Sentence examples for existence of smooth from inspiring English sources

Local existence of smooth solutions.

The existence of smooth kernels of optimal integral operators is proved and their properties are studied.

Next, by the standard continuous arguments, we can obtain the global existence of smooth solutions.

As usual, we first mollify the initial data to get the existence of smooth approximate solutions.

The local existence of smooth solutions to both systems is obtained.

We shall state the global existence of smooth solutions to the linearized problem (19) as follows.

The existence and uniqueness of local smooth solutions were proved firstly in [4], while the existence of global smooth solutions with small smooth initial data was shown in [5].

The existence and uniqueness of local smooth solutions were first obtained in [21]; moreover, the existence of global smooth solutions with small smooth initial data was shown in [20].

Under the assumptions of Theorem 1.1, the Cauchy problem (2.1), (2.3) admits a unique global smooth solution on ([0,infty) times mathbb{R}^{m}). Noting (1.4), by the maximum principle we obtain the result that, on the existence domain of smooth solution, we have r t,theta)> 0. (2.46).

In this paper, we are going to show the long time existence of the smooth solution for the porous medium equations in a smooth bounded domain:(0.1){ut="△umin Ω×[0,∞),u x,0)="u0>0in Ω,u x,t)= 0for x∈∂Ω where m>1 is the permeability.

From the existence we obtain the existence of a smooth optimal mapping solving the transportation problem as formulated in Corollary 1.2 in [15].