Sentence examples for using the above estimates from inspiring English sources

Using the above estimates, we get (4.26).

Using the above estimates we obtain (3.23).

Using the above estimates, we obtain, and (3.6).

Using the above estimates (3.4) and (3.6), we can obtain the general uniform decay estimate of the energy functional.

Using the above estimates of Case (i) and Case (ii), we have, for any (fin L^{varphi}), int_{{mathbb{R}}^{n}}varphi bigl x, biglvert Tf(x bigrvert bigr),dxlesssim int _{{mathbb{R}}^{n}}varphi bigl x, biglvert f(x bigrvert bigr),dx, which, together with Lemma 3.5, implies that T is bounded on (L^{varphi}).

By the compactness of the injection of V into H, using the above estimates and the Ascoli-Arzelà theorem, we deduce that there exist a subsequence { u m } m ≥ 1 (we relabel the same) and u ∈ W 1, 2 ( 0, T ; V ) ∩ L 2 q ( − h, T ; V ) for any T > 0 with u = φ in ( − h, 0 ).

Using the above estimate in (2.29), the inequality (1.5) for follows.

Using the above estimate, the inequality (|cdot|_{0,Omega} lesssim |cdot|_{B^{frac{alpha}{2}}(Omega)}), and Lemma 3.1, we get biglVert q^{ast}-q_{L}^{ast}bigrVert _{0,Omega}lesssimeta _{1}+eta_{2}.

Using the above estimate to the second term on the right-hand side of equation (3.7) yields ∫ R ( Λ q + 1 u ) Λ q + 1 ( u n + 1 u x ) d x = c ∥ u ∥ L ∞ n ∥ u x ∥ L ∞ ∥ u ∥ H q + 1 2. (3.9).

end{aligned} (31) Using the above estimate inequality (31), we obtain begin{aligned} biglVert u x,t -U x,t -U xrVert _{infty }leq m_{0}h^{4}+biglVert hat{U}(x,t)-U(x,t) bigrVert _{infty }leqnd{aligned} (32) The collocation conditions are begin{aligned} Lu(x_{i},t)=LU(x_{i},t)=f(x_{i},t),quad i=0,1, ldots,N.

Now, put, and and use the above estimates to obtain,,, and, so that (3.16).