Energy methods: uniqueness, stability and decay

An energy method multiplies the equation by the solution (or its time derivative), integrates over space, and integrates by parts to get an identity for the rate of change of a positive quantity. For the heat equation \( u_t = k\,u_{xx} \) on \( (0, L) \) with \( u = 0 \) at both ends: \[ \frac{d}{dt}\,\frac12\int_0^Lu^2\,dx = \int_0^Luu_t\,dx = k\int_0^Luu_{xx}\,dx = -k\int_0^Lu_x^2\,dx \le 0. \] For \( u = e^{-t}\sin x \) on \( (0, \pi) \), \( k = 1 \), the example gives \( \int_0^\pi u^2 = \frac\pi2e^{-2t} \), whose half has derivative \( -\frac\pi2e^{-2t} = -\int_0^\pi u_x^2 \), as it must.

Uniqueness and stability. The difference \( w \) of two solutions with initial data \( f_1, f_2 \) solves the same problem, so \( \int w(x,t)^2\,dx \le \int(f_1 - f_2)^2\,dx \) for all \( t \ge 0 \). Equal data give equal solutions; nearby data (in mean square) give nearby solutions. The same argument proves uniqueness for Poisson's equation (\( \int|\nabla w|^2 = 0 \) forces \( w \) constant, hence zero) and, with \( E = \frac12\int(u_t^2 + c^2u_x^2) \), for the wave equation.

Decay. The Poincaré inequality \( \int_0^Lu^2\,dx \le \frac{L^2}{\pi^2}\int_0^Lu_x^2\,dx \) (for \( u(0) = u(L) = 0 \), with equality for \( \sin(\pi x/L) \)) turns the identity into \( \frac{d}{dt}\int u^2 \le -\frac{2k\pi^2}{L^2}\int u^2 \), and Gronwall's inequality gives \( \int u^2 \le e^{-2k\pi^2t/L^2}\int f^2 \): exponential decay at the rate of the first eigenvalue, with no series used.

Picture it: energy is a ball rolling downhill in the space of all temperature profiles. The identity says it always rolls down; Poincaré says the slope is never gentler than a fixed amount, so it reaches the bottom exponentially fast.

Think it: for Navier-Stokes, multiply by \( u \) and integrate. The pressure term vanishes because \( \nabla\cdot u = 0 \), and so does the nonlinear term, \( \int(u\cdot\nabla)u\cdot u\,dx = \int u\cdot\nabla\big(\tfrac12|u|^2\big)\,dx = 0 \). What is left is the energy inequality \( \frac12\|u(t)\|_{L^2}^2 + \nu\int_0^t\|\nabla u\|_{L^2}^2\,ds \le \frac12\|u_0\|_{L^2}^2 \). It holds for all finite-energy data and all time. Other global bounds can be derived from it, but in three dimensions every such bound is supercritical in the sense of the scaling lesson.