Asymptotic summation preserving divergence freedom

Statement

On a space-time domain in $\mathbb R^3\times\mathbb R$ , let $q>0$ satisfy the hypotheses of shrinking-cutoff summation. Suppose smooth representatives $A_j,B_j$ satisfy its increasing-order derivative estimates and

\nabla_x\cdot B_j=0,\qquad B_j\cdot\nabla_xq=0.

For a smooth divergence-free base $u_0$ , cutoff scales can be chosen so that

u=u_0+\sum_{j\ge1}\left(\nabla_x\times(\chi(a_jq)A_j)+\chi(a_jq)B_j\right)

is smooth, locally finite, and divergence-free. Its difference from the uncut finite states has the same kind of increasing-order tail estimate, with a derivative loss independent of the stage.

Constraint and estimates

Each curl has zero divergence. The direct term has divergence $\chi\nabla\cdot B_j+a_j\chi'(a_jq)\nabla q\cdot B_j=0$ . The curl product rule adds $a_j\chi'(a_jq)\nabla q\times A_j$ ; estimates for the potential through one extra derivative control this term. At stage $j$ , impose the finite list of output derivative bounds through order $j$ before choosing $a_j$ . The diagonal and tail arguments then proceed as for scalar summation.

Geometry and boundaries

An axisymmetric azimuthal field $B_j=b_j(r,z,t)e_\theta$ satisfies these two constraints when $q$ is independent of the angle and the field has a smooth Cartesian representative. Smooth zero extensions across lateral support boundaries are hypotheses on the representatives. Multiplying velocities by cutoffs after taking curl would generally introduce a divergence defect.