Active scalar equation

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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.

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.

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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 $s \in [0,1]$.[1]

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 $u = \nabla^\perp \Delta^{10} \theta$.

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.

References

  1. Chae, D.; Gancedo, F.; Córdoba, D.; Constantin, Peter; Wu, Jun (2011), "Generalized surface quasi-geostrophic equations with singular velocities", Arxiv preprint arXiv:1101.3537 
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