Energy dissipation and uniqueness

Consider uₜ=κΔu with κ>0 on a bounded smooth domain and homogeneous Dirichlet data. For a sufficiently regular solution, multiply by u and integrate by parts. The boundary term vanishes because u has zero trace. The derivative of half the squared L² norm equals minus κ times the squared gradient norm. With homogeneous Neumann data, the same boundary term vanishes because the normal derivative is zero. Periodic data also cancel paired boundary contributions. Nonhomogeneous conditions generally contribute work or heat exchange and must be retained.

The identity extends to the weak energy class through time regularization or the Hilbert-space chain rule. In that setting the squared L² norm is absolutely continuous and its derivative is the dual pairing with uₜ almost everywhere. It is not necessary to assert that u is classically differentiable at every time. For bounded measurable symmetric uniformly elliptic divergence coefficients, the dissipation becomes the integral of ∇u dotted with A∇u, still bounded below by the ellipticity constant times gradient energy.

If two weak solutions share the same forcing, boundary data, and initial trace, their difference w solves the homogeneous equation with zero data. Its energy is nonincreasing and initially zero. Since energy is nonnegative, it remains zero, so the solutions agree in the energy space. This argument does not require a heat-kernel expression or an eigenfunction expansion. It also explains continuous dependence: a nonzero initial difference cannot grow in L² for this homogeneous linear diffusion problem.

If the forcing differs, pair the forcing difference with w and use a dual norm. With the H⁻¹ norm dual to the H₀¹ gradient norm, Young's inequality absorbs half the gradient term and leaves a controlled forcing contribution. Integrating yields a stability bound. For equations with lower-order terms, additional estimates and a Gronwall factor may appear. An estimate for one particular equation should not be generalized unchanged to a reaction term capable of amplifying solutions. Uniqueness concerns a specified solution class and prescribed data.

On a bounded domain with homogeneous Dirichlet data, Poincaré's inequality bounds the squared gradient norm below by λ₁ times the squared solution norm, where λ₁>0 is the first Dirichlet Laplacian eigenvalue. The energy identity then gives E(t)≤E(0)exp(−2κλ₁t). The L² norm itself decays with exponent κλ₁. Confusing squared norm and norm creates a factor-of-two error. This rate is sharp when the initial profile is a first eigenfunction.

For Neumann or periodic flow, a constant profile has zero gradient and does not decay. The mean is conserved, so apply Poincaré only after subtracting that mean, on a connected bounded domain or a connected periodic torus. A disconnected insulated domain conserves a separate mean on each component; one global average is insufficient to remove every zero mode. Energy dissipation alone says that variation costs energy. Identifying the equilibrium and proving a positive spectral gap are the extra steps needed to quantify convergence toward it.