A classical model for sources and sinks in a two-dimensional perfect incompressible fluid occupying a bounded domain dates back to Yudovich's paper [44] in 1966. In this model, on the one hand, the normal component of the fluid velocity is prescribed on the boundary and is nonzero on an open subset of the boundary, corresponding either to sources (where the flow is incoming) or to sinks (where the flow is outgoing). On the other hand the vorticity of the fluid which is entering into the domain from the sources is prescribed.

In this paper, we investigate the existence of weak solutions to this system by relying on {\it a priori} bounds of the vorticity, which satisfies a transport equation associated with the fluid velocity vector field. Our results cover the case where the vorticity has a $L^p$ integrability in space, with $p $ in $[1,+\infty]$, and prove the existence of solutions obtained by compactness methods from viscous approximations. More precisely we prove the existence of solutions which satisfy the vorticity equation in the distributional sense in the case where $p > \frac43$, in the renormalized sense in the case where $p > 1$, and in a symmetrized sense in the case where $p =1$.

In this paper, we consider non-homogeneous fractional equations in Orlicz spaces, with a source depending on the spatial variable, the unknown function and its fractional gradient. The latter is adapted to the Orlicz framework. The main contribution of the article is to establish the equivalence between weak and viscosity solutions for such equations.

We show that uniqueness results of the kind obtained for KdV and Schrödinger equations ([7], [28]), are not valid for the dispersion generalized--Benjamin--Ono equation in the weighted Sobolev spaces $$ H^s(\mathbb R)\cap L^2(x^{2r}dx), $$ for appropriated $s$ and $r$. In particular, we obtain that the uniqueness result proved for the dispersion generalized--Benjamin--Ono equation ([13]), is not true for all pairs of solutions $u_1\neq 0$ and $u_2\neq 0$. To achieve these results, we employ the techniques present in our recent work [6]. We also improve some Theorems established for the dispersion generalized--Benjamin--Ono equation and for the Benjamin--Ono equation ([13], [12]).

In the setting of a compact Riemannian manifold of dimension $N\ge3$, we provide a structural description of the limiting behavior of the energy measures of solutions to the parabolic Ginzburg-Landau equation. In particular, we provide a decomposition of the limiting energy measure into a diffuse part, which is absolutely continuous with respect to the volume measure, and a concentrated part supported on a codimension $2$ rectifiable subset. We also demonstrate that the time evolution of the diffuse part is determined by the heat equation while the concentrated part evolves according to a Brakke flow. This paper extends the work of Bethuel, Orlandi, and Smets from [8].