Degenerate parabolic stochastic partial differential equations

4314 M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336
dn ε 1 (x, t, ξ) = σ (x)∇u
ε 2 dδ u ε (x,t)(ξ) dx dt,
dn ε 2 (x, t, ξ) = ε|∇uε | 2 dδ u ε (x,t)(ξ) dx dt,
and define m ε = n ε 1 + nε 2
. Then, using the same approach as in Section 2, one can verify that the
pair ( f ε = 1 u ε >ξ , m ε ) satisfies the kinetic formulation of (25): let ϕ ∈ Cc ∞(TN ×R), t ∈ [0, T ],
then it holds true P-a.s.
f ε (t), ϕ − f 0 , ϕ t
− f ε (s), b ε (ξ) · ∇ϕ ds
=
−
t
0
t
0
0
f ε (s), div A(x)∇ϕ ds − ε
δu ε =ξ Φ ε (u ε ) dW, ϕ + 1 2
t
0
t
0
f ε (s), ϕ ds
Note, that by taking limit in ε we lose this precise structure of n 2 .
4.2. Energy estimates
δu ε =ξ G 2 , ∂ ξ ϕ ds − m ε , ∂ ξ ϕ ([0, t)). (26)
In this subsection we shall establish the so-called energy estimate that makes it possible to
find uniform bounds for approximate solutions and that will later on yield a solution by invoking
a compactness argument.
Lemma 4.2. For all ε ∈ (0, 1), for all t ∈ [0, T ] and for all p ∈ [2, ∞), the solution u ε satisfies
the inequality
E∥u ε (t)∥ p L p (T N ) ≤ C 1 + E∥u 0 ∥ p L p (T N )
. (27)
Proof. According to Theorem 4.1, the process u ε is an L p (T N )-valued continuous
semimartingale so we can apply the infinite-dimensional Itô formula [8, Theorem 4.17] for the
function f (v) = ∥v∥ p L p (T N ) . If q is the conjugate exponent to p then f ′ (v) = p|v| p−2 v ∈
L q (T N ) and
f ′′ (v) = p(p − 1)|v| p−2 Id ∈ L L p (T N ), L q (T N ) .
Therefore
t
∥u ε (t)∥ p L p (T N ) = ∥u 0∥ p L p (T N ) − p
|u ε | p−2 u ε div B ε (u ε ) dx ds
0 T
N t
+ p |u ε | p−2 u ε div A(x)∇u ε dx ds
0 T
N t
+ εp |u ε | p−2 u ε u ε dx ds
0 T N
+ p t
|u ε | p−2 u ε g ε
k≥1 0 T N k (x, uε ) dx dβ k (s)
+ 1 t
2 p(p − 1)
0
T N |u ε | p−2 G 2 ε (x, uε ) dx ds. (28)