## Caccioppoli inequality

A Caccioppoli inequality is of the form

$\displaystyle r^p\int_{B_{\frac{r}{2}}(x_0)}|\nabla u|^pdx\leq C\int_{B_r(x_0)}\bigg|u(x)-(u)_{B_{r}(x_0)}\bigg|^pdx\quad\quad (1)$

where $C$ is a constant independent of $r$ and $u$, $1, and $\displaystyle (u)_{B_{r}(x_0)}=\frac{1}{|B_r(x_0)|}\int_{B_r(x_0)}udx$.

Caccioppoli inequality can be considered as the reverse Poincare inequality

$\displaystyle \int_{B_r(x_0)}\bigg|u(x)-(u)_{B_{r}(x_0)}\bigg|^pdx\leq Cr^p\int_{B_{r}(x_0)}|\nabla u|^pdx$

holds for $u\in W^{1,p}(B_r(x_0))$.

Of course $(1)$ does not hold for any $u\in W^{1,p}(B_r(x_0))$, but if $u$ satisfies some second order elliptic equation or certain variational problems, $u$ can satisfies $(1)$.

For example, if $u$ is the weak solution of the divergence form

$\displaystyle \text{div}(A\nabla u)=0$ or $(a^{ij}\partial_ju)_i=0$ in $B_r(x_0)$

here $a^{ij}$ are just bounded measurable and strictly elliptic. $\nu I\leq A\leq \nu^{-1}I$.

Choose a cut off function $\eta\in C^\infty_c(B_r(x_0))$ and $\eta=1$ in $B_{\frac{r}{2}}(x_0)$ and $\displaystyle |\nabla\eta|\leq \frac{4}{r}$. Let $v=\eta^2(u-\overline{u})\in H^1_0(B_r(x_0))$, here we use $\overline{u}=(u)_{B_{r}(x_0)}$ for abbreviation. Then

$\displaystyle \int_{B_r(x_0)}a^{ij}\partial_ju\partial_iv=0$

$\displaystyle \int_{B_r(x_0)} a^{ij}\partial_ju\eta^2\partial_iu=-\int_{B_r(x_0)} 2a^{ij}\partial_ju\eta\nabla\eta(u-\overline{u})$

$\displaystyle \nu\int_{B_r(x_0)}\eta^2|\nabla u|^2dx\leq C\int_{B_r(x_0)} |\partial_ju\eta\nabla\eta(u-\overline{u})|$

$\displaystyle \int_{B_r(x_0)}\eta^2|\nabla u|^2dx\leq C(\nu)\int_{B_r(x_0)}|\nabla\eta(u-\overline{u})|^2dx\leq \frac{C}{r^2}\int_{B_r(x_0)} |u-\overline{u}|^2dx$

Noticing that $\eta=1$ on $B_{\frac{r}{2}(x_0)}$, we can get $(1)$

$\textbf{Remark:}$ If $u$ satisfies $(a^{ij}\partial_ju)_i+b^i\partial_iu=0$, we can also get the same conclusion.

MA6000A: Theory of Partial Differential Equations. Roger Moser

• Hafid Younsi  On October 28, 2013 at 6:03 pm

Dear Professor ;

I am interested in your Website and in the Caccioppoli inequality and in the Navier-Stokes equations.

I write to you for asked if we can apply the Caccioppoli inequality for the unsteady Stokes system on the weak solutions of the unsteady 3D Navier-Stokes system.

Or if there existes anothre type Caccioppoli inequality or reverse Poincare Inequality for the unsteady 3D Navier-Stokes equations.

Best regards,

Hafid

• Sun's World  On October 29, 2013 at 1:32 am

Dear Hafid,

I am not any professor. I am just a graduate student in Rutgers. I can not say anything about Caccioppoli inequality for unsteady 3D Navier-stokes equations. I just began to learn Navier-Stokes equations, mostly on 2D case. But i am interested in your research. May be there is something I can help you. If you possible, can you give me some references to read?