2578 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS
76.6 ExamplesHere we give some examples. The first is a standard example, the porous media equation,which is discussed well in [116]. For stochastic versions of this example, see [108]. Thegeneralization to stochastic equations does not require the theory developed here. We willshow, however, that it can be considered in terms of the theory of this paper without muchdifficulty using an approach proposed in [23]. These examples involve operators which arenot monotone, in the usual way but they can be transformed into equations which do fit theabove theory.
Example 76.6.1 The stochastic porous media equation is
ut −∆
(u |u|p−2
)= f , u(0) = u0, u = 0 on ∂U
where here U is a bounded open set in Rn,n ≤ 3 having Lipschitz boundary. One canconsider a stochastic version of this as a solution to the following integral equation
u(t)−u0 +∫ t
0(−∆)
(u |u|p−2
)ds =
∫ t
0ΦdW +
∫ t
0f ds (76.6.62)
where here
Φ ∈ L2([0,T ]×Ω,L2
(Q1/2U,H
))∩L2
(Ω,L∞
([0,T ] ,L2
(Q1/2U,H
))),
H = L2 (U) and the equation holds in the manner described above in H−1 (U) . Assumep≥ 2 and f ∈ L2 ((0,T )×Ω,H).
One can consider this as an implicit integral equation of the form
(−∆)−1 u(t)−(−∆)−1 u0+∫ t
0u |u|p−2 ds=(−∆)−1
∫ t
0ΦdW +(−∆)−1
∫ t
0f ds (76.6.63)
where −∆ is the Riesz map of H10 (U) to H−1 (U). Then we can also consider (−∆)−1 as a
map from L2 (U) to L2 (U) as follows.
(−∆)−1 f = u where −∆u = f , u = 0 on ∂U.
Thus we let W = L2 (U) = H and V = Lp (U) . Let B≡ (−∆)−1 on L2 (U) as just described.Let A(u) = u |u|p−2. It is obvious that the necessary coercivity condition holds. In addition,there is a strong monotonicity condition which holds. Therefore, if u0 ∈ L2
(Ω,L2 (U)
)and
F0 measurable, Theorem 76.4.7 applies and we can conclude that there exists a uniquesolution to the integral equation 76.6.63 in the sense described in this theorem. Here u ∈Lp ([0,T ]×Ω,Lp (U)) and is progressively measurable, the integral equation holding forall t for ω off a set of measure zero. Since A satisfies for some δ > 0 an inequality of theform
⟨Au−Av,u− v⟩ ≥ δ ∥u− v∥pLp(U)
it follows easily from the above methods that the solution is also unique. In fact, thisfollows right away from Theorem 76.4.7 because
⟨−∆−1u,u
⟩= ∥u∥2
H−1 .