Take the real $L^2$ <inner product> of the equation with $u$. The homogeneous <Dirichlet boundary conditions> and <integration by parts> give
$$
\begin{aligned}
\frac12\frac d{dt}\|u(t)\|_{L^2}^2
&=\int_{-1}^1u u_{xx}\,dx+\alpha\int_{-1}^1u u_x\,dx\\
&=-\int_{-1}^1|u_x|^2\,dx
+\frac\alpha2[u^2]_{-1}^{1}\\
&=-\|u_x\|_{L^2}^2\leq0.
\end{aligned}
$$
Thus $\|u(t)\|_{L^2}\leq\|u_0\|_{L^2}$, and applying the same estimate to the difference of two solutions gives continuous dependence on the initial data. The constant-advection term is <skew-symmetric matrix>[skew-symmetric] under these boundary conditions and contributes no energy. Hence
$$
\boxed{\text{the problem is well posed in }L^2\text{ for every }\alpha\in\mathbb R.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 341 / 4 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-341/4/a/solution