Sentence examples for applying regularity from inspiring English sources

So it is divided into two steps: first of all applying the standard energy estimate on the temporal derivatives of the solution, and then spatial derivatives by applying regularity theory of Stokes equation repeatedly.

Finally, inverting the transformed functions and applying the regularity criterion of the transformed function, the required expansion formula is derived.

By applying the regularity argument of elliptic problem (1.2 - 1.3 1.2 - 1.3lear that y ∈ H2, so that p∈ H1.

Applying the regularity theory and strong maximum principle of elliptic equations, we see that u 0 − is one positive solution of problem (1.3).

Applying the regularity theory and strong maximum principle of elliptic equations, we find that u 0 + is one positive solution of problem (1.3).

Among the areas where researchers have applied the regularity lemma is the field of machine learning in artificial intelligence.

In order to prove Theorem 1.1, we want to apply the regularity result given by Theorem 4.2.

In this section, we will apply the regularity lifting lemma to obtain the integrable intervals of the solutions of system (1.1).

An important point is that the way the functions w k have been defined gives that they belong to the Sobolev space W 1, p ( B j ) and this allows to apply the regularity theory available for such solutions.

In fact, applying the standard elliptic regularity result to the equation (2.3), we obtain biglVert nabla^{3}ubigrVert _{L^{2}}leq Cbigl( Vert rho u_{t}Vert _{H^{1}}+Vert rho vcdot nabla u Vert _{H^{1}}+Vert nabla pVert _{H^{1}}+biglVert nabla(rhopi bigrVert _{H^{1}}bigr), (2.42) we could estimate the right-hand side of (2.42) item by item.

Corollaries 2.6 and 3.2 can be applied to obtain regularity for (3.10).