## Special inequality of O. A. Ladyzhenskaya

$\mathbf{Problem:}$ If $u=u(x,,y)$ is a smooth function of compact support in $\mathbb{R}^2$, then

$\displaystyle \int_{\mathbb{R}^2}u^4(x,y)dxdy\leq 2\int_{\mathbb{R}}u^2(x,y)dxdy\int_{\mathbb{R}^2}|\nabla u|^2dxdy$

$\mathbf{Proof:}$ Because $\displaystyle u^2(x,y)=2\int_{-\infty}^x uu_xdx=2\int_{-\infty}^y uu_ydy$

we have

$\displaystyle \max\limits_{x}u^2(x,y)\leq 2 \int_{-\infty}^\infty |uu_x|dx$

$\displaystyle \max\limits_{y}u^2(x,y)\leq 2 \int_{-\infty}^\infty |uu_y|dy$

Using the Schwarz’s inequality

$\displaystyle \int_{\mathbb{R}^2}u^4dxdy\leq \int_{-\infty}^{\infty}\max\limits_{x}u^2dy\int_{-\infty}^{\infty}\max\limits_{y}u^2dx$

$\displaystyle \leq 4\int_{\mathbb{R}^2}|uu_x|dxdy\int_{\mathbb{R}^2}|uu_y|dxdy$

$\displaystyle \leq 2\int_{\mathbb{R}^2}u^2dxdy\int_{\mathbb{R}^2}|\nabla u|^2dxdy$

$\text{Q.E.D}\hfill \square$

$\mathbf{Remark:}$ From this inequality and approximation argument, we know $W^{1,2}(\mathbb{R}^2)\hookrightarrow L^4(\mathbb{R}^2)$.

Excerpt from Mathematical theory of viscous incompressible flow. Ladyzhenskaya. Also it contains a similar inequality in $\mathbb{R}^3$.