This question is from Tao's Nonlinear Dispersive Equation, Exercise 2.55.
Show that smooth solutions $u\in C_{t,\text{loc}}^\infty \mathcal{S}_x(\mathbb{R}\times\mathbb{R}\to\mathbb{R})$ to the Airy equation $\partial_tu+\partial_x^3u=0$ obey the estimation $$\int_0^T \int_{|x|\leq R}u_x^2dxdt\lesssim \frac{T+R^3}{R}\int_\mathbb{R}u(0,x)^2dx$$ for all $T>0$ and $R>0$.
My attempt: I followed the hint presented in the problem; I established the conservation law $\partial_t(u^2)=-\partial_x^3(u^2)+3\partial_x(u_x^2)$ by calculation, and obtained the $L_x^2$-conservation law by integrating both sides wrt $x$. Also, the hint in the book says that I should integrate the conservation law against a suitable cutoff function which equals 1 for $x>2R$, zero for $x<-2R,$ and increases steadily for $-R<x<R$. However, when I did the integration and used integration by parts, a lot of messy dummy terms not useful for the estimation appears. How should I find a 'suitable' cutoff function here?
Thanks in advance!
Let $\phi$ be a smooth cutoff function which is zero for $x<-2$, 1 for $x>2,$ and steadily increasing in $-2<x<2.$ Then observe that $\phi'(x)>0$ in $[-1,1],$ so let $c=\min_{-1\leq x\leq 1}\phi'(x)>0.$
Now multiplying the both sides of the conservation law by $\phi_R(x)=\phi(x/R)$ and integrating in space give \begin{align*} \partial_t\int_\mathbf{R}u(x,t)^2\phi(x/R)dx&=-\int_\mathbf{R}\partial_x^3(u^2)\phi(x/R)dx+3\int_\mathbf{R}\partial_x(u_x^2)\phi(x/R)dx\\&=\frac{1}{R^3}\int_\mathbf{R}u(x,t)^2\phi'''(x/R)dx-\frac{3}{R}\int_\mathbf{R}u_x(x,t)^2\phi'(x/R)dx. \end{align*}
Observe that $$\int_\mathbf{R}u_x^2\phi'(x/R)dx\geq\int_{-R}^Ru_x^2\phi'(x/R)dx\geq c\int_{-R}^R u_x^2dx.$$
Therefore,\begin{align*} \int_0^T\int_{|x|\leq R}u_x^2dx&\lesssim R\int_\mathbf{R}u(x,0)^2\phi(x/R)dx-R\int_\mathbf{R}u(x,T)^2\phi(x/R)dx\\&+\frac{1}{R^2}\int_0^T\int_\mathbf{R}u(x,t)^2\phi'''(x/R)dx\\&\lesssim R\int_\mathbb{R}u(x,0)^2dx+\frac{T}{R^2}\int_\mathbf{R}u(x,t)^2dx=\frac{T+R^3}{R^2}\int_\mathbb{R}u(x,0)^2dx \end{align*} where the last equality follows from the $L_x^2$ conservation law. This completes the proof.