Now for any (phi in G) there exists a (m > 0) such that begin{aligned} phi =phi _{n_1}circ phi _{n_2}circ cdots circ phi _{n_m} end{aligned}where (1 le n_j le n) for every (1 le j le m).
A morphism from (langle [n'],i'_cdot rangle ) to (langle [n],i_cdot rangle ) in ((Delta ^L_M I ^perp ) is given by a map (f:[n] rightarrow [n']) in (Delta ) and a collection of maps (l_j:i'_{f j)} rightarrow i_j), (0 le j le n) in the class L such that the diagram Open image in new window commutes for every (0 le j le j' le n).
end{aligned} (12 Moreover, the a priori estimate begin{aligned} left| psi _{t, varphi }(s;,cdot,) right| _{L^q} ; le ; left| varphi right| _{L^q}, cdot, e^{ Vert [{mathrm{ div}}_{mathbf x}, {mathbf g}]^- Vert _{L^infty }+ Vert u Vert _{L^infty }),cdot,t}, end{aligned} (13)holds for every (0 le s le t le T) and (varphi in C^1_c({mathbb R}^n)).
+ Vert u - widehat{u}Vert _{L^q} big ) end{aligned}hold for every (0 le s le t le T) and (varphi in C^1_c({{mathbb R}^{N}})) with a constant C which depends on (N), p, ( Vert partial _{mathbf x}, {mathbf g}Vert _{L^infty }), ( Vert partial _{mathbf x}, widehat{{mathbf g}}Vert _{L^infty }), ( Vert nabla _{mathbf x}, uVert _{L^infty }) and ( Vert nabla _{mathbf x}, widehat{u}Vert _{L^infty }).
Since (Z_{d-1} y_i,y'_i)) and (D^mathcal {I}(y_i)) hold for every (1 le i le h), by the induction assumption, it follows that (D^{mathcal {I}'}(y'_i)) holds for every (1 le i le h).
Since (phi _i(Omega _G) subset bar{Omega }_{phi _i}), (phi _i(p) in Omega _{phi _i}) for every (1 le i le n).
Let (G= langle phi _1,phi _2, ldots,phi _m rangle) where each (phi _i in mathcal {E}_k) for every (1 le i le m).
A partition (mathbb {Y}= {Y_1, Y_2, ldots,Y_n}) is consistent with a set (X) if, for every (1 le i le n), (Y_i) is not split by (X).
Let (G=langle phi _1,phi _2,ldots,phi _mrangle) where each (phi _i in mathcal {E}_k) for every (1 le i le m).
A partition (P = {Y_1,ldots,Y_n}) is consistent with a set X if, for every (1 le i le n), (Y_i) is not split by X.
