Sentence examples for smooth solutions for from inspiring English sources

There exist global smooth solutions for the 2D Euler equation with symmetry [19].

In Section 4, we get the existence of second order smooth solutions for 2D Euler equation with symmetry outside a core region.

In (Chen and Tan [2010], [2012]) we established the global existence and decay rates of the smooth solutions for the Cauchy problem.

end{aligned} (1.4) They obtained the existence of the global smooth solutions for the initial value problem of (1.4) and discussed the convergence behavior of solutions as (betarightarrow0).

First we obtained the existence and uniqueness for local smooth solutions for the periodic initial value problem (1 - 5) via the a priori estimates and the Galerkin method.

Kawashima [13] obtained the smooth solutions for two-dimensional compressible MHD equations when the initial data is a small perturbation of a given constant state.

So we get the existence of one order global smooth solution for (3) and (4).

We then use the Szegö projection to show there is a smooth solution for the ¯∂b problem given smooth data.

We prove the global existence and the decay estimates of small smooth solution for the 2-D MHD equations without magnetic diffusion.

Thus, the further problem at the center of the mathematical theory concerning equations (1.1) is whether or not it has a global in time smooth solution for any prescribed smooth initial data, which is still a challenging open problem.

Yet, just like the 3D Navier-Stokes equations and the 3D MHD equations, whether there exists a global smooth solution for the 3D generalized MHD equations (1) or not is an open problem.