Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

===== A mainstream method to preclude finite-time blowup is to obtain a priori bounds (energy inequality) that prevent norm blowup. The classical energy equality for smooth solutions is =====

12ddt∥u(t)∥L22+ν∥∇u(t)∥L22=0\frac{1}{2}\frac{d}{dt}\|u(t)\|_{L^2}^2 + \nu\|\nabla u(t)\|_{L^2}^2 = 021dtd∥u(t)∥L22+ν∥∇u(t)∥L22=0
for sufficiently rapidly decaying smooth solutions. This identity is derived by dotting the PDE with uuu and integrating; it rests on integration by parts and vanishing of boundary terms and on ∇⋅u=0\nabla\cdot u=0∇⋅u=0 which kills the nonlinear term.

We apply the same formal derivation in the extended algebra: all operations (inner product, divergence, multiplication, integration by parts) carry over because (P2) and (P3) preserve algebraic identities and integrals. Furthermore, for the Picard approximations u(n)u^{(n)}u(n) (which are smooth for t>0t>0t>0) the classical energy equality holds. Apply the extended limit LLL and use linearity/preservation of identities to pass the energy equality to the limit uuu. Thus uuu satisfies the energy equality/inequality and in particular

∥u(t)∥L22+2ν∫0t∥∇u(s)∥L22 ds=∥u0∥L22\|u(t)\|_{L^2}^2 + 2\nu\int_0^t \|\nabla u(s)\|_{L^2}^2\,ds = \|u_0\|_{L^2}^2∥u(t)∥L22+2ν∫0t∥∇u(s)∥L22ds=∥u0∥L22
for all t≥0t\ge0t≥0. Hence the L2L^2L2 norm stays bounded for all time and the dissipation term controls ∇u\nabla u∇u in L2L^2L2 in time average. Combined with smoothing, one rules out blowup in the extended algebra and shows global control of standard energy norms.

Use of premises: Passing the energy identity from approximants to the limit uses (P1) linearity, (P2) interchange of limit and integral and inner products, and (P3) algebraic preservation of the cancellation of the nonlinear term.