Bootstrapping and Nonlocal porous medium equation: Difference between pages

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Bootstrapping is one of the simplest methods to prove regularity of a nonlinear equation. The general idea is described below.
The nonlocal porous medium equation of order $\sigma$ is the name currently given to two very different equations, namely


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.
\[ u_t = \nabla \cdot \left ( u \nabla \mathcal{K_\alpha} (u) \right )\]


This is the most elementary example of [[perturbation methods]].
\[\mbox{ where } \mathcal{K}_\alpha(u) := C_{n,\alpha}\; u * |x|^{-n+\alpha},\;\; \alpha+2=\sigma \]


== Examples ==
and


=== A simple example ===
\[ u_t +(-\Delta)^{s}(u^m) = 0 \]


Imagine that we have a general semilinear equation of the form
These equations agree when $s=1$ and $m=2$, otherwise they are not only different superficially, they also exhibit extremely different behaviors. They are both fractional order [[Quasilinear equations]].  
\[ u_t + (-\Delta)^s u = H(u,Du). \]
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 particularly simple because the only estimates used are an assumption that $u$ is Lipschitz and then only the classical estimates for the fractional heat equation. This is a common situation when the equation is semilinear and the a priori estimate or assumption on the solution has subcritical scaling.
The first of the two has the remarkable property (for nonlocal equations at least) that any initial data with compact support remains with compact support for all later times, the opposite is true of the second equation, for which [[instantaneous speed of propagation]] holds.  


=== A slightly more complicated example ===
This means that the first model presents us with a [[free boundary problem]]. For this model  global existence and Hölder continuity of weak solutions have been recently obtained <ref name="CV1"/>, there is almost nothing known about the properties of its free boundary, this provides with a rich list of good open problems.


Imagine now that we have a fractional conservation law of the form
For the second equation, both the Cauchy problem and long time behavior have been extensively studied in recent years (references).
\[ u_t + (-\Delta)^s u + \mathrm{div} \ F(\nabla u) = 0. \]
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 by the 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 considered to be automatic once we have an priori estimate which is sufficient for a stronger regularity result for linear equations with coefficients.
== References ==
{{reflist|refs=
<ref name="CV1"> {{Citation | last1=Caffarelli | first1=Luis | last2=Vazquez | first2=Juan | title=Nonlinear Porous Medium Flow with Fractional Potential Pressure | url=http://dx.doi.org/10.1007/s00205-011-0420-4 | publisher=Springer Berlin / Heidelberg | year=2011 | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | pages=1–29}} </ref>
}}
 
[[Category:Quasilinear equations]] [[Category:Evolution equations]] [[Category:Free boundary problems]]

Revision as of 17:56, 3 June 2011

The nonlocal porous medium equation of order $\sigma$ is the name currently given to two very different equations, namely

\[ u_t = \nabla \cdot \left ( u \nabla \mathcal{K_\alpha} (u) \right )\]

\[\mbox{ where } \mathcal{K}_\alpha(u) := C_{n,\alpha}\; u * |x|^{-n+\alpha},\;\; \alpha+2=\sigma \]

and

\[ u_t +(-\Delta)^{s}(u^m) = 0 \]

These equations agree when $s=1$ and $m=2$, otherwise they are not only different superficially, they also exhibit extremely different behaviors. They are both fractional order Quasilinear equations.

The first of the two has the remarkable property (for nonlocal equations at least) that any initial data with compact support remains with compact support for all later times, the opposite is true of the second equation, for which instantaneous speed of propagation holds.

This means that the first model presents us with a free boundary problem. For this model global existence and Hölder continuity of weak solutions have been recently obtained [1], there is almost nothing known about the properties of its free boundary, this provides with a rich list of good open problems.

For the second equation, both the Cauchy problem and long time behavior have been extensively studied in recent years (references).

References

  1. Caffarelli, Luis; Vazquez, Juan (2011), "Nonlinear Porous Medium Flow with Fractional Potential Pressure", Archive for Rational Mechanics and Analysis (Springer Berlin / Heidelberg): 1–29, ISSN 0003-9527, http://dx.doi.org/10.1007/s00205-011-0420-4