MathDB

Let

\varepsilon

be positive constant and

u

satisfies that

\begin{cases} (\partial_t-\varepsilon\partial_x^2-\partial_y^2)u=0, & (t,x,y) \in \mathbb{R}_+ \times \mathbb{R} \times \mathbb{R}_+,\\ \partial_y u\vert_{y=0}=\partial_x h, &\\u\vert_{t=0}=0. & \end{cases}

Here

h(t,x)

is a smooth Schwartz function. Define the operator

e^{a\langle D\rangle}

\mathcal{F}_x(e^{a\langle D\rangle} f)(k)=e^{a\langle k\rangle} \mathcal{F}_x(f)(k), \langle k\rangle=1+\vert k\vert, where

\mathcal{F}_x

stands for the Fourier transform in

x

. Show that

\int_0^T \|e^{(1-s)\langle D\rangle} u\|_{L_{x,y}^2}^2 ds \le C \int_0^T \|e^{(1-s)\langle D\rangle} h\|_{H_x^{\frac{1}{4}}}^2 ds

with constant

C

independent of

\varepsilon, T

and

h

Problem Statement