eldorado.tu-dortmund.de/server/api/core/bitstreams/21a28a31-aee2-4ab7-b656-32a29ff2217c/content
t)‖q−1 Lq(Γ)
≤ c8
(
‖F (·, t)‖Lq(Γ) + ‖w(·, t)‖Lq(Γ×(0,δ)) + ‖w0‖C2(Γ)
)
‖w(·, t)‖q−1 Lq(Γ)
≤ c8
(
‖w(·, t)‖q Lq(Γ) +
(
‖F (·, t)‖Lq(Γ) + ‖w0‖C2(Γ)
)
‖w(·, t)‖q−1 Lq(Γ)
)
≤ c8‖w(·, t)‖q Lq(Γ) + c8
(
‖F (·, t)‖q [...] t)‖q Lq(Γ) + ‖w0‖q
C2(Γ)
)
, (2.39)
where c8 = c8(q) > 0 changes from line to line and we used Hölder’s and Young’s In- equality. We apply Gronwall’s Lemma A.9 to (2.39) to find a bound for the Lq-norm [...] Lions-Aubin’s Lemma 3.7 to (Vh, uh, vh) to show Lemma 3.9 (ii). With Lemma 3.8 (ii) we find that (Vh, uh, vh) ∈ L2M1. With Lemma 3.8 (ii) and (iii) we obtain
Vh ∈ H1(0, T ; (H1(Ω))∗) and uh, vh ∈ H1(0, T ; …