No abstract

We are concerned with the one‐fluid Euler–Poisson system for the electrons on the cubes in three dimensions. With the bootstrap estimates, we prove long‐term stability of periodic solutions of this system with small smooth irrotational perturbations of a constant background. Our main result is that the solutions are well‐posed for a time at least ε−12+normalOfalse(1Nfalse)R54−normalOfalse(1Nfalse)$$ {\varepsilon}^{-\frac{1}{2}+\mathrm{O}\left(\frac{1}{N}\right)}{R}^{\frac{5}{4}-\mathrm{O}\left(\frac{1}{N}\right)} $$, where ε$$ \varepsilon $$ is the size of the initial data, the sufficiently large R≫1$$ R\gg 1 $$ is the side length of the cubes, and N$$ N $$ is the index of the classical Sobolev spaces HN$$ {H}^N $$.

In this paper, we are concerned with the existence and uniqueness of global weak solutions for the weakly dissipative Dullin-Gottwald-Holm equation describing the unidirectional propagation of surface waves in shallow water regime:                                        ut − α2uxxt + c0ux + 3uux + γuxxx + λ(u − α2uxx) = α2(2uxuxx + uuxxx).Our main conclusion is that on c0 = − γ/α2 and λ ≥ 0, if the initial data satisfies certain sign conditions, then we show that the equation has corresponding strong solution which exists globally in time, finally we demonstrate the existence and uniqueness of global weak solutions to the equation.