Degenerate parabolic stochastic partial differential equations

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M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336 4315 If we define H ε (ξ) = ξ 0 |ζ |p−2 B ε (ζ ) dζ then the second term on the right hand side vanishes due to the boundary conditions t −p |u ε | p−2 u ε div B(u ε ) t dx ds = p div H ε (u ε ) dx ds = 0. 0 T N 0 T N The third term is nonpositive as the matrix A is positive-semidefinite t p |u ε | p−2 u ε div A(u ε )∇u ε dx ds 0 T N t = −p |u ε | p−2 ∇u ε ∗ A(x) ∇u ε dx ds ≤ 0 0 T N and the same holds for the fourth term as well since A is only replaced by εI . The last term is estimated as follows 1 t t 2 p(p − 1) |u ε | p−2 G 2 0 T N ε (x, uε ) dx ds ≤ C |u ε | p−2 1 + |u ε | 2 dx ds 0 T N t ≤ C 1 + ∥u ε (s)∥ . p L p (T N ) ds Finally, expectation and application of the Gronwall lemma yield (27). Corollary 4.3. The set {u ε ; ε ∈ (0, 1)} is bounded in L p (Ω; C([0, T ]; L p (T N ))), for all p ∈ [2, ∞). Proof. Continuity of trajectories follows from Theorem 4.1. To verify the claim, a uniform estimate of E sup 0≤t≤T ∥u ε (t)∥ p is needed. We repeat the approach from the preceding L p (T N ) lemma, only for the <strong>stochastic</strong>ally forced term we apply the Burkholder–Davis–Gundy inequality. We have T E sup ∥u ε (t)∥ p L 0≤t≤T p (T N ) ≤ E∥u 0∥ 1 p L p (T N ) + C + E∥u ε (s)∥ p L 0 p (T N ) ds t + p E sup |u ε | p−2 u ε g ε 0 T N k (x, uε ) dx dβ k (s) 0≤t≤T k≥1 and using the Burkholder–Davis–Gundy and the Schwartz inequality, the assumption (2) and the weighted Young inequality in the last step yield t E sup |u ε | p−2 u ε g ε 0≤t≤T k≥1 0 T N k (x, uε ) dx dβ k (s) T 2 1 2 ≤ C E |u ε | p−1 |g ε T N k (x, uε )| dx ds 0 k≥1 T ≤ C E |uε | p 2 0 2 L 2 (T N ) k≥1 |uε | p−2 2 |g ε k (·, u ε (·))| T ≤ C E ∥u ε ∥ p 1 + ∥u ε ∥ p ds L p (T N ) L p (T N ) 0 1 2 0 2 L 2 (T N ) ds □ 1 2
4316 M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336 Therefore 1 2 ≤ C E sup 0≤t≤T ∥u ε (t)∥ p L p (T N ) 1 + ≤ 1 2 E sup ∥u ε (t)∥ 1 p L 0≤t≤T p (T N ) + C + T 0 T 0 ∥u ε (s)∥ p L p (T N ) ds 1 2 E∥u ε (s)∥ p L p (T N ) ds . T E sup ∥u ε (t)∥ p L 0≤t≤T p (T N ) ≤ C 1 + E∥u 0 ∥ p L p (T N ) + E∥u ε (s)∥ p L 0 p (T N ) ds and the corollary follows from (27). 4.3. Compactness argument □ To show that there exists u : Ω × T N × [0, T ] → R, a kinetic solution to (1), one needs to verify the strong convergence of the approximate solutions u ε . This can be done by combining tightness of their laws with the pathwise uniqueness, which was proved above. First, we need to prove a better spatial regularity of the approximate solutions. Towards this end, we introduce two seminorms describing the W λ,1 -regularity of a function u ∈ L 1 (T N ). Let λ ∈ (0, 1) and define T N p λ |u(x) − u(y)| (u) = dx dy, T N |x − y| N+λ pϱ λ (u) = sup 1 0 0 such that for all t ∈ [0, T ] E∥u ε (t)∥ W s,1 (T N ) ≤ C T,s,u 0 1 + E∥u0 ∥ W ς,1 (T N ) . (31) Proof. Proof of this statement is based on Proposition 3.2. We have E (T N ) 2 ϱ τ (x − y) f ε (x, t, ξ) f¯ ε (y, t, ξ) dξ dx dy R
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Degenerate parabolic stochastic partial differential equations

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