Bootstrapping

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Bootstrapping is one of the simplest methods to prove regularity of a nonlinear equation. The general idea is described below.

Assume that $u$ is a solution to some nonlinear equation of any kind. By being a solution to the nonlinear equation, it is also a solution to a linear equation whose coefficients depend on $u$. Typically this is some form of linearization of the equation. If an a priori estimate is known on $u$, then that provides some assumption on the coefficients of the linear equation that $u$ satisfies. The linear equation, in turn, may provide a new regularity estimate for the solution $u$. If this regularity estimate is stronger than the original a priori estimate, then we can start over and repeat the process to obtain better and better regularity estimates.

This is the most elementary example of perturbation methods.

Examples

A simple example

Imagine that we have a general semilinear equation of the form \[ u_t + (-\Delta)^s u = H(u,Du). \] Where $H$ is some smooth function and $s \in (1/2,1]$. Assume that a solution $u$ is known to be Lipschitz. Therefore, $u$ coincides with the solution $v$ of the linear equation \begin{align*} v(0,x) &= u(0,x) \\ v_t + (-\Delta)^s v &= H(u,Du). \end{align*} Since the right hand side $H(u,Du)$ is bounded, then the solution v must be $C^{2s}$ in space. Since $2s > 1$, then we improved our regularity on $u$ (which is the same as $v$). But now $H(u,Du) \in C^{2s-1}$ and therefore $v \in C^{4s-1}$. Continuing the iteration, we obtain that $u \in C^\infty$.

The above example is relatively simple because the only estimates used are an assumption that $u$ is Lipschitz and the classical estimates for the fractional heat equation. The bootstrapping method usually works when the equation is semilinear and the a priori estimate or assumption on the solution has subcritical scaling.

A slightly more complicated example

Imagine now that we have a fractional conservation law of the form \[ u_t + (-\Delta)^s u + \mathrm{div} \ F(\nabla u) = 0. \] Where $F$ is some smooth vector valued function and $s \in (0,1/2)$. Assume that a solution $u$ is known to be $C^\alpha$ for some $\alpha>1-2s$. As before, $u$ coincides with the solution $v$ of a linear equation whose coefficients depend on $u$. However, the equation is now more complicated. \begin{align*} v(0,x)&=u(0,x) \\ v_t + (-\Delta)^s v + b(x,t) \cdot \nabla v &= 0 \end{align*} where $b(x,t) = F'(u)$. Since $F$ is smooth and $u \in C^\alpha$ in space, we have that $b \in C^\alpha$ in space, which implies that $v \in C^{1,\alpha}$ in space applying estimates for linear drift-diffusion equations. Therefore $u \in C^{1,\alpha}$. Differentiating the equation and repeating the procedure we get $u \in C^{2,\alpha}$, $u \in C^{3,\alpha}$, etc...

The procedure is slightly more complicated because the linear equation has variable coefficients and a less standard estimate for linear equations is used. Still the outline of the idea is the same. Bootstrap arguments are more or less automatic once we have an priori estimate which is sufficient for a stronger regularity result for linear equations with coefficients.

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