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.

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