2D Boussinesq Equations Research Articles

Global Regularity for Several Incompressible Fluid Models with Partial Dissipation

This paper examines the global regularity problem on several 2D incompressible fluid models with partial dissipation. They are the surface quasi-geostrophic (SQG) equation, the 2D Euler equation and the 2D Boussinesq equations. These are well-known models in fluid mechanics and geophysics. The fundamental issue of whether or not they are globally well-posed has attracted enormous attention. The corresponding models with partial dissipation may arise in physical circumstances when the dissipation varies in different directions. We show that the SQG equation with either horizontal or vertical dissipation always has global solutions. This is in sharp contrast with the inviscid SQG equation for which the global regularity problem remains outstandingly open. Although the 2D Euler is globally well-posed for sufficiently smooth data, the associated equations with partial dissipation no longer conserve the vorticity and the global regularity is not trivial. We are able to prove the global regularity for two partially dissipated Euler equations. Several global bounds are also obtained for a partially dissipated Boussinesq system.

The global attractor of the 2d Boussinesq equations with fractional Laplacian in subcritical case

We prove global well-posedness of strong solutions and existence of the global attractor for the 2D Boussinesq system in a periodic channel with fractional Laplacian in subcritical case. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and temperature.

On the regularity criteria of the 2D Boussinesq equations with partial dissipation

In this paper, we consider the two-dimensional (2D) Boussinesq equations with partial dissipation. The issue of whether the 2D Boussinesq equations always possess global (in time) classical solutions can be difficult when there is only partial dissipation. Therefore, the goal of this paper is to establish some regularity criteria of the corresponding systems.

Global solutions for the incompressible rotating stably stratified fluids

We consider the initial value problem of the 3D incompressible Boussinesq equations for rotating stratified fluids. We establish the dispersive and the Strichartz estimates for the linear propagator associated with both the rotation and the stable stratification. As an application, we give a simple proof of a unique existence of global in time solutions to our system using just a contraction mapping principle.

Global Regularity and Long-time Behavior of the Solutions to the 2D Boussinesq Equations without Diffusivity in a Bounded Domain

New results are obtained for global regularity and long-time behavior of the solutions to the 2D Boussinesq equations for the flow of an incompressible fluid with positive viscosity and zero diffusivity in a smooth bounded domain. Our first result for global boundedness of the solution $(u, \theta) \in D(A)\times H^1$ improves considerably the main result of the recent article [7]. Our second result on global regularity of the solution $(u, \theta) \in V \times H^1$ for both bounded domain and the whole space ${\mathbb R}^2$ is a new one. It has been open and also seems much more challenging than the first result. Global regularity of the solution $(u, \theta) \in D(A) \times H^2$ is also proved.

A logarithmically improved regularity criterion of smooth solutions for the 3D Boussinesq equations

In this note, we consider the three-dimensional (3D) incompressible Boussinesq equations. We obtain the logarithmically improved regularity criterion of smooth solutions in terms of the velocity field. This result improves some previous works.

Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation

As a continuation of the previous work [48], in this paper we focus on the Cauchy problem of the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation. We give an elementary proof of the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. The argument is based on the nonlinear lower bounds for the fractional Laplacian established in [13]. Consequently, this result significantly improves the recent works [13,45,48].

Global Regularity Results of the 2D Boussinesq Equations with Fractional Laplacian Dissipation

As a continuation of the previous work [40], in this paper we focus on the Cauchy problem of the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation. We give an elementary proof of the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. The argument is based on the nonlinear lower bounds for the fractional Laplacian established in [12]. Consequently, this result significantly improves the recent works [12, 38, 40].

A note on global regularity results for 2D Boussinesq equations with fractional dissipation

In this paper we study the Cauchy problem for the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Invoking the energy method and several commutator estimates, we get the global regularity result of the 2D Boussinesq equations as long as $1-\alpha<\beta< \min\Big\{\frac{\alpha}{2},\,\, \frac{3\alpha-2}{2\alpha^{2}-6\alpha+5}, \,\,\frac{2-2\alpha}{4\alpha-3}\Big\}$ with $0.77963\thickapprox\alpha_{0}<\alpha<1$. As a result, this result is a further improvement of the previous two works \cite{MX,YXX}.

