Degenerate parabolic stochastic partial differential equations

endowed with the norm
M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336 4297
∥v∥ 2 U 0
= k≥1
α 2 k
k 2 ,
v = α k e k .
k≥1
Note that the embedding U ↩→ U 0 is Hilbert–Schmidt. Moreover, trajectories of W are P-a.s. in
C([0, T ]; U 0 ) (see [8]).
In the present paper, we use the brackets ⟨·, ·⟩ to denote the duality between the space of
distributions over T N × R and Cc ∞(TN × R). We denote similarly the integral
⟨F, G⟩ =
T N
R
F(x, ξ)G(x, ξ) dx dξ,
F ∈ L p (T N × R), G ∈ L q (T N × R),
where p, q ∈ [1, ∞] are conjugate exponents. The differential operators of gradient ∇,
divergence div and Laplacian ∆ are always understood with respect to the space variable x.
As the next step, we introduce the kinetic formulation of (1) as well as the basic definitions
concerning the notion of kinetic solution. The motivation for this approach is given by the
nonexistence of a strong solution and, on the other hand, the nonuniqueness of weak solutions,
even in simple cases. The idea is to establish an additional criterion – the kinetic formulation –
which is automatically satisfied by any strong solution to (1) and which permits to ensure the
well-posedness.
Definition 2.1 (Kinetic Measure). A mapping m from Ω to the set of nonnegative finite measures
over T N × [0, T ] × R is said to be a kinetic measure provided
(i) m is measurable in the following sense: for each ψ ∈ C 0 (T N × [0, T ] × R) the mapping
m(ψ) : Ω → R is measurable,
(ii) m vanishes for large ξ: if BR c = {ξ ∈ R; |ξ| ≥ R} then
lim E m T N × [0, T ] × B c
R = 0, (5)
R→∞
(iii) for any ψ ∈ C 0 (T N × R)
ψ(x, ξ) dm(x, s, ξ) ∈ L 2 (Ω × [0, T ])
T N ×[0,t]×R
admits a predictable representative. 1
Definition 2.2 (Kinetic Solution). Assume that, for all p ∈ [1, ∞),
and
u ∈ L p (Ω × [0, T ], P, dP ⊗ dt; L p (T N ))
(i) there exists C p > 0 such that
E ess sup ∥u(t)∥ p L
0≤t≤T
p (T N ) ≤ C p, (6)
(ii) σ ∇u ∈ L 2 (Ω × [0, T ]; L 2 (T N )).
Let n 1 be a mapping from Ω to the set of nonnegative finite measures over T N × [0, T ] × R
defined for any Borel set D ∈ B(T N × [0, T ] × R) as 2
1 Throughout the paper, the term representative stands for an element of a class of equivalence.
2 We will write shortly dn1 (x, t, ξ) = |σ (x)∇u| 2 dδ u(x,t) (ξ) dx dt.