Dirichlet convection-diffusion contraction ($u_t=u_{xx}+2\kappa u_x$)

For zero endpoint values and real constant drift, integration by parts gives $\frac{1}{2}d\Vert u{\Vert }_{L^2}^2/dt=-\Vert u_x{\Vert }_{L^2}^2$. The drift contributes only a zero boundary term in the real part. Thus the solution is contractive, uniformly in the drift. Multiplication by $e^{\kappa x}$ converts the equation to a heat equation with constant killing rate ${\kappa }^2$, providing existence and uniqueness as well as the direct energy estimate.