Active scalar equation

A general class of equations is often referred to as active scalars. It consists of solving the Cauchy problem for the transport equation \begin{align} \theta(x,t) &= \theta_0(x) \\ \partial_t \theta + u \cdot \nabla \theta &= 0 \end{align} where the vector field $u$ is related to $\theta$ by some operator.

There exists no differential operator which maps a scalar function $\theta$ to a divergence free vector field $u$, which commutes with translations and rotations, for which the equation is known to develop singularities in finite time. This is an open problem even for high order operators like $u = \nabla^\perp \Delta^{10} \theta$.

The case $u = \nabla^\perp (-\Delta)^{-1} \theta$, in two space dimensions, corresponds to the vorticity formulation of the 2D Euler equation. The case $u = \nabla^\perp (-\Delta)^{-1/2} \theta$, in two space dimensions, corresponds to the inviscid surface quasi-geostrophic equation. If we consider the full range of exponents $u = \nabla^\perp (-\Delta)^{-s} \theta$, the equation is known to be well posed in the classical sense if $s \geq 1$. For any $s<1$, the possible break down of classical solutions in finite time is an open problem.

As a model for the equations of fluid dynamics, the vector field $u$ is taken to be divergence free. The opposite case when $u$ is the gradient of a potential (or strictly speaking its dual equation) is studied in the context of the aggregation equation and the nonlocal porous medium equation.

General properties

In two space dimensions, under the general choice $u = \nabla^\perp (-\Delta)^{-s} \theta$, the vector field $u$ is divergence free. Therefore the transport equation enjoys all properties of divergence free flows: all $L^p$ norms of $\theta$ are conserved, the distribution function of $\theta$ is conserved, the set of points where the trajectories cross has measure zero, etc...

On the other hand, the following $s$-dependent energy is preserved by the flow: \[||\theta||_{\dot H^{-s}}^2 = \int \theta (-\Delta)^{-s} \theta \ dx.\]

The equation reduces to 2D Euler if $s=1$, and to inviscid SQG if $s=1/2$. The operator determining the velocity is more singular the smaller $s$ is. On the other hand, the conserved energy becomes a stronger quantity. The problem is known to be locally well posed for $s \in [0,1]$.

2D Euler (well posedness)

The usual Euler equation refers to the system \begin{align} \partial_t u + u \cdot \nabla u &= -\nabla p \\ \mathrm{div} \ u &= 0 \end{align} where $u$ is a vector valued function and $p$ is a scalar function.

In 2D, the vorticity $\omega(x,y) = \partial_x u_2 - \partial_y u_1$ satisfies the active scalar equation \[ \partial_t \omega + u \cdot \nabla \omega = 0 \] where $u = \nabla^\perp (-\Delta)^{-1} \omega$.

The equation is (borderline) well posed for the following reason. The $L^\infty$ norm of $\omega$ is clearly preserved since it is a transport equation. In order to obtain higher regularity estimates on $\omega$ we need to estimate the rate by which the trajectories of the flow by $u$ approach each other. The most usual way to do this is by estimating the Lipschitz norm of $u$. The fact that $\omega \in L^\infty$ uniformly in time does not immediately imply that $u$ is Lipschitz. Instead it implies the borderline weaker condition $u \in LogLip$. Thus, in particular $u$ satisfies the Osgood condition and the flow trajectories are uniquely defined. From this property of the flow one can easily derive higher regularity estimates for $\omega$ that grow doubly exponentially in time.

Inviscid surface quasi-geostrophic equation

The inviscid SQG equation corresponds to the choice $u = \nabla^{\perp} (-\Delta)^{-1/2} \theta$. In this case the velocity is given by an operator of order zero applied to $\theta$, which always gives a divergence free drift. From the $L^\infty$ a priori estimate on $\theta$, the vector field $u$ stays bounded in $BMO$.

This is a case which attracts a lot of interest. The known results coincide with the general case of active scalar equations for $s$ in the range $(0,1)$. The classical well posedness of the equation for large time is still an open problem.