Degenerate parabolic stochastic partial differential equations

where
Z(s) =
M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336 4321
1
Γ (1 − α)
s
0
(s − r) −α Φ ε u ε (r) dW (r).
Therefore using the Burkholder–Davis–Gundy and the Young inequality and the estimate (2), we
have
·
E
Φ ε (u ε q
) dW
≤ C E∥Z∥ q L q (0,T ;L 2 (T N ))
0
≤ C
T
0
0
T
C α−1/q ([0,T ];L 2 (T N ))
q
t
2
1
E
(t − s) 2α ∥Φε (u ε )∥ 2 L 2 (U;L 2 (T N )) ds dt
1 + ∥u ε (s)∥ q ds
L 2 (T N )
≤ CT q 2 (1−2α) E
0
≤ CT q (1−2α)
2 1 + ∥u ε ∥ q ≤ C
L q (Ω;L q (0,T ;L 2 (T N )))
and the claim follows.
□
Corollary 4.7. For all ϑ > 0 there exist β > 0 and C > 0 such that for all ε ∈ (0, 1)
E∥u ε ∥ C β ([0,T ];H −ϑ (T N )) ≤ C. (35)
Proof. If ϑ > 2, the claim follows easily from (34) by the choice β = λ. If ϑ ∈ (0, 2) the proof
follows easily from interpolation between H −2 (T N ) and L 2 (T N ). Indeed,
E sup ∥u ε (t)∥ H −ϑ (T N ) ≤ C E sup ∥u ε (t)∥ 1−θ sup ∥u ε (t)∥ θ H
0≤t≤T
0≤t≤T
−2 (T N )
L
0≤t≤T
2 (T N )
p 1
p q 1
q
≤ C E sup ∥u ε (t)∥ 1−θ
H
0≤t≤T
−2 (T N )
≤ C 1 + E sup ∥u ε (t)∥ (1−θ)p
H
0≤t≤T
−2 (T N )
1
p
E
sup ∥u ε (t)∥ θ L
0≤t≤T
2 (T N )
1 + E sup ∥u ε (t)∥ θq
L
0≤t≤T
2 (T N )
where the exponent p for the Hölder inequality is chosen in order to satisfy (1 − θ)p = 1,
i.e. since θ = 2−ϑ
2 , we have p = ϑ 2 . The first parenthesis can be estimated using (34)
while the second one using (27). Similar computations yield the second part of the norm of
C β ([0, T ]; H −ϑ (T N )). Indeed,
E
sup
0≤s,t≤T
s≠t
≤ C E
≤ C
∥u ε (t) − u ε (s)∥ H −ϑ (T N )
|t − s| β
sup
0≤s,t≤T
s≠t
1 + E sup
0≤s,t≤T
s≠t
∥u ε (t) − u ε (s)∥ 1−θ
H −2 (T N )
|t − s| β sup ∥u ε (t) − u ε (s)∥ θ L 2 (T N )
0≤s,t≤T
s≠t
∥u ε (t) − u ε (s)∥ (1−θ)p 1
p
1
H −2 (T N )
|t − s| βp 1 + E sup ∥u ε (t)∥ θq q
L
0≤t≤T
2 (T N )
1
q