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| Bootstrapping is one of the simplest methods to prove regularity of a nonlinear equation. The general idea is described below.
| | A quasilinear equation is one that is linear in all but the terms involving the highest order derivatives (whether they are of fractional order or not). For instance, the following equations are quasilinear |
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| 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.
| | \[ \mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0, u_t = \mbox{div} \left ( u^p \nabla u\right ), (-\Delta)^{s}+H(x,u,\nabla u)=0 (s>1/2) \] |
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| This is one of the simplest examples of a [[perturbation method]].
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| == Example ==
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| Imagine that we have a general semilinear equation of the form
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| \[ u_t + (-\Delta)^s u = H(u,Du). \] | |
| 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
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| \begin{align*} | |
| v(0,x) &= u(0,x) \\
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| v_t + (-\Delta)^s v &= H(u,Du).
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| \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 $u \in C^\infty$.
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Revision as of 17:08, 3 June 2011
A quasilinear equation is one that is linear in all but the terms involving the highest order derivatives (whether they are of fractional order or not). For instance, the following equations are quasilinear
\[ \mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0, u_t = \mbox{div} \left ( u^p \nabla u\right ), (-\Delta)^{s}+H(x,u,\nabla u)=0 (s>1/2) \]