The Cauchy problem for the following Boussinesq system, $$ \begin{cases} u_t + v_x + uu_x = 0\\ v_t - u_{xxx} + u_{x} + (uv)_x = 0 \end{cases} $$ is considered. It is showed that this problem is locally well-posed in $H^s(\mathbb{R}) \times H^{s-1}(\mathbb{R})$ for any $s>3/2$. The proof involves parabolic regularization and techniques of Bona-Smith. It is also determined that the special solitary-wave solutions of this system are orbitally stable for the entire range of the wave speed. Combining these facts we can extend globally the local solution for data sufficiently close to the solitary wave.

In this paper, we study the existence of steady flows of an inviscid compressible barotropic fluid through a bounded simply connected domain $\Omega \in \mathbb{R}^3$. We prove, under suitable conditions on the data, the existence and local uniqueness of a subsonic steady compressible flow satisfying a given mass flow per unit surface on the whole boundary of the domain, as well as two additional conditions on the inflow boundary.

We consider a simple model arising in the control of noise. We assume that the two-dimensional cavity $\Omega=(0,1)\times (0,1)$ is occupied by an elastic, inviscid, compressible fluid. The potential $\phi$ of the velocity field satisfies the linear wave equation. The boundary of $\Omega$ is divided in two parts, $\Gamma_{0}$ and $\Gamma_{1}$. The first one, $\Gamma_{0}$, is flexible and occupied by a dissipative vibrating string. The transversal displacement of the string, $W$, satisfies a non homogeneous one-dimensional wave equation. On $\Gamma_{0}$ the continuity of the normal velocities of the fluid and the string is imposed. The subset $\Gamma_{1}$ of the boundary is assumed to be rigid and therefore, the normal velocity of the fluid vanishes. This constitutes a non homogeneous dissipative system of two coupled wave equations in dimensions two and one respectively. The non homogeneous term acting on the flexible part of the boundary (an elastic force or an exterior source of noise) is assumed to be periodic. We are interested in the existence of periodic solutions of this system. Due to the localization of the damping term in a relatively small part of the boundary and to the effect of the hybrid structure of the system, the existence of periodic solutions holds for a restricted class of non homogeneous terms. Some resonance-type phenomena are also exhibited.

We show the well-posedness in $H^{\frac 12 }$ of the Cauchy problem for a certain class of one dimensional nonlinear Schrödinger equations with the derivative nonlinearity. This is an improvement of results in $H^1$ by N. Hayashi and T. Ozawa [2,3,4,20]. Our results can cover the derivative nonlinear Schrödinger equation. Our proof is based on the Fourier restriction norm method and the gauge transformation.

We extend to dimension $3$ a result previously known in higher dimensions, concerning the topological effect of critical points at infinity on the level sets of a functional associated to an elliptic problem with critical nonlinearity.