Турбулентный конвективный теплоперенос в тонких вертикальных слоях жидкости

Turbulent Rayleigh-Benard convection in bounded vertical layers is studied in numerical simulations. The influence of layer thickness on convective heat transfer is investigated in detail. In this work we study the ability of two-dimensional and quasi-two-dimensional mathematical models to describe the characteristics of heat transfer in the thin layers. These models require much less computing resources in comparison to three-dimensional direct numerical simulations of turbulent convection. Three statements of the problem are considered: two-dimensional (2D), three-dimensional (3D) and quasi-two-dimensional (Q2D). The Q2D model uses the modified 2D equations, which are derived from the 3D Boussinesq equations under the assumption of a laminar transverse velocity profile. The 2D and Q2D simulations were performed using our own code, whereas the 3D simulations were carried out using open source CFD software OpenFOAM Extend 3.1. It is shown that the flow structure strongly affects the Nusselt number value. In thin layers the convective heat transfer increases despite a decrease in the intensity of the large-scale flow. It is shown, that the friction on lateral boundaries, even in frame of a rough model of linear friction, which was used in the Q2D simulations strongly affects the structure of the turbulent convective flow, changes its local and integral characteristics. The comparison with 3D numerical results demonstrates that the Q2D results qualitatively reproduce the dependence of the Nusselt number on the layer thickness, but underestimate the values of the Nusselt numbers by 30%. Also 2D results reveal poor correlation with the real structure of fluid flow in the domain for any aspect ratio.Received 26.01.2016; accepted 10.02.2016

Numerical Evaluation of 2D Ground States

A ground state is defined as the positive radial solution of the multidimensional nonlinear problem −Δu(x)+u(x)−f(u(x))=0, x∈ℝn; lim|x|→∞|u(x)|=0,$\varepsilon \, \propto \,k_ \bot ^{1 - \xi }$ with the function f being either f (u ) =a |u |p –1 u or f (u ) =a |u |p u +b |u |2p u . The numerical evaluation of ground states is based on the shooting method applied to an equivalent dynamical system. A combination of fourth order Runge-Kutta method and Hermite extrapolation formula is applied to solving the resulting initial value problem.The efficiency of this procedure is demonstrated in the 1D case, where the maximal difference between the exact and numerical solution is ≈ 10–11 for a discretization step 0:00025. As a major application, we evaluate numerically the critical energy constant. This constant is defined as a functional of the ground state and is used in the study of the 2D Boussinesq equations.

Regularity results for the 2D Boussinesq equations with critical or supercritical dissipation

Regularity results for the 2D Boussinesq equations with critical or supercritical dissipation

Regularity criteria for the 2D Boussinesq equations with supercritical dissipation

Regularity criteria for the 2D Boussinesq equations with supercritical dissipation

Global-wellposedness of 2D Boussinesq equations with mixed partial temperature-dependent viscosity and thermal diffusivity

In this paper, we study the global well-posedness of 2D anisotropic nonlinear Boussinesq equations with horizontal temperature-dependent viscosity and vertical thermal diffusivity in the whole space. Due to lacking vertical viscosity and horizontal thermal diffusivity, there is no smooth effect in those directions. Besides, the nonlinearity of temperature-dependent viscosity gives rise to new difficulties. To solve it, we make full use of the incompressible condition and anisotropic inequalities to obtain the H1 estimates of velocity field and H1+s estimates of temperature for any s∈(0,1/2]. In the end, we build up a uniqueness criterion which together with the a priori estimates admits a unique global solution without any smallness assumptions.

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

We prove the global well-posedness of the two-dimensional Boussinesq equations with only vertical dissipation. The initial data $(u_0,\theta_0)$ are required to be only in the space $X=\{f\in L^2(\mathbb R^2)\,|\,\partial_xf\in L^2(\mathbb R^2)\}$, and thus our result generalizes that in [C. Cao, J. Wu, Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Rational Mech. Anal., Vol. 208 (2013), 985-1004], where the initial data are assumed to be in $H^2(\mathbb R^2)$. The assumption on the initial data is at the minimal level that is required to guarantee the uniqueness of the solutions. A logarithmic type limiting Sobolev embedding inequality for the $L^\infty(\mathbb R^2)$ norm, in terms of anisotropic Sobolev norms, and a logarithmic type Gronwall inequality are established to obtain the global in time a priori estimates, which guarantee the local solution to be a global one.

