State the Lax-Milgram lemma.

Let $\mathbf{V}=\mathbf{V}\left(x_{1}, x_{2}, x_{3}\right)$ be a smooth vector field which is $2 \pi$-periodic in each coordinate $x_{j}$ for $j=1,2,3$. Write down the definition of a weak $H_{p e r}^{1}$ solution for the equation

$$-\Delta u+\sum_{j} V_{j} \partial_{j} u+u=f$$

to be solved for $u=u\left(x_{1}, x_{2}, x_{3}\right)$ given $f=f\left(x_{1}, x_{2}, x_{3}\right)$ in $H^{0}$, with both $u$ and $f$ also $2 \pi$-periodic in each co-ordinate. [In this question use the definition

$$H_{p e r}^{s}=\left\{u=\sum_{m \in \mathbb{Z}^{3}} \hat{u}(m) e^{i m \cdot x} \in L^{2}:\|u\|_{H^{s}}^{2}=\sum_{m \in \mathbb{Z}^{3}}\left(1+\|m\|^{2}\right)^{s}|\hat{u}(m)|^{2}<\infty\right\}$$

for the Sobolev spaces of functions $2 \pi$-periodic in each coordinate $x_{j}$ and for $\left.s=0,1,2, \ldots\right]$

If the vector field is divergence-free, prove that there exists a unique weak $H_{p e r}^{1}$ solution for all such $f$.

Supposing that $\mathbf{V}$ is the constant vector field with components $(1,0,0)$, write down the solution of $(*)$ in terms of Fourier series and show that there exists $C>0$ such that

$$\|u\|_{H^{2}} \leqslant C\|f\|_{H^{0}}$$