitable scaling of the noise, weak solutions to the stochastic 2D Boussinesq equations converge weakly to the unique solution of the deterministic viscous Boussinesq system. Consequently, the transport noise asymptotically regularises the inviscid 2D Boussinesq system and enhances dissipation in the limit.

On the instability of the possible blow-up solutions to 2D Boussinesq system

Motivated by [10,11], we first derive the bicharacteristic-amplitude equations related to the WKB approximation of 2D Boussinesq system. Then we apply WKB method to prove the instability to the possible blow-up solutions of this system as time approaching to the blow-up time. In particular, this result explains the reason why the prediction of finite time singularities to 2D Boussinesq system via direct numerical experiments is so difficult.

Stability of stationary solutions to 2D Boussinesq equations with partial dissipation on a flat strip

The global existence and stability of perturbation of 2D Boussinesq equations with partial dissipation near hydrostatic equilibrium on a flat strip Ω≔T×[0,1] are obtained with the help of a time-weight energy estimate. Meanwhile, we also establish the explicit decay rates of the solution of 2D Boussinesq equations.

Stability of Couette flow for 2D Boussinesq system with vertical dissipation

This paper establishes the nonlinear stability of the Couette flow for the 2D Boussinesq equations with only vertical dissipation. The Boussinesq equations concerned here model buoyancy-driven fluids such as atmospheric and oceanographic flows. Due to the presence of the buoyancy forcing, the energy of the standard Boussinesq equations could grow in time. It is the enhanced dissipation created by the linear non-self-adjoint operator y∂x−ν∂yy in the perturbation equation that makes the nonlinear stability possible. When the initial perturbation from the Couette flow (y,0) is no more than the viscosity to a suitable power (in the Sobolev space Hb with b>43), we prove that the solution of the 2D Boussinesq system with only vertical dissipation on T×R remains close to the Couette at the same order. A special consequence of this result is the stability of the Couette for the 2D Navier-Stokes equations with only vertical dissipation.

Remarks on Type I Blow-Up for the 3D Euler Equations and the 2D Boussinesq Equations

In this paper, we derive kinematic relations for quantities involving the rate of strain tensor and the Hessian of the pressure for solutions of the 3D Euler equations and the 2D Boussinesq equations. Using these kinematic relations, we prove new blow-up criteria and obtain conditions for the absence of type I singularity for these equations. We obtain both global and localized versions of the results. Some of the new blow-up criteria and type I conditions improve previous results of Chae and Constantin (Int Math Res Notices rnab014, 2021).

Incorporating wave-breaking data in the calibration of a Boussinesq-type wave model

Wave breaking is one of the main forcing mechanisms in coastal hydrodynamics, driving mean water levels and currents. Its accurate representation and modeling is therefore essential to develop a thorough understanding and predictive capabilities. In this work, the performance of a 1D Boussinesq Type Model is assessed by expanding the traditional approach of calibrating the models against profiles and time series of wave quantities data, with remotely sensed observations of the spatio-temporal occurrence of breaking events, on a laboratory setting considering one regular and one irregular wave condition over a fixed barred beach. A re-analysis of calibrated parameters from a wave breaking model based on the turbulent roller analogy is performed using observed wave-by-wave breaking events. The calibration process aims at matching the initiation/cessation of individual breaking events, and the histograms of breaking occurrence at each location. Results confirm that it is possible to find a physically sound parameter space that yields minimal differences in profiles of phase averaged quantities such as the root-mean-square wave height, wave setup, skewness, and asymmetry; nevertheless, not all of these parameter combinations succeed in representing the spatio-temporal occurrence of breaking events. These results suggest that models can reproduce the overall behavior of wave breaking dissipation, although this does not necessarily mean that the process itself is appropriately modeled. This could in turn influence the modeling capability of other quantities derived from these models, such as mean currents. It is thus considered that incorporating spatio-temporal wave breaking data is essential in assessing model performances.

Stability and exponential decay for the 2D anisotropic Boussinesq equations with horizontal dissipation

The hydrostatic equilibrium is a prominent topic in fluid dynamics and astrophysics. Understanding the stability of perturbations near the hydrostatic equilibrium of the Boussinesq system helps gain insight into certain weather phenomena. The 2D Boussinesq system focused here is anisotropic and involves only horizontal dissipation and horizontal thermal diffusion. Due to the lack of the vertical dissipation, the stability and precise large-time behavior problem is difficult. When the spatial domain is \({\mathbb {R}}^2\), the stability problem in a Sobolev setting remains open. When the spatial domain is \({\mathbb {T}}\times {\mathbb {R}}\), this paper solves the stability problem and specifies the precise large-time behavior of the perturbation. By decomposing the velocity u and temperature \(\theta \) into the horizontal average \(({\bar{u}}, {\bar{\theta }})\) and the corresponding oscillation \(({\widetilde{u}}, {\widetilde{\theta }})\), and deriving various anisotropic inequalities, we are able to establish the global stability in the Sobolev space \(H^2\). In addition, we prove that the oscillation \(({\widetilde{u}}, {\widetilde{\theta }})\) decays exponentially to zero in \(H^1\) and \((u, \theta )\) converges to \(({\bar{u}}, {\bar{\theta }})\). This result reflects the stratification phenomenon of buoyancy-driven fluids.

Stability of Couette flow for 2D Boussinesq system in a uniform magnetic field with vertical dissipation

In this paper, we establish the nonlinear stability of Couette flow in a uniform magnetic field for the Boussinesq equations with magnetohydrodynamics convection in the domain T×R with only vertical dissipation. This extends the result for full dissipation case in the reference Bian et al. (2020) to partial dissipation case and the result for Boussinesq system in the reference Deng et al. (2020) to the full system in the presence of the magnetic field.