Global regularity results for the 2D Boussinesq equations with partial dissipation

The two-dimensional (2D) incompressible Boussinesq equations model geophysical fluids and play an important role in the study of the Raleigh–Bernard convection. Mathematically this 2D system retains some key features of the 3D Navier–Stokes and Euler equations such as the vortex stretching mechanism. The issue of whether the 2D Boussinesq equations always possess global (in time) classical solutions can be difficult when there is only partial dissipation or no dissipation at all. This paper obtains the global regularity for two partial dissipation cases and proves several global a priori bounds for two other prominent partial dissipation cases. These results take us one step closer to a complete resolution of the global regularity issue for all the partial dissipation cases involving the 2D Boussinesq equations.

Initial boundary value problem for 2D Boussinesq equations with temperature-dependent diffusion

We consider the initial-boundary value problem (IBVP) of 2D inviscid heat conductive Boussinesq equations with nonlinear heat diffusion over a bounded domain with smooth boundary. Under slip boundary condition of velocity and the homogeneous Dirichlet boundary condition for temperature, we show that there exists a unique global smooth solution to the IBVP for H3initial data. Moreover, we show that the temperature converges exponentially to zero as time goes to infinity, and the velocity and vorticity are uniformly bounded in time.

Remarks on global regularity of the 2D Boussinesq equations with fractional dissipation

In this paper, we are interested in the study of the Cauchy problem to the two-dimensional (2D) incompressible Boussinesq equations with fractional dissipation. By making use of the nonlinear lower bounds for the fractional Laplacian established in Constantin and Vicol (2012), we establish the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. This result significantly improves the recent works of Constantin and Vicol (2012) and Yang et al. (2014).

Blowup in stagnation-point form solutions of the inviscid 2d Boussinesq equations

The 2d Boussinesq equations model large scale atmospheric and oceanic flows. Whether its solutions develop a singularity in finite-time remains a classical open problem in mathematical fluid dynamics. In this work, blowup from smooth nontrivial initial velocities in stagnation-point form solutions of this system is established. On an infinite strip Ω={(x,y)∈[0,1]×R+}, we consider velocities of the form u=(f(t,x),−yfx(t,x)), with scalar temperature θ=yρ(t,x). Assuming fx(0,x) attains its global maximum only at points xi⁎ located on the boundary of [0,1], general criteria for finite-time blowup of the vorticity −yfxx(t,xi⁎) and the time integral of fx(t,xi⁎) are presented. Briefly, for blowup to occur it is sufficient that ρ(0,x)≥0 and f(t,xi⁎)=ρ(0,xi⁎)=0, while −yfxx(0,xi⁎)≠0. To illustrate how vorticity may suppress blowup, we also construct a family of global exact solutions. A local-existence result and additional regularity criteria in terms of the time integral of ‖fx(t,⋅)‖L∞([0,1]) are also provided.

Heat transport by coherent Rayleigh-Bénard convection

Steady but generally unstable solutions of the 2D Boussinesq equations are obtained for no-slip boundary conditions and Prandtl number 7. The primary solution that bifurcates from the conduction state at Rayleigh number Ra ≈ 1708 has been calculated up to Ra ≈ 5.106 and its Nusselt number is Nu ∼ 0.143 Ra0.28 with a delicate spiral structure in the temperature field. Another solution that maximizes Nu over the horizontal wavenumber has been calculated up to Ra = 109 and scales as Nu ∼ 0.115 Ra0.31 for 107 &amp;lt; Ra ≤ 109, quite similar to 3D turbulent data that show Nu ∼ 0.105 Ra0.31 in that range. The optimum solution is a simple yet multi-scale coherent solution whose horizontal wavenumber scales as 0.133 Ra0.217. That solution is unstable to larger scale perturbations and in particular to mean shear flows, yet it appears to be relevant as a backbone for turbulent solutions, possibly setting the scale, strength, and spacing of elemental plumes.