American Institute of Mathematical Sciences

We obtain the local well-posedness of a moving boundary problem that describes the swelling of a pocket of water within an infinitely thin elongated pore (i.e. on \begin{document}$ [a, +\infty), \ a>0 $\end{document}). Our result involves fine a priori estimates of the moving boundary evolution, Banach fixed point arguments as well as an application of the general theory of evolution equations governed by subdifferentials.

We consider the 1D transport equation with nonlocal velocity field:

\begin{document}$ \begin{equation*} \label{intro eq} \begin{split} &\theta_t+u\theta_x+\nu \Lambda^{\gamma}\theta = 0, \\ & u = \mathcal{N}(\theta), \end{split} \end{equation*} $\end{document}

where \begin{document}$ \mathcal{N} $\end{document} is a nonlocal operator and \begin{document}$ \Lambda^{\gamma} $\end{document} is a Fourier multiplier defined by \begin{document}$ \widehat{\Lambda^{\gamma} f}(\xi) = |\xi|^{\gamma}\widehat{f}(\xi) $\end{document}. In this paper, we show the existence of solutions of this model locally and globally in time for various types of nonlocal operators.

The purpose of this work is to carry out investigations of a generalized two-phase model for porous media flow. The momentum balance equations account for fluid-rock resistance forces as well as fluid-fluid drag force effects, in addition, to internal viscosity through a Brinkmann type viscous term. We carry out detailed investigations of a one-dimensional version of the general model. Various a priori estimates are derived that give rise to an existence result. More precisely, we rely on the energy method and use compressibility in combination with the structure of the viscous term to obtain \begin{document}$ H^1 $\end{document}-estimates as well upper and lower uniform bounds of mass variables. These a priori estimates imply existence of solutions in a suitable functional space for a global time \begin{document}$ T>0 $\end{document}. We also derive discrete schemes both for the incompressible and compressible case to explore the role of the viscosity term (Brinkmann type) as well as the incompressible versus the compressible case. We demonstrate similarities and differences between a formulation that is based, respectively, on interstitial velocity and Darcy velocity in the viscous term. The investigations may suggest that interstitial velocity seems more natural to use in the formulation of momentum balance than Darcy velocity.

We consider stochastic mean field games for which the state space is a network. In the ergodic case, they are described by a system coupling a Hamilton-Jacobi-Bellman equation and a Fokker-Planck equation, whose unknowns are the invariant measure \begin{document}$ m $\end{document}, a value function \begin{document}$ u $\end{document}, and the ergodic constant \begin{document}$ \rho $\end{document}. The function \begin{document}$ u $\end{document} is continuous and satisfies general Kirchhoff conditions at the vertices. The invariant measure \begin{document}$ m $\end{document} satisfies dual transmission conditions: in particular, \begin{document}$ m $\end{document} is discontinuous across the vertices in general, and the values of \begin{document}$ m $\end{document} on each side of the vertices satisfy special compatibility conditions. Existence and uniqueness are proven under suitable assumptions.

We study systems of elliptic equations \begin{document}$ -\Delta u(x)+F_{u}(x, u) = 0 $\end{document} with potentials \begin{document}$ F\in C^{2}({\mathbb{R}}^{n}, {\mathbb{R}}^{m}) $\end{document} which are periodic and even in all their variables. We show that if \begin{document}$ F(x, u) $\end{document} has flip symmetry with respect to two of the components of \begin{document}$ x $\end{document} and if the minimal periodic solutions are not degenerate then the system has saddle type solutions on \begin{document}$ {\mathbb{R}}^{n} $\end{document}.

In this paper we study the optimal reinforcement of an elastic membrane, fixed at its boundary, by means of a network (connected one-dimensional structure), that has to be found in a suitable admissible class. We show the existence of an optimal network, and observe that such network carries a multiplicity that in principle can be strictly larger than one. Some numerical simulations are shown to confirm this issue and to illustrate the complexity of the optimal network when the total length becomes large.

In recent years, opinion dynamics has received an increasing attention and various models have been introduced and evaluated mainly by simulation. In this study, we introduce a model inspired by the so-called "bounded confidence" approach where voters engaged in an electoral decision with two options are influenced by individuals sharing an opinion similar to their own. This model allows one to capture salient features of the evolution of opinions and results in final clusters of voters. We provide a detailed study of the model, including a complete taxonomy of the equilibrium points and an analysis of their stability. The model highlights that the final electoral outcome depends on the level of interaction in the society, besides the initial opinion of each individual, so that a strongly interconnected society can reverse the electoral outcome as compared to a society with looser exchange.