= Dirichlet convection-diffusion contraction
{c}
{title2=$u_t=u_{xx}+2\kappa u_x$}
For zero endpoint values and real constant drift, integration by parts gives $\tfrac12 d\|u\|_{L^2}^2/dt=-\|u_x\|_{L^2}^2$. 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>.
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