LocalizedSplitSkew #
Ported from scottnarmstrong/Superdiffusion at 7f20476385d4af418f110303085ad0740c9eba46. Module imports, support namespace and local option directives adapted for Lean Pool.
The weighted energy inequality with a split antisymmetric part #
The antisymmetric part of the coefficient is split as k = ks + kl, with ks
merely bounded and kl continuously differentiable. The two parts are paid
for differently.
- The rough part
ksenters the flux cross term and is bounded there by its size on the transition layer, exactly as in the unsplit estimate. - The smooth part
klis a divergence-free drift: for every zero-trace test the fluxkl grad zpairs as-(test) ⟨div kl, grad z⟩, so the solution of the equation with coefficientaalso solves the equation with coefficientnu • I + ksand the extra forcing⟨div kl, grad z⟩. The cross term this produces is then integrated by parts and carries‖div kl‖rather than‖kl‖.
The resulting energy inequality keeps the drift term as an exact quantity on the left, so no absolute value is committed to before the weight is chosen.
Main results #
The smooth antisymmetric part as a drift #
The smooth antisymmetric part is a drift. A zero-trace solution of the
equation with coefficient a = nu • I + ks + kl, with kl a C¹
antisymmetric field, is a solution of the equation with coefficient
nu • I + ks and the extra forcing ⟨div kl, grad z⟩.
The energy inequality with a split antisymmetric part #
The weighted energy inequality with a split antisymmetric part. The rough part is charged to its size on the layer through the Young inequality; the smooth part appears as the exact drift term on the left, carrying only the divergence of the smooth part and no gradient of the solution.