Spatial Sobolev regularity for stochastic Burgers equations with additive trace class noise

Strong convergence rates for full-discrete approximations of stochastic Burgers equations with multiplicative noise

“…and this verifies (87). Moreover, Lemma 4.8 in Jentzen, Lindner & Pusnik[36],Lemma 4.3(v) in Jentzen, Lindner & Pusnik[36], and (223) prove that there exists a C ∈ (0, ∞) such that for all N ∈ AE, n ∈ AE, and all x, y ∈ H it holds thatF N (x) − F N (y) H −1/2 = P N (F (P N x) − F (P N y)) H −1/2 ≤ |c 1 | (P N x) 2 − (P N y) 2 H ≤ |c 1 | P N (x − y) H P N (x + y) L ∞ ((0,1),Ê) ≤ C x − y H x + y H 1/2In addition, the fact that inf n∈AE |λ n | ≥ 1, Lemma 4.13(ii) in Jentzen, Lindner & Pusnik[36] (with y 0, c 0 1) and (223) show for all N ∈ AE, n ∈ AE, and all x ∈ D n thatF N (x) H −1/2 ≤ F N (x) H = P N F (P N x) H ≤ |c 1 | √ 3 P N x 2 H 1/2 ≤ |c 1 | √ 3 T γ 1 n −γ 1 ,(249)which verifies (89). Furthermore, (223) imply for all n ∈ AE and all (t, x)∈ [0, T ] × D n that V (t, x) H ≤ x 2 H + 1 ≤ x 2 H 1/2 + 1 ≤ (1 + T γ 1 )n −γ 1 (250)and this shows (90).…”

“…255), (222), (251), (253), (254) then show for all p ∈ [2, ∞) that sup t∈[0,T ]sup N ∈AE sup n∈AE Y N,n t L p (È;H 1/2 ) < ∞. (256)Thus, Lemma 4.13(ii) in Jentzen, Lindner & Pusnik[36] (with w 0, c 0 1) demonstrates for allp ∈ [2, ∞) that sup t∈[0,T ] sup N ∈AE sup n∈AE F N (Y N,n t ) L p (È;H) ≤ |c 1 | sup t∈[0,T ] L p (È;Ê) < ∞(257)Moreover, (281) ensures for all q ∈ [4, ∞] that supt∈[0,T ] sup N ∈AE X N t L q (È;H) < ∞. (290)In addition, (277) implies for all N ∈ AE and alln ∈ AE ∩ [T, ∞) that ½ H\Dn (Y N,n τ N,n ) 2p L 2p (È;Ê) = È Y N,nτ N,n 290) and (291) yield that there exists a C ∈ (0, ∞) such that for alln ∈ AE ∩ [T, ∞) H\Dn (Y N,n τ N,n ) L p (È;H) ≤ sup N ∈AE sup t∈[0,T ] ( X N t L 2p (È;H) + Y N,n t L 2p (È;H) ) ½ H\Dn (Y N,n τ N,n ) L 2p (È;Ê) 4C 2 T (p+1)/2p n − 1 /2 .…”

“…which implies (97) and (98). Moreover, Lemma 4.23 in Jentzen, Lindner & Pusnik [36] and the fact that ε ≤ 1 verify that for all N ∈ AE and all (s, x), (t, y) ∈ [0, T ] × P N (H) it holds that…”

“…Furthermore, (223) imply for all n ∈ AE and all (t, x)∈ [0, T ] × D n that V (t, x) H ≤ x 2 H + 1 ≤ x 2 H 1/2 + 1 ≤ (1 + T γ 1 )n −γ 1 (250)and this shows (90). Next note that Corollary 4.22 (with α 1 91 120 ) in Jentzen, Lindner & Pusnik[36] imply that there exists a C ∈ (0, ∞) such that for all x ∈ H and all N ∈ AE it holds thatF N (x) H −91/120 ≤ F (P N (x)) H −91/120 ≤ C P N (x) 2 H ≤ C x 2 H . (251)Thus, Lemma 3.9 (with α 0), (249), Lemma 4.23 in Jentzen, Lindner & Pusnik[36], and (251) ensure for all p ∈ [2, ∞) that C ∈ (0, ∞) such that for all x ∈ H and all N ∈ AE it holds thatF N (x) H −1/3 ≤ C P N (x) H 4/15 (1 + P N (x) H 4/15 ) ≤ 2C(1 + x 2 H 4/15 ).…”

“…Next note that Corollary 4.22 (with α 1 91 120 ) in Jentzen, Lindner & Pusnik[36] imply that there exists a C ∈ (0, ∞) such that for all x ∈ H and all N ∈ AE it holds thatF N (x) H −91/120 ≤ F (P N (x)) H −91/120 ≤ C P N (x) 2 H ≤ C x 2 H . (251)Thus, Lemma 3.9 (with α 0), (249), Lemma 4.23 in Jentzen, Lindner & Pusnik[36], and (251) ensure for all p ∈ [2, ∞) that C ∈ (0, ∞) such that for all x ∈ H and all N ∈ AE it holds thatF N (x) H −1/3 ≤ C P N (x) H 4/15 (1 + P N (x) H 4/15 ) ≤ 2C(1 + x 2 H 4/15 ). Lemma 3.6, (251), (254), and the fact that ∀x ∈ H : x 29/120 ≤ x H 4/15 imply for all p ∈ [2, ∞) that…”