{"schema_version":"dataset-v1","id":"navier-stokes-critical-scaling","version":1,"title":"Where smoothness is guaranteed","description":"Mixed norms L^p in time and L^q in space, plotted at (3/q, 2/p). Under the Navier–Stokes scaling u_λ(t,x) = λu(λ²t, λx), the norm of L^p_t L^q_x scales like λ^{1 − 2/p − 3/q}, so 2/p + 3/q = 1 is scale-invariant. Exact exponent pairs along that line and along 2/p + 3/q = 3/2, where Leray–Hopf weak solutions are bounded.","license":"CC0-1.0 (exact values)","problems":["navier-stokes"],"statements":[{"problem":"navier-stokes","statement":"navier-stokes-prodi-serrin","role":"Controlling a norm at or below the line 2/p + 3/q = 1 gives smoothness."},{"problem":"navier-stokes","statement":"navier-stokes-l3-criterion","role":"The endpoint L∞L³: a bounded L³ norm rules out blowup."},{"problem":"navier-stokes","statement":"navier-stokes-leray-hopf-bound","role":"Every Leray–Hopf weak solution controls the norms on 2/p + 3/q = 3/2."},{"problem":"navier-stokes","statement":"navier-stokes-unforced-r3","role":"Closing the gap for all smooth data would answer alternative (A)."}],"figure":{"family":"region-plot","title":"Where smoothness is guaranteed","caption":"A solution bounded in a norm on or below the scale-invariant line is smooth. Every weak solution is only known to be bounded on the higher line. The band between them is the gap: no general solution is known to have any norm there.","x_label":"3/q (space)","y_label":"2/p (time)","x_range":[0,1.5],"y_range":[0,1],"y_scale":"linear","x_ticks":[[0,"0 · q = ∞"],[0.5,"½ · q = 6"],[1,"1 · q = 3"],[1.5,"3/2 · q = 2"]],"y_ticks":[[0,"0 · p = ∞"],[0.5,"½ · p = 4"],[1,"1 · p = 2"]],"layers":[{"key":"gap","label":"Open: the gap between what is controlled and what is needed","kind":"polygon","status":"open","statement":"navier-stokes-unforced-r3","x":null,"y_max":null,"series":null,"points":[[1,0],[1.5,0],[0.5,1],[0,1]]},{"key":"regularity","label":"Proved: bounded here ⇒ smooth (Ladyzhenskaya–Prodi–Serrin)","kind":"polygon","status":"proved","statement":"navier-stokes-prodi-serrin","x":null,"y_max":null,"series":null,"points":[[0,0],[1,0],[0,1]]},{"key":"energy","label":"Proved: every weak solution is bounded here (energy)","kind":"segment","status":"proved","statement":"navier-stokes-leray-hopf-bound","x":null,"y_max":null,"series":null,"points":[[0.5,1],[1.5,0]]},{"key":"endpoint","label":"Proved: L∞L³ endpoint (Escauriaza–Seregin–Šverák)","kind":"point","status":"proved","statement":"navier-stokes-l3-criterion","x":null,"y_max":null,"series":null,"points":[[1,0]]}],"supersedes":null},"provenance":{"method":"Exact rational exponent pairs (3/q, 2/p) on 2/p + 3/q = 1 for q ∈ {∞, 12, 6, 4, 10/3, 3} and on 2/p + 3/q = 3/2 for q ∈ {6, 5, 4, 3, 5/2, 2}; layers are the regions those statements describe.","software":{"python":"3.12.13"},"precision":"Exact rationals, stored as floating point.","generator":"openmath/scripts/build-datasets.py","generated_on":"2026-10-21"},"data":{"series":[{"key":"critical_line","label":"2/p + 3/q = 1 (scale-invariant)","x":[0,0.25,0.5,0.75,0.9,1],"y":[1,0.75,0.5,0.25,0.1,0]},{"key":"energy_line","label":"2/p + 3/q = 3/2 (Leray–Hopf bounds)","x":[0.5,0.6,0.75,1,1.2,1.5],"y":[1,0.9,0.75,0.5,0.3,0]}],"marks":[]},"sha256":"0db7ec759c0134d021228e50123adf657da2bd661c52eb64b09bfa82ee2f23e7"}