We prove the local well-posedness for the two-dimensional Zakharov-Kuznetsov equation in $H^s(\mathbb R ^2)$, for $s\in [1,2]$, on the background of an $L^\infty(\mathbb R ^3)$-function $\Psi(t,x,y)$, with $\Psi(t,x,y)$ satisfying some natural extra conditions. This result not only gives us a framework to solve the ZK equation around a Kink, for example, but also around a periodic solution, that is, to consider localized non-periodic perturbations of periodic solutions. Additionally, we show the global well-posedness in the energy space $H^1(\mathbb R ^2)$.

We consider a parabolic semilinear non-autonomous problem $(\tilde P)$ for a fractional time dependent operator $\mathcal{B}^{s,t}_\Omega$ with Wentzell-type boundary conditions in a possibly non-smooth domain $\Omega\subset\mathbb{R}^N$. We prove existence and uniqueness of the mild solution of the associated semilinear abstract Cauchy problem $(P)$ via an evolution family $U(t,\tau)$. We then prove that the mild solution of the abstract proble $(P)$ actually solves problem $(\tilde P)$ via a generalized fractional Green formula.

We study the regularity properties of the weak solutions $u:\Omega \subseteq \mathbb R ^n\to\mathbb R $ to problems of the type \begin{equation*} \begin{cases} -{\rm div}\, a(x,D u)+b(x)\phi'(|u|) {\frac{u}{|u|}} =f & \text{in}\,\,\Omega \\ u=0 & \text{on}\,\,\partial\Omega \end{cases} \end{equation*} with $\Omega\subset\mathbb R ^n$ a bounded open set and where the function $a(x,\xi)$ satisfies growth conditions with respect to the second variable expressed through an N-function $\phi$. We prove that, under a suitable interplay between the lower order terms and the datum $f$, which is assumed only to belong to $L^1(\Omega)$, the solutions are bounded in $\Omega$. Next, if $a(x,\xi)$ depends on $x$ through a Hölder continuous function, we take advantage from the boundedness of the solution $u$ to prove the higher differentiability and the higher integrability of its gradient, under mild assumptions on the data.

We consider the existence and uniqueness of time-periodic solutions to the Navier-Stokes equation in the whole space. We decompose periodic solutions into steady and purely periodic parts, and we analyze the equations they should satisfy. Based on the analysis of the purely periodic solutions represented by the Fourier transform to the Stokes equation, their additional regularity in time can be obtained and we use it to construct a time-periodic solution of the Navier-Stokes equation. Furthermore, we show that if the time-periodic solution is sufficiently small in an appropriate sense, then the Navier-Stokes equation admits no other solution in the same class.