Bootstrapping

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 either 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 one of the simplest examples of a perturbation method.

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. Assume that a solution $u$ is known to be Lipschitz and $s>1/2$. 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*} whose solutions are $C^{2s}$ 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 process, we obtain that $u \in C^\infty$.