Drift-diffusion equations

A drift-(fractional)diffusion equation refers to an evolution equation of the form \[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0,\] where $b$ is any vector field. The stationary version can also be of interest \[ b \cdot \nabla u + (-\Delta)^s u = 0.\]

This type of equations appear under several contexts. It is often useful to apply regularity results about drift-diffusion equations to semilinear equations from fluid dynamics (for example the surface quasi-geostrophic equations). For this reason, the assumption that $\mathrm{div} \ b = 0$ is taken sometimes. The equation can also be derived as the flow of a vector field with $\alpha$-stable white noise.

There are a number of regularity results depending of the assumptions on the vector field $b$ and the values of the exponent $s$. The type of regularity results available for drift-diffusion equations can be separated into two categories depending on whether or not they are proved using perturbation methods. Perturbative results are characterized by an assumption on the vector field $b$ that implies that at small scales its flow is negligible in comparison with fractional diffusion.

Scaling

The terms supercritical, critical, and subcritical are often used to denote whether the diffusion part of the equation controls the regularity or not. Given a quantitative assumption on the vector field $b$, one can check if it is subcritical, critical, or supercritical by checking the effect of scaling. More precisely, we know that the rescaled function $u_\lambda(t,x) = u(\lambda^{2s}t,\lambda x)$ satisfies the equation \[ \partial_t u_\lambda + \lambda^{2s-1} b(\lambda^{2s}t,\lambda x) \cdot \nabla u + (-\Delta)^s u = 0.\]

If an a priori estimate on $b$ improves with the scaling $\lambda^{2s-1} b(\lambda^{2s}t,\lambda x)$ for $\lambda <1$, the equation is in the subcritical regime. Typically, a regularity result for a drift-diffusion equation with subcritical assumptions on $b$ would be obtained using perturbation methods. Sometimes a strong regularity result holds and the solutions are necessarily classical.

If an a priori estimate on $b$ is invariant by the scaling $\lambda^{2s-1} b(\lambda^{2s}t,\lambda x)$, the equation is in the critical regime. There are some regularity results for drift-diffusion equation with critical assumptions on $b$ but the proofs are more delicate and cannot be obtained via perturbation methods.

If an a priori estimate on $b$ deteriorates with the scaling $\lambda^{2s-1} b(\lambda^{2s}t,\lambda x)$ for $\lambda <1$, the equation is in the supercritical regime. There is no regularity result available for any kind of supercritical assumption on $b$. In this case, the transport part of the equation is expected to dominate the equation.

Pertubative results

Kato classes

The Kato class is by definition that the limit of some quantity related to $b$ goes to zero as the scale becomes smaller, effectively making the drift term a small perturbation of the fractional diffusion. The assumption is tailor made in order to make bootstrapping arguments possible. Using this idea, it has been shown that the corresponding heat kernel is comparable to the heat kernel of the fractional heat equation .

$C^{1,\alpha}$ estimates

Under subcritical assumptions on $b$, one can sometimes show $C^{1,\alpha}$ estimates. These are strong regularity estimates that imply that the solutions are classical.

The result is the following. Assume that $b \in L^\infty(C^{1-2s+\alpha})$ for some $\alpha>0$, then the solution $u$ belongs to $L^\infty(C^{1,\alpha})$. This result has been proved for the case $s \leq 1/2$ , and it has also been proved for a stationary eigenvalue problem if $s \geq 1/2$ .

Scale invariant results

Divergence-free vector fields

If the vector field $b$ is divergence free, the method of De Giorgi-Nash-Moser can be adapted to show that the solution $u$ becomes immediately Holder continuous. This is a variation of the Holder estimates for integro-differential equations in variational form.

In the case $s=1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(BMO)$ .

In the case $s<1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(C^{1-2s})$ .

Vector fields with arbitrary divergence

For any bounded vector field $b$, one method for obtaining Holder estimates for integro-differential equations can be used to show that $u$ becomes immediately Holder continuous.

In the case $s=1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(L^\infty)$ , which is a marginally stronger assumption than the one needed if $b$ was assumed divergence-free.

In the case $s<1/2$, the vector field $b$ is required to belong to the scale invariant class $L^\infty(C^{1-2s})$ , which is the same assumption than the one needed if $b$ was assumed divergence-free. Therefore, if $s<1/2$, there is no known advantage in assuming $\mathrm{div} \ b =0$.