I'm doing a module on fluid mechanics currently and have been given the following question to solve:
$$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} - \nu \nabla^2 \mathbf{u} + \nabla p = -e^t\mathbf{u} + \Omega e^3 \times u \qquad (1)$$
with $\nabla \cdot \mathbf{u}=0, \nu > 0$ and $\Omega \in \mathbb{R}$ where $e^3 = (0,0,1)^T$.
(a) Let $\mathbf{u}$ be a smooth solution of (1) with corresponding pressure p that decays (along with derivatives) sufficiently rapidly at infinity. Using energy methods, show that the kinetic energy $E(t) := \int_{\mathbb{R}}|\mathbf{u}(x,t)|^2dx$ decays exponentially in time as $t \rightarrow \infty$.
(b) Let $\mathbf{u}$ and $\mathbf{v}$ be two smooth solutions of (1) with corresponding pressures $p$ and $q$ respectively, all of whom decay (along with derivatives) efficiently rapidly at infinity, and suppose $\mathbf{u}$ and $\mathbf{v}$ have the same initial data $u_0 \in \mathbf{L}^2(\mathbb{R}^3)$, i.e. $ u(x,0) \equiv v(x,0) \equiv u_0(x)$. Using energy methods, prove that $u = v$ for all time (i.e. that smooth solutions of (1) are unique).
At this point I'm not entirely sure how to start this question. For instance, with (a), I assume that we try to use equation (1) to find some information about our solution $u$ which we can then use to show the exponential decay of $E(t). However, I don't understand how exactly the information from (1) can become useful to this end.
This is my first post here, so I apologise if it isn't formatted correctly/isn't appropriate. Any help is appreciated. Thank you!