Incompressible Boussinesq Equations Research Articles

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}.

The vanishing diffusivity limit for the 2-D Boussinesq equations with boundary effect

In this paper, we tackle the issue of the vanishing diffusivity limit of the 2-D incompressible Boussinesq equations in the half plane. Our main purpose is to study the boundary layer effect and the convergence rates as the thermal diffusion parameter ϵ goes to zero. Under the homogeneous Dirichlet boundary condition of velocity and the nonhomogeneous Dirichlet boundary condition in the x-direction for temperature, we show that the boundary layer thickness is of the order O(ϵβ) with (0<β<23). In contrast with Jiang et al. (2011), the BL-thickness we got is thinner than that in Jiang et al. (2011). Moreover, we prove that as diffusivity tends to zero, the solutions of the fully viscous equations converge strongly to those of zero diffusion equations. We also obtain the convergence rates of the vanishing diffusivity limit.

An Analytical Solution for Three-Dimensional Sea–Land Breeze

Abstract Based on the hydrostatic, incompressible Boussinesq equations in the planetary boundary layer (PBL), the three-dimensional sea–land breeze (SLB) circulation has been elegantly expressed as functions of the surface temperature distribution. The horizontal distribution of the horizontal or vertical motion is determined by the first or second derivative of the surface temperature distribution. For symmetric land–sea and temperature distribution, the full strength of the sea breeze occurs inland but not at the coastline, and the maximum updraft associates with the heating center. Setting the temperature difference between land and sea (TDLS), which varies with the island size, there would exist an optimal island size corresponding to the strongest SLB circulation that weakens with both a larger and smaller island size. Each velocity component approaches a peak at a certain vertical level. Both the peak value and the corresponding vertical level link with the vertical scale of the surface temperature: the more significant the influence of the surface temperature vertically, the stronger the SLB circulation at a higher vertical level it induces. The Weather Research and Forecasting (WRF) Model's ideal simulation for the two-dimensional sea breeze is applied to verify the theory. Two cases, land breeze and sea breeze, further support the theory's results despite a certain slight discrepancy due to the highly simplified theoretical equations.

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.

Exact solutions of atmospheric (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq equations with viscosity

The symmetry reduction equations, similarity solutions, sub-groups and exact solutions of the (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq equations with viscosity (INHBV equations), which describe the atmospheric gravity waves, are researched in this paper. Calculation on symmetry shows that the equations are invariant under the Galilean transformations, scaling transformations, rotational transformations and space-time translations. Three types of symmetry reduction equations and similar solutions for the (3+1)-dimensional INHBV equations are proposed. Traveling wave solutions of the INHBV equations are demonstrated by means of symmetry method. The evolutions on the wind velocities and temperature perturbation are demonstrated by figures.

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).

A Stabilized Crank–Nicolson Mixed Finite Volume Element Formulation for the Non-stationary Incompressible Boussinesq Equations

At first, a semi-discrete Crank---Nicolson (CN) formulation with respect to time for the non-stationary incompressible Boussinesq equations is presented. Then, a fully discrete stabilized CN mixed finite volume element (SCNMFVE) formulation based on two local Gauss integrals and parameter-free is established directly from the semi-discrete CN formulation with respect to time. Next, the error estimates for the fully discrete SCNMFVE solutions are derived by means of the standard CN mixed finite element method. Finally, some numerical experiments are presented illustrating that the numerical errors are consistent with theoretical results, the computing load for the fully discrete SCNMFVE formulation are far fewer than that for the stabilized mixed finite volume element (SMFVE) formulation with the first time accuracy, and its accumulation of truncation errors in the computational process is far lesser than that of the SMFVE formulation with the first time accuracy. Thus, the advantage of the fully discrete SCNMFVE formulation for the non-stationary incompressible Boussinesq equations is shown sufficiently.

Blow-Up Criterion of Weak Solutions for the 3D Boussinesq Equations

The Boussinesq equations describe the three-dimensional incompressible fluid moving under the gravity and the earth rotation which come from atmospheric or oceanographic turbulence where rotation and stratification play an important role. In this paper, we investigate the Cauchy problem of the three-dimensional incompressible Boussinesq equations. By commutator estimate, some interpolation inequality, and embedding theorem, we establish a blow-up criterion of weak solutions in terms of the pressurepin the homogeneous Besov spaceḂ∞,∞0.

Proper orthogonal decomposition-based reduced-order stabilized mixed finite volume element extrapolating model for the nonstationary incompressible Boussinesq equations

In this study, we employ a proper orthogonal decomposition (POD) method to establish a POD-based reduced-order stabilized mixed finite volume element (SMFVE) extrapolating model with very few degrees of freedom for the nonstationary incompressible Boussinesq equations. We provide error estimates of the POD-based reduced-order SMFVE solutions and the algorithm implementation for the POD-based reduced-order SMFVE extrapolating model. In addition, we present some numerical experiments that verify the validity and reliability of the POD-based reduced-order SMFVE extrapolating model.

A POD-based reduced-order finite difference extrapolating model for the non-stationary incompressible Boussinesq equations

A proper orthogonal decomposition (POD) method is used to establish a POD-based reduced-order finite difference (FD) extrapolating model with fully second-order accuracy for the non-stationary incompressible Boussinesq equations (NSIBEs). The error estimates of the POD-based reduced-order FD solutions obtained from the POD-based reduced-order FD extrapolating model are provided. The algorithm implementation for the POD-based reduced-order FD extrapolating model is given. A numerical experiment shows that the numerical results are consistent with the theoretical conclusions. Moreover, it is shown that the POD-based reduced-order FD extrapolating model is feasible and efficient for finding the numerical solutions for NSIBEs. MSC:76M20, 65M12, 65M15.

