Sentence examples for imposed on the initial from inspiring English sources

Under certain assumptions imposed on the initial data and external force f, we obtain the global existence and regularity.

We show the regularity in (H^{2}) and (H^{4}) under certain assumptions imposed on the initial data.

In the paper, we shall analyze the spherical symmetric model and establish the regularity in (H^{2}) and (H^{4}) under certain assumptions imposed on the initial data.

The existence of global weak solutions to (1) is proved by Zhang and Yin [10] under certain conditions imposed on the initial value.

Under certain assumptions imposed on the initial data, we show the L 1 -stability of weak solutions for the atmospheric equations.

Compared with that of [2], the minimal regularity index and the smallness assumptions we imposed on the initial data are weaker.

Lin and Liu [6] established the L 2 -stability of peakons for Degasperis-Procesi Eq. (4) under certain assumptions imposing on the initial value.

The existence and uniqueness results for global weak solutions of the standard Camassa-Holm equation have been proved by Constantin and Escher [6], Constantin and Molinet [7] and Danchin [8, 9] under the sign condition imposing on the initial value.

No restrictions are imposed on the magnitude of the initial conditions, which results in a coupling between the shock and the induced waves, governed by an infinite system of ordinary differential equations.

The proposed control strategy appears to be robust in the sense that its application does not require any knowledge of the initial conditions imposed on the structure, and its performance is better than passive solutions, especially for the system induced in the first mode.

To lighten the burden imposed on the designer, we developed an initial hull structural modeling system in our previous study.