Erratum to: An Incompressible 2D Didactic Model with Singularity and Explicit Solutions of the 2D Boussinesq Equations

An incompressible 2D didactic model with singularity and explicit solutions of the 2D Boussinesq equations

We give an example of a well posed, finite energy, 2D incompressible active scalar equation with the same scaling as the surface quasi-geostrophic equation and prove that it can produce finite time singularities. In spite of its simplicity, this seems to be the first such example. Further, we construct explicit solutions of the 2D Boussinesq equations whose gradients grow exponentially in time ...

The 2D Incompressible Boussinesq Equations

Explicit Solutions of Generalized Nonlinear Boussinesq Equations

By considering the Adomian decomposition scheme, we solve a generalized Boussinesq equation. The method does not need linearization or weak nonlinearly assumptions. By using this scheme, the solutions are calculated in the form of a convergent power series with easily computable components. The decomposition series analytic solution of the problem is quickly obtained by observing the existence ...

The 2d Euler-boussinesq Equations with a Logarithmically Supercritical Velocity

This paper establishes the global existence and uniqueness of solutions to a generalized 2D Euler-Boussinesq systems of equations with a logarithmically supercritical velocity.

The 2d Boussinesq-navier-stokes Equations with Logarithmically Supercritical Dissipation

This paper studies the global well-posedness of the initial-value problem for the 2D Boussinesq-Navier-Stokes equations with dissipation given by an operator L that can be defined through both an integral kernel and a Fourier multiplier. When the symbol of L is represented by |ξ| a(|ξ|) with a satisfying lim|ξ|→∞ a(|ξ|) |ξ|σ = 0 for any σ > 0, we obtain the global well-posedness. A special cons...