Eventual Regularity of the Two-Dimensional Boussinesq Equations with Supercritical Dissipation

This paper studies solutions of the two-dimensional incompressible Boussinesq equations with fractional dissipation. The spatial domain is a periodic box. The Boussinesq equations concerned here govern the coupled evolution of the fluid velocity and the temperature and have applications in fluid mechanics and geophysics. When the dissipation is in the supercritical regime (the sum of the fractional powers of the Laplacians in the velocity and the temperature equations is less than 1), the classical solutions of the Boussinesq equations are not known to be global in time. Leray–Hopf type weak solutions do exist. This paper proves that such weak solutions become eventually regular (smooth after some time \(T>0\)) when the fractional Laplacian powers are in a suitable supercritical range. This eventual regularity is established by exploiting the regularity of a combined quantity of the vorticity and the temperature as well as the eventual regularity of a generalized supercritical surface quasi-geostrophic equation.

Global well-posedness for a class of 2D Boussinesq systems with fractional dissipation

The incompressible Boussinesq equations not only have many applications in modeling fluids and geophysical fluids but also are mathematically important. The well-posedness and related problem on the Boussinesq equations have recently attracted considerable interest. This paper examines the global regularity issue on the 2D Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Attention is focused on the case when the thermal diffusion dominates. We establish the global well-posedness for the 2D Boussinesq equations with a new range of fractional powers of the Laplacian.

Generalized 2D Euler–Boussinesq equations with a singular velocity

This paper studies the global (in time) regularity problem concerning a system of equations generalizing the two-dimensional incompressible Boussinesq equations. The velocity here is determined by the vorticity through a more singular relation than the standard Biot–Savart law and involves a Fourier multiplier operator. The temperature equation has a dissipative term given by the fractional Laplacian operator −Δ. We establish the global existence and uniqueness of solutions to the initial-value problem of this generalized Boussinesq equations when the velocity is “double logarithmically” more singular than the one given by the Biot–Savart law. This global regularity result goes beyond the critical case. In addition, we recover a result of Chae, Constantin and Wu [8] when the initial temperature is set to zero.

Blow-up criteria of the 3D boussinesq equations

In this paper, we are concerned with the three-dimensional incompressible Boussinesq equations. We obtain a logarithmically improved blow-up criteria for the system in the Morrey-Campanato space.

The Two-Dimensional Incompressible Boussinesq Equations with General Critical Dissipation

The two-dimensional incompressible Boussinesq equations with partial or fractional dissipation have recently attracted considerable attention and the global regularity issue has been extensively investigated. This paper aims at the global regularity in the case when the dissipation is critical. The critical dissipation refers to $\alpha +\beta=1$ when $\Lambda^\alpha \equiv (-\Delta)^{\frac{\alpha}{2}}$ and $\Lambda^\beta$ represent the fractional Laplacian dissipation in the velocity and the temperature equations, respectively. When $\alpha=1$ and $\beta =0$ or when $\alpha=0$ and $\beta=1$, the global regularity was obtained in [T. Hmidi, S. Keraani, and F. Rousset, J. Differential Equations, 249 (2010), pp. 2147--2174; T. Hmidi, S. Keraani, and F. Rousset, Comm. Partial Differential Equations, 36 (2011), pp. 420--445]. However, the approaches there do not apply to the situation when $\alpha+\beta=1$ with both $\alpha>0$ and $\beta>0$. The novelty here is to reduce the critical Boussinesq system to a critical active scalar equation or, more precisely, the generalized critical surface quasi-geostrophic equation. When $\alpha$ is restricted to a suitable range, the global regularity of the critical Boussinesq system can be obtained by exploiting the global regularity of this scalar equation and the global bound for a combined quantity of the vorticity and the temperature.

Exact solutions of (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq equations

The symmetries and the exact solutions of the (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations, which describe atmospheric gravity waves, are studied in this paper. The calculation on symmetry shows that the equations are invariant under the Galilean transformations, the scaling transformations, and the space—time translations. Three types of symmetry reduction equations and similar solutions for the (3+1)-dimensional INHB equations are proposed. Traveling and non-traveling wave solutions of the INHB equations are demonstrated. The evolutions of the wind velocities in latitudinal, longitudinal, and vertical directions with space—time are demonstrated. The periodicity and the atmosphere viscosity are displayed in the (3+1)-dimensional INHB system.

Regularity Criteria of the 3D Boussinesq Equations in the Morrey-Campanato Space

In this paper, we consider the three-dimensional (3D) incompressible Boussinesq equations. We obtain some regularity criteria for the three-dimensional (3D) Boussinesq equations in the Morrey-Campanato space.

Global Well–Posedness of the 3D Primitive Equations with Partial Vertical Turbulence Mixing Heat Diffusion

The three–dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly called the primitive equations. To overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection (or density stratification) equation. In this paper we prove the global regularity of strong solutions to this model in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-penetration and stress-free boundary conditions on the solid, top and bottom boundaries. Specifically, we show that short time strong solutions to the above problem exist globally in time, and that they depend continuously on the initial data.

Blow‐up criteria of smooth solutions to the 3D Boussinesq equations

In this paper, we investigate three‐dimensional incompressible Boussinesq equations and establish some logarithmically improved blow‐up criteria of smooth solutions to the Cauchy problem for the incompressible Boussinesq equations. Copyright © 2011 John Wiley &amp; Sons, Ltd.