A conservative positive-pressure SPH system can close a particle gap

The polynomial kernel is nonnegative and even. At s=±1 its value and first two inside derivatives vanish, so extension by zero is C²; its integral is (35/32)·2(1−1+3/5−1/7)=1. Each density is positive because its self term W_ε(0)>0. The chosen internal energies give p_i=(γ−1)ρ_ie_i=p_0ρ_i²>0 and A_i=p_0 at t=0. By [F1] and [F5], a unique local solution exists; pressure positivity persists for a short interval by continuity.

For 0<s<1, |W′(s)|=(105/16)s(1−s²)². Direct substitution gives |W′(2/5)|/|W′(9/10)|=28224/3249>2. Also |q_3−q_1|=13ε/10>ε, so the 1–3 kernel-gradient term vanishes.

Let r(t)=q_2(t)−q_1(t). From [F1] and step 1.2, the accelerations at t=0 are q̈_1=−2p_0ε^(−2)|W′(9/10)| and q̈_2=2p_0ε^(−2)(|W′(9/10)|−|W′(2/5)|). Therefore r̈(0)=2p_0ε^(−2)(2|W′(9/10)|−|W′(2/5)|)<0.

Since ṙ(0)=0 and r̈ is continuous, ṙ(t)<0 and r(t)<r(0) for all sufficiently small t>0. Thus the adjacent gap closes initially under strictly positive pressure.

The exact pair antisymmetry and matched internal-energy identity apply by [F2] and [F4], so total momentum and H are conserved throughout this same local motion. The example therefore refutes the claimed gap-stability implication while retaining those invariants.