The goal of this paper is to derive traffic flow macroscopic models from microscopic models. At the microscopic scales, we consider a Bando model, of the type following the leader, i.e., the acceleration of each vehicle depends on the distance to the vehicle in front of it. We take into account the possibility that each driver can have different characteristics such as sensibility to other drivers or optimal velocities. After rescaling, we prove that the solution of this system of ODEs converges to the solution of a macroscopic homogenized Hamilton-Jacobi equation which can be seen as a LWR (Lighthill-Whitham-Richards) model.

The energy-critical defocusing nonlinear Schrödinger equation on $3$-dimensional rectangular tori is considered. We prove that the global well-posedness result for the standard torus of Ionescu and Pausader extends to this class of manifolds, namely, for any initial data in $H^1$ the solution exists globally in time.

We prove the existence of the scattering operator in the neighborhood of the origin in $\mathbf{H}^{\omega ,1}\cap \mathbf{H}^{\mu },$ where $\mu > \omega =1+\frac{2}{n}$ for the semirelativistic Hartree type equation \begin{equation*} i\partial _{t}u=\sqrt{M^{2}-\Delta }u+F \big ( u \big ) ,\text{ } \big ( t,x \big ) \in {\mathbf{R}}\times {\mathbf{R}}^{n}, \end{equation*} where $F ( u ) = ( | x | ^{-\gamma }\ast | u | ^{2} ) u,1 < \gamma < 2,n\geq 3$ and $\ast $ denotes the convolution.

The stochastic three-dimensional compressible Navier-Stokes equations are considered in a bounded domain with multiplicative noise. The global existence of martingale solution is established through the Galerkin approximation method, stopping time, compactness method and the Jakubowski-Skorokhod theorem. A martingale solution is a weak solution for the fluid variables and the Brownian motion on a probability space. The initial data is arbitrarily large and satisfies a natural compatibility condition.

We prove that, for every $\lambda \gt 0 $ and $u_0 \geq 0$, the following evolution problem has a solution for small value of $T$, \begin{equation} \label{zbep1} \begin{cases} u > 0 \text{ in }\;\Omega_T = \Omega \times (0,T) \;, \\ \displaystyle \; u_t - \div (M(x)\nabla u)= \frac{\lambda}{u^\gamma}+ \mu \, u^{p} \; \text{ in }\;\Omega_T, \\ u=0 \text{ on }\;\partial\Omega \times (0,T),\\ u(0)=u_0 \text{ in }\; \Omega, \end{cases} \end{equation} where $\gamma > 0$, $\mu\geq0$ and $p > 0$. Moreover, we show the existence of a solution for every value of $T$ for suitable small data $\lambda$ and $u_0$ if $p > 1$ and for every data if $0 < p < 1$.

We prove local regularity results for the spatial gradient of weak solutions to nonlinear problems under the assumption that the involved operator becomes appropriately parabolic at infinity.

In this paper, we consider the existence and some qualitative properties of solutions to quasi-variational inequalities of the form $$ \begin{cases} \langle {\mathcal A}(u) , v-u \rangle + \langle F(u,u) , v-u\rangle + J(v,u) - J(u,u) \ge 0,\;\forall v\in W^{1,p}(\Omega) , \\ u\in D(J(\cdot, u)) , \end{cases} $$ where $\Omega$ is a bounded domain in ${\mathbb R}^N$, ${\mathcal A}$ is a second-order elliptic operator of Leray--Lions type on $W^{1,p}(\Omega)$, $F(\cdot , \cdot)$ is a multivalued bifunction, and $J(\cdot , u)$ is a convex functional depending on $u$. We propose concepts of sub- and supersolutions for this problem and study the existence and enclosure of solutions, and also the existence of extremal solutions between its sub- and supersolutions. Properties and examples of the involved mappings $F$ and $J$ and of sub-supersolutions of the above quasi-variational inequality are also presented.